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Mathematics Glossary

Key terms from the Mathematics course, linked to the lesson that introduces each one.

5,095 terms.

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-3i: Real part is 0, imaginary part is -3 (this is 0 + (-3)i); Lesson 1190 — Standard Form of Complex Numbers Lesson 1207 — Complex Numbers as Points in the Plane
(±a, 0): Lesson 1158 — Graphing Ellipses Centered at the Origin Lesson 1171 — Graphing Hyperbolas Centered at the Origin
(0, ±a): Lesson 1158 — Graphing Ellipses Centered at the Origin Lesson 1171 — Graphing Hyperbolas Centered at the Origin
(a - b): it matches the sign between the cubes; Lesson 339 — Sum and Difference of Cubes Lesson 387 — Difference of Cubes Pattern Lesson 2730 — Introduction to Congruence Notation
(a + b): or **(a - b)** — it matches the sign between the cubes; Lesson 339 — Sum and Difference of Cubes Lesson 388 — Sum of Cubes Pattern
(A ᵀ)⁻¹ = (A ⁻¹)ᵀ: Lesson 1306 — Properties of Inverse Matrices Lesson 2145 — Inverses of Products and Transposes
(a, b): means "all numbers between a and b, *excluding both endpoints*"; Lesson 222 — Interval Notation for Solutions Lesson 1207 — Complex Numbers as Points in the Plane Lesson 1813 — Computing Limits Using Direct Substitution Lesson 1819 — Properties and Composition of Continuous Functions Lesson 1836 — Linear Approximation in Two Variables Lesson 2826 — The Range of a Continuous Random Variable
(a, b]: means "exclude a, but *include* b"; Lesson 222 — Interval Notation for Solutions Lesson 2826 — The Range of a Continuous Random Variable
(AB) ⁻¹ = B ⁻¹A ⁻¹: Lesson 1306 — Properties of Inverse Matrices Lesson 2145 — Inverses of Products and Transposes
(AB)C = A(BC): Lesson 1286 — Matrix Multiplication is Associative Lesson 2101 — Properties of Matrix Multiplication
(cos θ, sin θ): .; Lesson 992 — Coordinates on the Unit Circle Lesson 1000 — Using the Unit Circle to Find Exact Values Lesson 1004 — Trigonometric Functions in Each Quadrant Lesson 1010 — Using the Unit Circle to Evaluate Trig Functions Lesson 1062 — Proof and Geometric Interpretation of Sum Formulas
(f g)(x): , which reads as "f composed with g at x" or simply "f of g of x.; Lesson 558 — Function Composition Basics Lesson 1408 — Composition of Functions Review
(h, k): are the **exact coordinates of the vertex**—no calculation needed once you're in this form!; Lesson 407 — Finding the Vertex by Completing the Square Lesson 423 — Vertex Form of a Quadratic Function Lesson 424 — Finding the Vertex from Vertex Form Lesson 916 — Standard Form of a Circle Lesson 917 — Finding Center and Radius from Circle Equations Lesson 1136 — Standard Form of a Circle Lesson 1137 — Graphing Circles from Standard Form Lesson 1148 — Parabolas with Vertex at (h, k): Vertical Axis (+3 more)
(n+1)st derivative: of your function evaluated at some mystery point *c*; Lesson 1785 — The Lagrange Form of the Remainder Lesson 1786 — Using the Lagrange Remainder to Bound Error
(x - a): is how far you've moved from *a*; Lesson 1469 — Linear Approximation Formula Lesson 1836 — Linear Approximation in Two Variables
(x - c): , identify **c**.; Lesson 349 — Synthetic Division Setup Lesson 351 — The Remainder Theorem
(x - h): and **(y - k)**, so if you see **(x + 1)**, that means **h = -1** (because x - (-1) = x + 1).; Lesson 916 — Standard Form of a Circle Lesson 917 — Finding Center and Radius from Circle Equations
(x + 1): , that means **h = -1** (because x - (-1) = x + 1).; Lesson 916 — Standard Form of a Circle Lesson 1668 — Revolving Around Vertical Lines (x = k)
(x, y): = any other point on the line (the variable); Lesson 257 — Point-Slope Form: y - y₁ = m(x - x₁)Lesson 912 — Point-Slope Form and Writing Line Equations Lesson 916 — Standard Form of a Circle Lesson 1002 — Extending Trig Functions to All Angles Lesson 1136 — Standard Form of a Circle Lesson 1207 — Complex Numbers as Points in the Plane Lesson 1813 — Computing Limits Using Direct Substitution Lesson 1836 — Linear Approximation in Two Variables
(x₁, y₁): and **(x₂, y₂)**, the slope **m** is:; Lesson 245 — Calculating Slope from Two Points Lesson 257 — Point-Slope Form: y - y₁ = m(x - x₁)Lesson 779 — Right Triangles on the Coordinate Plane Lesson 912 — Point-Slope Form and Writing Line Equations
(x₂, y₂): , the slope **m** is:; Lesson 245 — Calculating Slope from Two Points Lesson 779 — Right Triangles on the Coordinate Plane
(y - k): , so if you see **(x + 1)**, that means **h = -1** (because x - (-1) = x + 1).; Lesson 916 — Standard Form of a Circle Lesson 917 — Finding Center and Radius from Circle Equations
[a, b): means "include a, but *not* b"; Lesson 222 — Interval Notation for Solutions Lesson 2826 — The Range of a Continuous Random Variable
[a, b]: means "all numbers from a to b, *including both endpoints*"; Lesson 222 — Interval Notation for Solutions Lesson 1356 — The Intermediate Value Theorem Lesson 1666 — Revolving Around the y-axis Lesson 2826 — The Range of a Continuous Random Variable Lesson 2894 — Introduction to Continuous Uniform Distribution
{v₁, v₂, ..., v }: is linearly dependent if you can find scalars c₁, c₂, .; Lesson 2066 — Linear Dependence and Redundant Vectors Lesson 2211 — Orthogonal Bases
∂f/∂x: This emphasizes "the rate of change of *f* with respect to *x*.; Lesson 1823 — Notation for Partial Derivatives Lesson 1828 — Partial Derivatives of Functions with Three or More Variables
+ab: (sign is *opposite* the original); Lesson 339 — Sum and Difference of Cubes Lesson 387 — Difference of Cubes Pattern
|A|: (the absolute value of A); Lesson 1015 — Amplitude: Vertical Stretching with y = A·sin(x)Lesson 1033 — Amplitude: Vertical Stretch and Compression Lesson 1034 — Finding Amplitude from Equations and Graphs Lesson 1290 — What is a Determinant?
|det(A)|: gives the scaling factor; Lesson 1290 — What is a Determinant?Lesson 2149 — Determinant of a 2×2 Matrix
−2ab·cos(C): adjusts for the angle between the sides, accounting for how "spread out" or "squeezed together" the triangle is.; Lesson 1102 — Introduction to the Law of Cosines Lesson 1110 — The Law of Cosines and the Pythagorean Theorem
√(x²) = |x|: , not just x.; Lesson 160 — Radical Expressions with Variables Lesson 503 — Absolute Value Considerations in Simplifying Radicals
△ ABC ~ △ DEF: Lesson 726 — Introduction to Similarity Lesson 755 — SSS Similarity Criterion
0.5v: = 〈 1.; Lesson 1246 — Scalar Multiplication of Vectors Lesson 2049 — Scalar Multiplication of Vectors
1 person: works, it takes **12 hours**.; Lesson 125 — Inverse Variation Lesson 2819 — False Positives and Base Rate Fallacy
12 cm: , which indeed satisfy 5² + 12² = 13².; Lesson 437 — Geometry: Pythagorean Applications Lesson 856 — Perimeter of Polygons
180° = π radians: .; Lesson 983 — What is a Radian?Lesson 986 — Common Angle Conversions
2^n: elements.; Lesson 2531 — Introduction to Power Sets Lesson 2532 — Computing Power Sets
2×2 matrix: Lesson 1297 — Using Determinants to Test Invertibility Lesson 2164 — Determinant of Scalar Multiples
2v: = 2〈 3, 4〉 = 〈 2·3, 2·4〉 = 〈 6, 8〉; Lesson 1246 — Scalar Multiplication of Vectors Lesson 2049 — Scalar Multiplication of Vectors
2x: Lesson 170 — Writing Expressions with Variables Lesson 376 — Factoring Out the GCF from Each Group
2π radians: .; Lesson 988 — Sector Area Using Radians Lesson 990 — Coterminal Angles in Radians Lesson 998 — Coterminal Angles on the Unit Circle Lesson 1014 — Period of Sine and Cosine Functions
2π/|B|: instead of 2π.; Lesson 1029 — Transformations of Secant Lesson 1036 — Calculating Period from the B Coefficient
3 + 4i: Real part is 3, imaginary part is 4; Lesson 1190 — Standard Form of Complex Numbers Lesson 1199 — Complex Conjugates Lesson 1207 — Complex Numbers as Points in the Plane Lesson 1208 — The Argand Diagram
3 units right: Lesson 244 — Rise Over Run: Understanding Slope Geometrically Lesson 248 — Finding Slope from a Graph
30-60-90 triangle: have side lengths that follow predictable, fixed ratios every single time.; Lesson 791 — Introduction to Special Right Triangles Lesson 794 — The 30-60-90 Triangle: Side Ratios Lesson 796 — Recognizing Special Triangles from Angles
360° = 2π radians: , which means **180° = π radians**.; Lesson 983 — What is a Radian?Lesson 986 — Common Angle Conversions
3x = 12: .; Lesson 1307 — Using Inverses to Solve Matrix Equations Lesson 1316 — Solving with Matrix Inverses
3x²: means 3 × x × x (three times x-squared); Lesson 176 — Exponents in Variable Expressions Lesson 312 — Terms, Coefficients, and Constants Lesson 332 — Simplifying Products by Combining Like Terms
45-45-90 triangle: and the **30-60-90 triangle**—have side lengths that follow predictable, fixed ratios every single time.; Lesson 791 — Introduction to Special Right Triangles Lesson 792 — The 45-45-90 Triangle: Side Ratios Lesson 796 — Recognizing Special Triangles from Angles
5 units: Lesson 702 — Distance and the Ruler Postulate Lesson 775 — Finding the Hypotenuse Lesson 779 — Right Triangles on the Coordinate Plane
5x: .; Lesson 285 — Eliminating a Variable by Subtracting Equations Lesson 312 — Terms, Coefficients, and Constants Lesson 332 — Simplifying Products by Combining Like Terms
Δx: (the actual measurement error).; Lesson 1473 — Applications of Differentials in Measurement Lesson 1552 — Sigma Notation for Riemann Sums
λ̂ = 1/X̄: Lesson 3002 — Method of Moments for One Parameter Lesson 3006 — MLE for Common Distributions
λ̂ = X̄: Lesson 3002 — Method of Moments for One Parameter Lesson 3006 — MLE for Common Distributions
π radians = 180°: Lesson 984 — Converting Degrees to Radians Lesson 985 — Converting Radians to Degrees
π/4 radians: Lesson 990 — Coterminal Angles in Radians Lesson 1210 — Argument of a Complex Number
ρ(x, y, z): , then the total mass *M* is:; Lesson 1932 — Applications: Density and Total Mass Lesson 1956 — Computing Total Mass with Triple Integrals
φ(n): (read "phi of n"), counts the number of integers between 1 and *n* (inclusive) that are **relatively prime** to *n* — meaning they share no divisors with *n* except 1.; Lesson 2746 — Euler's Totient Function φ(n)Lesson 2763 — Computing φ(n) for RSA

A

A_i: is the matrix **A** with its *i*-th column replaced by the vector **b**.; Lesson 1317 — Cramer's Rule
AA: \*.; Lesson 2276 — The Spectral Theorem for Normal Matrices
AA (Angle-Angle) Similarity Criterion: states that **if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.; Lesson 753 — AA Similarity Criterion
AA ⁻¹: and **A ¹A** equal **I**.; Lesson 1300 — Definition of an Inverse Matrix
AA ⁻¹ = I: and **A ¹A = I**; Lesson 1300 — Definition of an Inverse Matrix Lesson 2139 — Properties of Invertible Matrices
AA similarity: with the original triangle—and therefore with each other!; Lesson 800 — The Altitude to the Hypotenuse
AA^T: are both symmetric matrices whose eigenvalues and eigenvectors directly reveal the SVD components.; Lesson 2284 — Computing SVD via Eigenvalue Decomposition
AAS: stands for **Angle-Angle-Side**.; Lesson 1094 — Solving AAS Triangles Lesson 1102 — Introduction to the Law of Cosines Lesson 1106 — Choosing Between Law of Sines and Law of Cosines
AB: exists only if **n = p**; Lesson 1281 — Matrix Multiplication Definition and Compatibility Lesson 1283 — Multiplying 2×2 Matrices Lesson 1303 — Verifying an Inverse by Multiplication Lesson 2100 — Matrix Multiplication: Definition Lesson 2159 — Determinant of a Product
AB ≠ BA: .; Lesson 1285 — Matrix Multiplication is Not Commutative Lesson 2101 — Properties of Matrix Multiplication
AB = I: (the identity matrix); Lesson 1303 — Verifying an Inverse by Multiplication Lesson 2139 — Properties of Invertible Matrices
AB/AC = AC/AD: Lesson 803 — Proving Geometric Mean Relations Using Similarity
About the x-axis: `Ix = ∬ _R y² ρ(x,y) dA`; Lesson 1961 — Computing Moments of Inertia
About the x-axis (I_x): Measures resistance to rotation around the x-axis.; Lesson 1960 — Moments of Inertia
About the y-axis: `Iy = ∬ _R x² ρ(x,y) dA`; Lesson 1961 — Computing Moments of Inertia
About the y-axis (I_y): Similarly, using x² (or x² + z² in 3D).; Lesson 1960 — Moments of Inertia
About the z-axis: `Iz = ∭ _V (x² + y²) ρ(x,y,z) dV`; Lesson 1961 — Computing Moments of Inertia
About the z-axis (I_z): Uses the distance from the z-axis: r² = x² + y².; Lesson 1960 — Moments of Inertia
Absolute convergence: asks a simple but powerful question: *What happens if we ignore all the signs and just look at the magnitudes?; Lesson 1755 — Absolute Convergence
Absolute Convergence Test: is a theorem that gives you a strategic advantage:; Lesson 1757 — The Absolute Convergence Test
absolute maximum: is the single highest point on the entire function (within a given domain)—the tallest mountain peak across the whole range.; Lesson 573 — Extrema: Local and Absolute Maxima and Minima Lesson 1358 — The Extreme Value Theorem Lesson 1508 — The Closed Interval Method Lesson 1871 — Global Extrema: Finding Absolute Maximum and Minimum
absolute minimum: is the single lowest point on the entire function—the deepest valley across everything.; Lesson 573 — Extrema: Local and Absolute Maxima and Minima Lesson 1358 — The Extreme Value Theorem Lesson 1508 — The Closed Interval Method Lesson 1871 — Global Extrema: Finding Absolute Maximum and Minimum
Absolute value: is exactly this distance measurement.; Lesson 42 — Absolute Value Lesson 249 — Comparing Steepness Using Slope Lesson 426 — Effects of Parameter a on Parabola Width Lesson 1203 — Absolute Value (Modulus) of Complex Numbers Lesson 1643 — Area Between a Curve and the x-axis
Absolute value check: Since we have an even root and even powers, the answer is technically 5|x|²|y|², which simplifies to 5x²y²; Lesson 504 — Combining Simplification Techniques
Absolute Value Family: The parent function is `f(x) = |x|`, forming a V-shape at the origin.; Lesson 588 — Transformations and Function Families
Absolutely integrable: ∫|f(x)|dx is finite; Lesson 2460 — Convergence of Fourier Series
AC Method: Multiply a·c = 2·3 = 6.; Lesson 364 — Factoring by Grouping After Splitting the Middle Term Lesson 373 — Applications and Strategy Selection for Trinomial Factoring
Acceleration: a(t) = v'(t) = s''(t) = -Aω² sin(ωt) or a(t) = -Aω² cos(ωt); Lesson 1428 — Applications: Rates of Change in Oscillatory Motion Lesson 1545 — Applications: Position from Velocity, Velocity from Acceleration Lesson 2426 — The One-Dimensional Wave Equation
Accidents: at an intersection per month; Lesson 2893 — Applications of the Poisson Distribution
Acoustics: Modeling sound propagation in rooms or open space; Lesson 2434 — The Three-Dimensional Wave Equation
Actual course: **v** + **w** = ⟨0, 200 ⟩ + ⟨30, 0⟩ = ⟨30, 200 ⟩; Lesson 1266 — Navigation Problems: Heading and Course
Acute: 0° < angle < 90° (sharp and small); Lesson 704 — Classifying Angles by Measure Lesson 731 — Classifying Triangles by Angles Lesson 789 — Acute and Obtuse Triangle Tests Lesson 1122 — The Ambiguous Case with Acute Angles
acute angle: is less than 90°; Lesson 703 — Measuring Angles and Angle Notation Lesson 704 — Classifying Angles by Measure
Acute Triangle: Lesson 731 — Classifying Triangles by Angles Lesson 788 — Using the Converse to Identify Right Triangles
Acute Triangle Test: Lesson 789 — Acute and Obtuse Triangle Tests
AD/CD = CD/BD: Lesson 803 — Proving Geometric Mean Relations Using Similarity
Add: when things are being combined, joined together, or increased; Lesson 18 — Word Problems with Addition and Subtraction Lesson 199 — Solving One-Step Addition/Subtraction Equations Lesson 350 — Performing Synthetic Division Lesson 362 — Factoring Trinomials with Leading Coefficient 1 Lesson 1053 — Sum and Difference Formulas for Cosine Lesson 1074 — Deriving Product-to-Sum Formulas Lesson 1199 — Complex Conjugates Lesson 2509 — Common Mistakes in Induction Proofs (+2 more)
Add 1: to get 1 + [f'(x)]²; Lesson 1676 — Computing Arc Length: Polynomial Functions
Add an edge: to your growing forest *if and only if* it doesn't form a cycle with edges already chosen; Lesson 2675 — Minimum Spanning Trees: Kruskal's Algorithm
Add back: the triple intersection (restores those wrongly-removed elements); Lesson 2618 — Inclusion-Exclusion for Three Sets
Add or subtract: depending on whether parts are added on or cut out; Lesson 871 — Composite Figures with Circles and Semicircles Lesson 1074 — Deriving Product-to-Sum Formulas
Add solutions together: The sum satisfies the original PDE and all boundary conditions simultaneously; Lesson 2440 — Superposition and Non-Homogeneous Boundaries
Add the absolute values: (or just add the areas, ensuring each integral has positive result); Lesson 1647 — Area with Multiple Regions Lesson 1650 — Area Enclosed by Intersecting Curves
Add the arguments: θ₁ + θ₂; Lesson 1221 — Multiplying Complex Numbers in Polar Form
Add the edge: (*v*, *u*) to your spanning tree; Lesson 2673 — Finding Spanning Trees: Depth-First Search
add the exponents: .; Lesson 142 — Product Rule for Exponents Lesson 165 — Multiplying Numbers in Scientific Notation
Add the numerators: – Combine the top numbers; Lesson 69 — Adding Fractions with Like Denominators Lesson 463 — Adding Rationals with Unlike Denominators
Add the probabilities: The probability that Y equals y is the sum of all probabilities P(X = x) where g(x) = y; Lesson 2868 — Functions of Random Variables: Discrete Case
Add the squares: Combine those two results; Lesson 775 — Finding the Hypotenuse
Add them: Total area = 15 + 8 = 23 ft²; Lesson 868 — Finding Area of Composite Figures by Addition
Add them all together: Lesson 893 — Nets and Surface Area
Add them together: Total = 2πr² + 2πrh; Lesson 888 — Surface Area of Cylinders
Add them up: Lesson 670 — Evaluating Finite Sums with Sigma Notation Lesson 872 — Irregular Composite Figures on Grids Lesson 1573 — Area and Signed Area
Add this vector: to your set (it will remain linearly independent); Lesson 2090 — Extending to a Basis
Adding: 2 ⅓ + 1 ½; Lesson 74 — Adding and Subtracting Mixed Numbers Lesson 608 — Comparing Exponential and Linear Growth Lesson 1053 — Sum and Difference Formulas for Cosine Lesson 2058 — Geometric Interpretation of Linear Combinations Lesson 2618 — Inclusion-Exclusion for Three Sets
Adding 0: Any number plus 0 stays the same (5 + 0 = 5); Lesson 10 — Addition Facts to 10
Adding 1: Goes to the next counting number (7 + 1 = 8); Lesson 10 — Addition Facts to 10
Adding back: the three-way intersection (+|A ∩B∩C|), since we subtracted it too many times; Lesson 2618 — Inclusion-Exclusion for Three Sets
Adding inside shifts LEFT: `f(x + 2)` moves the graph 2 units to the left; Lesson 579 — Horizontal Shifts of Functions
Adding like radicals: Lesson 158 — Adding and Subtracting Radicals
Adding parallel lines: Drawing a line parallel to an existing side through a key point can create corresponding angles or proportional segments.; Lesson 769 — Auxiliary Lines and Constructions
Adding vertically: means writing one polynomial above the other, lining up terms with the same degree in neat columns—just like you learned to add multi-digit numbers in elementary school.; Lesson 321 — Adding Polynomials Vertically
Adding/Subtracting and Rearranging: Lesson 2315 — Separable Equations with Algebraic Manipulation
Addition: "Maria has 15 apples.; Lesson 18 — Word Problems with Addition and Subtraction Lesson 36 — The Commutative Property Lesson 37 — The Associative Property Lesson 172 — Writing Expressions from Word Problems Lesson 183 — Order of Operations in Evaluation Lesson 285 — Eliminating a Variable by Subtracting Equations Lesson 1197 — Adding and Subtracting Complex Numbers Lesson 2640 — Operations on Generating Functions: Addition and Scalar Multiplication (+2 more)
Addition and Subtraction: – Work left to right, applying sign rules; Lesson 47 — Mixed Operations with Negative Numbers Lesson 81 — Operations with Multiple Fractions Lesson 513 — Mixed Operations with Radicals
Addition clues: altogether, total, combined, in all, more than (adding); Lesson 18 — Word Problems with Addition and Subtraction
Addition phrases: Lesson 188 — Translating Words into Mathematical Expressions
Addition Property of Equality: If you add the same number to both sides of an equation, the sides remain equal.; Lesson 198 — Addition and Subtraction Properties of Equality
Addition rule: E[X + Y] = E[X] + E[Y]; Lesson 2853 — Properties of Expectation: Linearity
Addition/Subtraction: Combining or comparing parts of the same whole (like pieces of pie, portions of a task, or parts of a distance); Lesson 82 — Word Problems Involving Fraction Operations
Addition/subtraction at the end: `+ 5` means a vertical shift up 5 units; Lesson 586 — Identifying Transformations from Equations
Addition/subtraction with x: `x - 3` means a horizontal shift right 3 units (opposite of the sign!; Lesson 586 — Identifying Transformations from Equations
Additional pitfalls to avoid: Lesson 2358 — Common Mistakes and Problem-Solving Strategies
additive identity: because it leaves a number's identity unchanged in addition.; Lesson 39 — Identity and Inverse Properties Lesson 2047 — Vector Equality and Zero Vector Lesson 2103 — Special Matrices: Identity and Zero
additive inverse: a number that adds with it to give zero:; Lesson 39 — Identity and Inverse Properties Lesson 1275 — Properties of Matrix Addition
Additivity: `T(u + v) = T(u) + T(v)`; Lesson 2259 — Definition of Linear Transformations Lesson 2922 — Properties and Applications of Chi- Squared
Adjacent: The two angles share a common side (ray) and a common vertex; Lesson 708 — Linear Pairs Lesson 948 — Identifying Opposite, Adjacent, and Hypotenuse Lesson 951 — The Tangent Ratio (TOA)Lesson 953 — Finding Side Lengths Using Cosine Lesson 954 — Finding Side Lengths Using Tangent Lesson 956 — Choosing the Appropriate Trig Ratio Lesson 962 — Solving When Given One Leg and One Acute Angle Lesson 970 — Using SOH-CAH-TOA with Depression Angles (+4 more)
Adjacent leg: tan(35°) = 12/adjacent, so adjacent = 12/tan(35°) ≈ 17.; Lesson 962 — Solving When Given One Leg and One Acute Angle
adjacent side: to the **hypotenuse**.; Lesson 950 — The Cosine Ratio (CAH)Lesson 953 — Finding Side Lengths Using Cosine Lesson 969 — Using SOH-CAH-TOA with Elevation Angles Lesson 977 — Finding Heights Using Angles of Elevation
Adjust your aim: and repeat until you hit the target; Lesson 2406 — Solving Boundary Value Problems: Shooting Method Concept
Adjusted R²: penalizes you for adding predictors that don't pull their weight.; Lesson 3076 — R² and Adjusted R² in Multiple Regression
Advantages: Lesson 281 — Comparing Graphing and Substitution Methods
After analysis: If ANOVA shows significance, use post-hoc tests to identify *which* groups differ; Lesson 3102 — ANOVA Applications and Experimental Design
After rationalizing, check for: Lesson 520 — Simplifying After Rationalizing
After we sample: , we flip the perspective: our observed mean is fixed, so we build an interval around it using that same margin (X̄ ± 1.; Lesson 3010 — The Logic Behind Confidence Intervals
Agriculture: Testing three fertilizer types across multiple plots; Lesson 3102 — ANOVA Applications and Experimental Design
AI = A: and **IA = A**; Lesson 1288 — Multiplying by the Identity Matrix Lesson 2103 — Special Matrices: Identity and Zero
Algebraic comparison: Determine where *f(x) - g(x) > 0*.; Lesson 1645 — Identifying Top and Bottom Functions Lesson 1753 — Verifying Conditions for the Alternating Series Test
Algebraic multiplicity: is how many times an eigenvalue appears as a root of the characteristic polynomial.; Lesson 2173 — Algebraic vs Geometric Multiplicity Lesson 2184 — Algebraic Multiplicity of Eigenvalues Lesson 2195 — When Diagonalization Fails Lesson 2196 — Geometric and Algebraic Multiplicity Lesson 2274 — Eigenspaces and Multiplicity for Symmetric Matrices
Algebraic test: f(-x) = f(x) for all x in the domain; Lesson 569 — Even and Odd Functions
Algebraically intensive: Requires careful manipulation and simplification; Lesson 281 — Comparing Graphing and Substitution Methods
Algorithm analysis: Determining which algorithm scales better; Lesson 1535 — Applications to Growth Rates and Asymptotic Behavior
Align indices: so every sum runs over the same power of x (often xⁿ); Lesson 2400 — Finding Power Series Solutions Near Ordinary Points
Align with given measurements: If dimensions are already provided that match certain shape boundaries, use those natural divisions rather than creating new ones.; Lesson 874 — Optimizing Composite Figure Problems
Alignment matters: Always line up terms by their degree (x³ with x³, x² with x², etc.; Lesson 346 — Dividing by Binomials Using Long Division
all: mathematical expressions, whether they contain numbers only or include variables.; Lesson 177 — Order of Operations with Variables Lesson 301 — Systems of Linear Inequalities Lesson 460 — Subtracting Rationals with Like Denominators Lesson 550 — Domain Restrictions from Square Roots Lesson 1085 — Factoring Trigonometric Equations Lesson 1110 — The Law of Cosines and the Pythagorean Theorem Lesson 1367 — Common Techniques and Restrictions on δ Lesson 1570 — The Fundamental Theorem of Calculus: Part 2 (+10 more)
All combinations: 2 + 6, 3 + 4, 8 + 2, and so on; Lesson 10 — Addition Facts to 10
All four triangular faces: = 4 × 15 = 60 cm²; Lesson 889 — Surface Area of Pyramids
All linear terms: → Plane; Lesson 1807 — Common Surfaces: Planes, Paraboloids, and Saddles
All negative terms: If every term is negative, like ∑(-1/n²), you have a **negative series**.; Lesson 1719 — Series with Positive and Negative Terms
All of R²: the entire plane itself; Lesson 2080 — Subspaces of R^2 and R^3
all of R³: Lesson 2062 — Geometric Span in R³: Lines, Planes, and All of R³ Lesson 2080 — Subspaces of R^2 and R^3
All positive terms: When every term in a series is positive (or non-negative), like ∑(1/n²), we call it a **positive series** or series with positive terms.; Lesson 1719 — Series with Positive and Negative Terms
all real numbers: Lesson 206 — Recognizing Identity and Contradiction Equations Lesson 222 — Interval Notation for Solutions Lesson 223 — Special Cases: All Reals and No Solution
All solutions: You've written something that's always true (like "a number equals itself"); Lesson 229 — Absolute Value Equations with No Solution or All Solutions
all three: conditions:; Lesson 737 — Triangle Inequality Theorem Lesson 2116 — Row Echelon Form: Definition and Properties Lesson 2810 — Applications: Reliability and Success Probabilities Lesson 3031 — Choosing Between z-Tests and Other Tests
All three terms squared: a², b², and c²; Lesson 340 — Product of (a + b + c)²
all values of x: .; Lesson 1624 — Solving for Coefficients: Substitution Method Lesson 2072 — Linear Independence in General Vector Spaces
All variables cancel out: Lesson 212 — Equations with No Solution
Almost sure convergence: (Strong Law) says: With probability 1, the *entire sequence* itself converges to the limit.; Lesson 2974 — Almost Sure Convergence vs. Convergence in Probability
Alphabetical order: Write `abc`, not `bca` (when variables multiply); Lesson 178 — Reading and Interpreting Algebraic Notation
Already normal: If the population itself is approximately normal, the CLT works perfectly even for small n (like n = 5); Lesson 2986 — Rule of Thumb for CLT Applicability
Alternate exterior angles: are pairs of angles that lie:; Lesson 716 — Alternate Exterior Angles Lesson 718 — Using Angle Relationships to Find Measures
alternate interior angles: .; Lesson 715 — Alternate Interior Angles Lesson 718 — Using Angle Relationships to Find Measures Lesson 721 — Applications of Parallel Line Properties Lesson 732 — Triangle Angle Sum Theorem Lesson 970 — Using SOH-CAH-TOA with Depression Angles Lesson 971 — Alternate Interior Angles in Elevation/Depression
alternating harmonic series: Lesson 1752 — The Alternating Series Test (Leibniz Test)Lesson 1756 — Conditional Convergence
alternating series: .; Lesson 1719 — Series with Positive and Negative Terms Lesson 1751 — Introduction to Alternating Series Lesson 1752 — The Alternating Series Test (Leibniz Test)
Alternating Series Test: (also called the **Leibniz Test**) states:; Lesson 1752 — The Alternating Series Test (Leibniz Test)
alternating signs: +, -, +.; Lesson 1292 — Determinant of a 3×3 Matrix: Expansion by Minors Lesson 1719 — Series with Positive and Negative Terms
Alternative hypothesis: H₁: μ ≠ μ₀; Lesson 3026 — Two-Tailed z-Tests Lesson 3027 — One-Tailed z-Tests
Alternative notations: you might see:; Lesson 1841 — The Gradient Vector: Definition and Notation
Altitude formula: altitude = √(segment₁ × segment₂); Lesson 804 — Finding Missing Lengths with Geometric Means
always: the middle letter.; Lesson 703 — Measuring Angles and Angle Notation Lesson 711 — Perpendicular Lines and Right Angles Lesson 2754 — Existence and Uniqueness in CRT
Always check: After separating variables by dividing by *h(y)*, find where *h(y) = 0*.; Lesson 2314 — Losing Solutions Through Division
always exists: .; Lesson 2244 — Uniqueness and Existence of QR Decomposition Lesson 2967 — Characteristic Functions (Optional)
Always positive: By definition, gcd(*a*, *b*) > 0; Lesson 2711 — Greatest Common Divisor (GCD): Definition Lesson 2926 — Properties and Applications of the F-Distribution Lesson 3042 — Chi-Squared Distribution Basics
Always round up: to the next whole number, since you can't survey a fraction of a person!; Lesson 3013 — Choosing Sample Size for Desired Margin of Error
Always start here: If lim(n→ ∞) a ≠ 0, the series diverges immediately.; Lesson 1750 — Strategy: Choosing the Right Convergence Test
Always use parentheses: when substituting a negative number.; Lesson 184 — Evaluating with Negative Numbers
always verify: your solutions don't create undefined expressions or extraneous roots.; Lesson 639 — Mixed Exponential-Logarithmic Equations Lesson 1107 — Law of Cosines with Obtuse Angles
Ambiguous statements: "This number is large.; Lesson 2468 — Propositions and Truth Values
amplitude: of the wave.; Lesson 1015 — Amplitude: Vertical Stretching with y = A·sin(x)Lesson 1018 — Vertical Shift: y = sin(x) + D and the Midline Lesson 1019 — Combined Transformations: y = A·sin(B(x - C)) + D Lesson 1033 — Amplitude: Vertical Stretch and Compression Lesson 1034 — Finding Amplitude from Equations and Graphs Lesson 2369 — Free Undamped Harmonic Motion
Amplitude (A): How far the phenomenon varies from its average (e.; Lesson 1022 — Applications: Modeling Periodic Phenomena Lesson 1042 — Applications: Modeling Periodic Phenomena
Amplitude = |A|: The wave oscillates A units above and below the midline; Lesson 1019 — Combined Transformations: y = A·sin(B(x - C)) + D
Analogy: If you slice a pizza into 8 pieces and take 5, you have `5/8` of a pizza—less than one complete pizza.; Lesson 62 — Proper, Improper, and Mixed Numbers Lesson 84 — Reading and Writing Decimals Lesson 230 — Introduction to Compound Inequalities Lesson 307 — Solving Linear Programming Problems Graphically Lesson 621 — The Product Rule for Logarithms Lesson 666 — Growth and Decay Applications Lesson 685 — Properties of Series: Linearity and Splitting Lesson 698 — Collinear and Coplanar Points (+96 more)
Ancient problems: The Chinese Remainder Theorem (which you'll learn next) was developed to solve counting problems involving different-sized groups; Lesson 2752 — Systems of Linear Congruences
angle: Lesson 943 — Rotational Symmetry Lesson 968 — Drawing Diagrams for Angle Problems Lesson 977 — Finding Heights Using Angles of Elevation Lesson 1129 — What Are Conic Sections?
angle bisector: is a ray that divides an angle into two angles of equal measure.; Lesson 710 — Angle Bisectors Lesson 738 — Medians, Altitudes, and Angle Bisectors Lesson 819 — Rhombuses: Properties and Characteristics
angle of depression: is the angle formed between a horizontal line and your line of sight when you look *downward* at something.; Lesson 966 — Defining Angles of Elevation and Depression Lesson 967 — Identifying Angles in Real-World Scenarios Lesson 970 — Using SOH-CAH-TOA with Depression Angles Lesson 976 — Angles of Depression: Definition and Setup
angle of elevation: is the angle formed between a horizontal line and your line of sight when you look *upward* at something.; Lesson 966 — Defining Angles of Elevation and Depression Lesson 967 — Identifying Angles in Real-World Scenarios Lesson 969 — Using SOH-CAH-TOA with Elevation Angles Lesson 970 — Using SOH-CAH-TOA with Depression Angles Lesson 975 — Angles of Elevation: Definition and Setup Lesson 977 — Finding Heights Using Angles of Elevation
Angle of rotation: How far to turn (measured in degrees); Lesson 930 — Introduction to Rotations in the Coordinate Plane
Angle Sum Property: If you know one acute angle, subtract it from 90° to find the other; Lesson 960 — Finding Missing Angles in Right Triangles
Angle Sum Violations: Lesson 1128 — Common Errors and Verification in SSA
Angle-Angle-Side: .; Lesson 1094 — Solving AAS Triangles
Angle-Side-Angle: .; Lesson 1095 — Solving ASA Triangles
Angles: at corresponding vertices are corresponding angles; Lesson 725 — Corresponding Parts of Congruent Figures
Angular equation: Lesson 2441 — Laplace's Equation in Polar Coordinates
Angular frequency: $\omega_0 = \sqrt{k/m}$ (radians per second); Lesson 2369 — Free Undamped Harmonic Motion
Angular limits: θ goes from α to β; Lesson 1919 — Computing Areas Using Polar Integrals
Angular range: α ≤ θ ≤ β; Lesson 1918 — Regions Bounded by Polar Curves
annulus: (plural: annuli).; Lesson 831 — Concentric Circles and Annuli Lesson 1916 — Integrating over Circular Regions
Another example: Lesson 32 — Parentheses and Grouping Symbols Lesson 154 — The Product Property of Radicals Lesson 314 — Standard Form of a Polynomial Lesson 517 — Conjugates and Binomial Denominators Lesson 543 — Function Notation with Multiple Variables Lesson 670 — Evaluating Finite Sums with Sigma Notation Lesson 1066 — Using Double Angle Formulas to Simplify Expressions Lesson 1436 — Combining Product, Quotient, and Exponential Rules (+3 more)
Another trap: `|x - 2| + 4 = 1`; Lesson 229 — Absolute Value Equations with No Solution or All Solutions
ANOVA: (Analysis of Variance), which tests whether multiple group means differ significantly.; Lesson 2926 — Properties and Applications of the F-Distribution
Answer: 59; Lesson 14 — Adding and Subtracting with Larger Numbers Lesson 15 — Addition with Regrouping (Carrying)Lesson 16 — Subtraction with Regrouping (Borrowing)Lesson 17 — Adding and Subtracting Three or More Numbers Lesson 68 — Ordering Multiple Fractions Lesson 85 — Comparing and Ordering Decimals Lesson 99 — Finding What Percent One Number Is of Another Lesson 117 — Unit Rates in Proportions (+3 more)
Answer or difference: (how many are left); Lesson 11 — Understanding Subtraction as Taking Away
Answer: -9: Lesson 47 — Mixed Operations with Negative Numbers
Answer: 13: Lesson 35 — Addition and Subtraction (Left to Right)
Answer: 16: Lesson 31 — Introduction to Order of Operations (PEMDAS/BODMAS)
Answer: 17: Lesson 35 — Addition and Subtraction (Left to Right)
Answer: 18: Lesson 31 — Introduction to Order of Operations (PEMDAS/BODMAS)Lesson 35 — Addition and Subtraction (Left to Right)
antisymmetric: relation allows one-way pairs freely, but restricts two-way pairs to identical elements only.; Lesson 2544 — Antisymmetric Relations Lesson 2546 — Combining Properties: Partial Orders Lesson 2549 — Inverse Relations
any: quadratic equation of the form ax² + bx + c = 0 using completing the square.; Lesson 405 — Solving General Quadratics by Completing the Square Lesson 502 — Simplifying nth Roots with Variables Lesson 1110 — The Law of Cosines and the Pythagorean Theorem Lesson 1570 — The Fundamental Theorem of Calculus: Part 2 Lesson 1775 — Maclaurin Series for e^x Lesson 1930 — Volume as a Triple Integral Lesson 2150 — Determinant of a 3×3 Matrix: Cofactor Expansion Lesson 2207 — Pythagorean Theorem in Inner Product Spaces (+3 more)
Any shape: (skewed, uniform, bimodal—doesn't matter!; Lesson 2982 — CLT for Non-Normal Populations
Appears inside another function: (like the argument of a power, trig function, exponential, etc.; Lesson 1580 — Identifying the Inner Function u
Applications: This property helps prove uniqueness of solutions, estimate function values, and understand why harmonic functions describe equilibrium states in physics (electrostatics, fluid flow, temperature).; Lesson 2444 — Mean Value Property and Applications
Applications include: Lesson 1178 — Applications of Hyperbolas: LORAN and Reflective Properties
Apply: the shortcut formula; Lesson 342 — Using Special Products to Simplify Expressions Lesson 849 — Tangent-Radius Perpendicularity Theorem Lesson 1114 — Choosing the Correct Angle in SAS Area Formula Lesson 1730 — Mixed Examples: Known Series Lesson 2274 — Eigenspaces and Multiplicity for Symmetric Matrices Lesson 2909 — Finding Probabilities with Normal Distributions
Apply algebra or logic: to transform the expressions; Lesson 2493 — Direct Proof Examples and Practice
Apply angle relationships: Use corresponding angles, alternate interior angles, or same-side interior angles; Lesson 771 — Proving Properties of Parallel Lines
Apply chain rule again: when differentiating `∂f/∂u` or `∂g/∂x`, since these may depend on `x` indirectly; Lesson 1859 — Second-Order Derivatives via the Chain Rule
Apply continuity correction: Since you're approximating a discrete distribution with a continuous one, adjust boundaries by ±0.; Lesson 2985 — Approximating Binomial Distributions
Apply CPCTC: State that since the triangles are congruent, the corresponding parts you care about must also be congruent; Lesson 747 — CPCTC: Corresponding Parts of Congruent Triangles
Apply De Moivre's Theorem: z^n = r^n(cos nθ + i sin nθ); Lesson 1226 — Using De Moivre's Theorem to Compute Powers
Apply elementary row operations: to convert it to row echelon form; Lesson 2127 — Computing Rank via Row Reduction
Apply equilibrium conditions: Lesson 1269 — Tension in Cables and Ropes
Apply formula: Area = √[12(12 - 7)(12 - 8)(12 - 9)]; Lesson 1116 — Applying Heron's Formula
Apply Gram-Schmidt: to v₁, v₂, .; Lesson 2224 — Gram-Schmidt for Subspaces
Apply grouping when needed: – Especially after using the AC method on trinomials; Lesson 381 — Applications and Mixed Practice
Apply inclusion-exclusion: to count items with *none* of the bad properties; Lesson 2623 — The Sieve Method in Practice
Apply initial conditions: Use Fourier sine series to determine coefficients A_n and B_n from u(x,0) and u_t(x,0); Lesson 2431 — Fourier Series Solutions for Bounded Domains
Apply integration by parts: once (or twice), choosing u and dv strategically; Lesson 1597 — Cyclic Integration by Parts
Apply known special limits: Lesson 1419 — Derivative of sin(x) from First Principles
Apply Limit Comparison: to compare growth rates with p-series or geometric series; Lesson 1744 — When the Ratio Test is Inconclusive
Apply order of operations: (parentheses, exponents, multiplication/division left-to-right, addition/subtraction left-to-right); Lesson 185 — Evaluating Expressions with Fractions
Apply product rule: where necessary (because terms like `(∂f/∂u)·(∂g/∂x)` require it); Lesson 1859 — Second-Order Derivatives via the Chain Rule
Apply Pythagorean identities: to replace `sin²θ + cos²θ` with 1; Lesson 1051 — Verifying Trigonometric Identities
Apply row operations: Use Gaussian elimination to convert the left side to reduced row echelon form (RREF); Lesson 2142 — The Gauss-Jordan Method for Finding Inverses
Apply that second method: (integration by parts, u-sub, trig identities, etc.; Lesson 1619 — Combining Trig Substitution with Other Techniques
Apply the appropriate formula: expand using the sum/difference formula; Lesson 1056 — Verifying Identities with Sum and Difference Formulas
Apply the appropriate method: from previous lessons; Lesson 2335 — Applications of Exact and Bernoulli Equations
Apply the chain rule: derivative of outside × derivative of inside; Lesson 1412 — Chain Rule with Radicals Lesson 1449 — The Basic Technique of Implicit Differentiation Lesson 1856 — Implicit Differentiation in Multiple Variables
Apply the continuity correction: Since the binomial is discrete but the normal is continuous, adjust boundaries by ±0.; Lesson 2913 — Normal Approximation to the Binomial
Apply the definition: `√[(x - c)² + y²] = e · |x - a²/c|`; Lesson 1183 — Deriving Conic Equations from Eccentricity
Apply the difference formula: sin(A − B) = sin(A)cos(B) − cos(A)sin(B); Lesson 1055 — Evaluating Non-Standard Angles Using Sum Formulas
Apply the Distributive Property: to eliminate parentheses; Lesson 216 — Mixed Practice: Complex Linear Equations
Apply the formula: directly instead of multiplying term-by-term; Lesson 343 — Special Products in Word Problems Lesson 1599 — Definite Integrals and Integration by Parts Lesson 1627 — Integrating Partial Fractions with Linear Terms Lesson 2794 — Conditional Probability with Venn Diagrams Lesson 2870 — The Jacobian Method for Transformations Lesson 3109 — Spearman's Rank Correlation
Apply the initial condition: by substituting *x = x₀* and *y = y₀* into your general solution; Lesson 2324 — Initial Value Problems for Linear ODEs
Apply the modification rule: if your guess overlaps with $y_c$; Lesson 2357 — Higher-Order ODEs and Undetermined Coefficients Lesson 2358 — Common Mistakes and Problem-Solving Strategies
Apply the power rule: to each variable term; Lesson 1393 — Combining Basic Rules for Polynomials Lesson 1542 — Antiderivatives of Polynomial Functions
Apply the product property: to separate perfect squares (or perfect nth powers); Lesson 504 — Combining Simplification Techniques
Apply the Pythagorean Theorem: Lesson 779 — Right Triangles on the Coordinate Plane
Apply the rule: Lesson 86 — Rounding Decimals
Apply the sign convention: for quadrant II: negate the x-coordinate, keep the y-coordinate positive; Lesson 995 — Second Quadrant Points
Apply the sign pattern: In Quadrant IV, cosine is positive (x > 0) and sine is negative (y < 0).; Lesson 997 — Fourth Quadrant Points Lesson 2150 — Determinant of a 3×3 Matrix: Cofactor Expansion
Apply the theorem: a^n = a^(q(p-1) + r) = (a^(p-1))^q · a^r ≡ 1^q · a^r ≡ a^r (mod p); Lesson 2744 — Computing Large Powers Using Fermat's Little Theorem
Apply the theorem again: d² = (√(l² + w²))² + h²; Lesson 780 — The Pythagorean Theorem in 3D
Apply trig substitution: to eliminate the radical; Lesson 1619 — Combining Trig Substitution with Other Techniques
Apply vertical stretch: The steepness changes based on |A|; Lesson 1027 — Transformations of Cotangent
Apply well-ordering: Since *S* is nonempty (by assumption), it has a smallest element, call it *m*; Lesson 2518 — Proofs Using Well-Ordering
Apply Zero Product Property: Either x + 4 = 0 or x - 4 = 0; Lesson 398 — Solving Special Forms: Difference of Squares
Applying Lagrange: Since *a* is relatively prime to *n*, it belongs to U(*n*), which has order φ(*n*).; Lesson 2749 — Proof of Euler's Theorem
Approximate decimal: π ≈ 3.; Lesson 830 — Circles and Pi (π)
arc: on the circle between where the two radii touch the edge; Lesson 833 — Central Angles and Their Measures Lesson 836 — Central Angles and Intercepted Arcs Lesson 864 — Area of Sectors and Arc Length
Arc Addition: means if you have two adjacent arcs (arcs that share an endpoint), you can find the total arc by simply adding their measures together.; Lesson 844 — Arc Addition and Subtraction
Arc Addition Postulate: If point B lies on arc AC, then:; Lesson 844 — Arc Addition and Subtraction
arc length: is the actual distance you travel along that curve—not the straight-line distance, but the path itself.; Lesson 837 — Arc Length Formula Lesson 864 — Area of Sectors and Arc Length Lesson 983 — What is a Radian?Lesson 987 — Arc Length Using Radians Lesson 1914 — The Polar Area Element dA = r dr dθ Lesson 1935 — Volume Elements in Cylindrical Coordinates
Arc Subtraction: works when you know the total arc and need to find one part—you subtract the known arc from the whole.; Lesson 844 — Arc Addition and Subtraction
Architecture: Calculating the area of a triangular gable end when you know all three dimensions of the roof support beams?; Lesson 1120 — Real-World Area Applications Lesson 1682 — Applications: Arc Length in Physics and Geometry
Area: is the space inside a shape.; Lesson 193 — Perimeter and Area Word Problems Lesson 760 — Perimeter and Area Ratios of Similar Triangles Lesson 829 — Area of a Circle Lesson 857 — Area of Rectangles and Squares Lesson 862 — Area of Circles Lesson 1477 — Geometric Related Rates: Expanding Circles Lesson 1930 — Volume as a Triple Integral
Area formula: `A = (1/2) × base × height`; Lesson 1479 — Related Rates with Triangles Lesson 2003 — Applications and Extensions of Green's Theorem
Area function: A = L × W; Lesson 435 — Area and Perimeter Optimization Problems
area of a rectangle: .; Lesson 126 — Joint Variation Lesson 1478 — Related Rates with Rectangles and Squares
Area of a Trapezoid: = ½ × (*b₁* + *b₂*) × *h*; Lesson 860 — Area of Trapezoids
Area of R: = ∬ ᴿ 1 dA; Lesson 1910 — Applications: Area of General Regions
Area problems: Finding dimensions when given total area expressions; Lesson 373 — Applications and Strategy Selection for Trinomial Factoring Lesson 420 — Applications: Projectile Motion and Area Problems
arg(z): or sometimes **θ**, is the angle measured **counterclockwise** from the positive real axis to the line connecting the origin to that point.; Lesson 1210 — Argument of a Complex Number Lesson 1218 — Principal Argument and Angle Conventions
Argand diagram: (also called the complex plane) is a visual way to represent complex numbers.; Lesson 1208 — The Argand Diagram
argument: of *z*, written as **arg(z)** or sometimes **θ**, is the angle measured **counterclockwise** from the positive real axis to the line connecting the origin to that point.; Lesson 1210 — Argument of a Complex Number Lesson 1215 — Converting from Rectangular to Polar Form Lesson 1217 — Finding the Argument (Angle)
Argument of trig/exponential: e^(3x) → u = 3x or cos(x²) → u = x²; Lesson 1580 — Identifying the Inner Function u
Arithmetic: 10, 7, 4, 1, -2, .; Lesson 651 — Definition of Arithmetic Sequences
Arithmetic (linear): Use when the change is independent of current population size.; Lesson 690 — Population Growth and Decay Models
Arithmetic errors in summation/integration: Double-check your calculations; Lesson 2842 — Verifying Valid PMFs and PDFs
Arithmetic progression: a(n) = a(n-1) + d (constant difference); Lesson 2627 — What is a Recurrence Relation?
arithmetic sequence: is a list of numbers where each term after the first is created by adding the same fixed value to the previous term.; Lesson 651 — Definition of Arithmetic Sequences Lesson 691 — Depreciation and Asset Value
Arithmetic sequences: model populations that grow or shrink by a *fixed amount* each time period.; Lesson 690 — Population Growth and Decay Models
Around the x-axis: If you rotate a curve from the xy-plane (say y = g(x)) around the x-axis:; Lesson 1801 — Surfaces of Revolution
Around the z-axis: If you have a curve defined by z = f(r) where r is the distance from the z-axis, rotating it around the z-axis gives:; Lesson 1801 — Surfaces of Revolution
Arrive at a contradiction: something that can't possibly be true; Lesson 767 — Proof by Contradiction
Arrive at the conclusion: – The statement you set out to prove; Lesson 768 — Direct Proof Techniques
Arrow direction: Points toward steepest increase of f at that location; Lesson 1850 — Gradient Fields and Visualization
Arrow length: Magnitude indicates how rapidly f is changing (longer = steeper); Lesson 1850 — Gradient Fields and Visualization
Arrow patterns: Reveal the function's landscape at a glance; Lesson 1850 — Gradient Fields and Visualization
Arrows: show logical connections and the flow of reasoning; Lesson 765 — Flow Chart Proof Format
As = s: , meaning the system doesn't change.; Lesson 2179 — Applications: Steady States and Dynamics
As a fraction: Lesson 105 — What is a Ratio?
As a geometric sequence: You record snapshots at hour 1, hour 2, hour 3.; Lesson 668 — Geometric Sequences vs Exponential Functions
As an exponential function: You model the population at *any* moment in time, including fractional hours like 2.; Lesson 668 — Geometric Sequences vs Exponential Functions
As k → ∞: The distribution approaches normality (thanks to the Central Limit Theorem); Lesson 3042 — Chi-Squared Distribution Basics
As Type I: For x from 0 to 2, y goes from the bottom edge (y = 0) to the slanted edge.; Lesson 1904 — Setting Up Integrals for Triangular Regions
As Type II: For y from 0 to 4, x goes from the left slanted edge to the right edge (x = 2).; Lesson 1904 — Setting Up Integrals for Triangular Regions
As x → -∞: F(x) → 0 (no probability accumulated before all outcomes); Lesson 2844 — Properties of Cumulative Distribution Functions
As x → +∞: F(x) → 1 (all probability eventually included); Lesson 2844 — Properties of Cumulative Distribution Functions
ASA: stands for **Angle-Side-Angle**.; Lesson 1095 — Solving ASA Triangles Lesson 1102 — Introduction to the Law of Cosines Lesson 1106 — Choosing Between Law of Sines and Law of Cosines
Ask yourself: Lesson 2827 — Distinguishing Discrete from Continuous
Assemble the series: using these coefficients; Lesson 2459 — Computing Fourier Series for Simple Functions
Assess what you know: Do you have SAS or SSS?; Lesson 1111 — Complex Problems Combining Triangle Laws
Assign parameters: to them (commonly *t*, *s*, *r*, etc.; Lesson 1315 — Systems with Infinitely Many Solutions
Association between variables: Spearman's Rank Correlation or Kendall's Tau; Lesson 3111 — Choosing and Interpreting Nonparametric Tests
associative: , **identity**, and **additive inverse** properties—just like regular number addition.; Lesson 1275 — Properties of Matrix Addition Lesson 2048 — Vector Addition Lesson 2098 — Matrix Addition and Subtraction Lesson 2524 — Union of Sets
associative property: .; Lesson 22 — The Commutative and Associative Properties of Multiplication Lesson 37 — The Associative Property
Assume: the angles are *not* congruent; Lesson 767 — Proof by Contradiction Lesson 2491 — Direct Proof: Basic Structure Lesson 2722 — Infinitude of Primes: Euclid's Proof
Assume ¬Q: (the negation of the conclusion); Lesson 2495 — Proof by Contrapositive: Method and Strategy
Assume it's fair: (H₀: the coin is fair); Lesson 3024 — The Logic of Hypothesis Testing
Assume negation: Suppose the statement you want to prove is false; Lesson 2518 — Proofs Using Well-Ordering
Assume the form: y = Σ(n=0 to ∞) a xⁿ; Lesson 2400 — Finding Power Series Solutions Near Ordinary Points
Assume the opposite: of what you want to prove; Lesson 767 — Proof by Contradiction Lesson 2497 — Proof by Contradiction: Basic Idea Lesson 3024 — The Logic of Hypothesis Testing
Asymmetric: Unlike the normal or t-distribution; Lesson 2926 — Properties and Applications of the F-Distribution
asymptotes: .; Lesson 1028 — Graphing the Secant Function Lesson 1174 — Hyperbolas Centered at (h, k)
Asymptotic approximations: Simplifying complex expressions for large values; Lesson 1535 — Applications to Growth Rates and Asymptotic Behavior
Asymptotic Behavior: Lesson 612 — Properties of e^x: Growth and Behavior
At 30° (π/6): The longer leg (adjacent) is √3/2, shorter leg (opposite) is 1/2; Lesson 994 — Reference Angles and the First Quadrant
At 60° (π/3): The roles flip—shorter leg is adjacent, longer is opposite; Lesson 994 — Reference Angles and the First Quadrant
At boundaries: Lesson 1347 — Exponential and Logarithmic Limits
at least one: of the propositions is true.; Lesson 2471 — Disjunction (OR)Lesson 2484 — Negating Quantified Statements Lesson 2620 — Counting with At Least One Property Lesson 2622 — Derangements and the Derangement Formula Lesson 2810 — Applications: Reliability and Success Probabilities
at most: $50 on groceries"; Lesson 225 — Applications of Linear Inequalities Lesson 2876 — Computing Binomial Probabilities
At most k objects: `(1 + x + x² + .; Lesson 2642 — Using Generating Functions to Count Distributions
At r = 10: dV = 4π(10)²(0.; Lesson 1473 — Applications of Differentials in Measurement
At x = 2: Lesson 443 — Evaluating Rational Expressions
At x = 5: Lesson 443 — Evaluating Rational Expressions
Augment: the flow by adding the bottleneck amount to each edge on the path; Lesson 2706 — The Ford-Fulkerson Algorithm
augmented matrix: by placing the identity matrix **I** right next to it: **[A | I]**; Lesson 1304 — The Augmented Matrix Method Lesson 1309 — Writing Systems as Augmented Matrices Lesson 1313 — Gaussian Elimination Process Lesson 2105 — Introduction to Gaussian Elimination Lesson 2107 — Augmented Matrices Lesson 2252 — Computing Transition Matrices
augmented matrix method: gives you a systematic, mechanical process.; Lesson 1304 — The Augmented Matrix Method Lesson 1305 — Row Reduction to Find Inverses of 3×3 Matrices
augmenting paths: (paths that start and end at unmatched vertices, alternating between edges not in the matching and edges in the matching).; Lesson 2702 — The Hungarian Algorithm for Maximum Bipartite Matching Lesson 2706 — The Ford- Fulkerson Algorithm
Auxiliary lines and constructions: are extra elements you deliberately draw into a diagram to reveal hidden congruences, similarities, or angle relationships.; Lesson 769 — Auxiliary Lines and Constructions
Av: = λ**v**, **Aw** = μ**w**, then **v** · **w** = 0.; Lesson 2272 — The Spectral Theorem for Symmetric Matrices
Av = λv: and discover what makes a vector "eigen" (special) for a given matrix.; Lesson 2168 — Definition of Eigenvectors and Eigenvalues Lesson 2169 — Finding Eigenvalues by Hand for 2×2 Matrices Lesson 2172 — Eigenspaces and Their Dimension Lesson 2183 — Roots of the Characteristic Polynomial as Eigenvalues
average: of the two angles, `(A + B)/2`, and the **half-difference**, `(A - B)/2`.; Lesson 1076 — Sum-to-Product Formulas Lesson 3079 — Confidence and Prediction Intervals
average cost: if a company spends $500 on setup plus $3 per item, the average cost per item is `C(x) = (500 + 3x)/x`.; Lesson 494 — Applications of Rational Functions Lesson 1513 — Cost and Revenue Optimization
Average Power/Current: If power consumption `P(t)` varies throughout the day, find average power over 24 hours.; Lesson 1698 — Applications of Average Value
average rate: at which events occur in the given interval.; Lesson 2889 — Introduction to the Poisson Distribution Lesson 2893 — Applications of the Poisson Distribution
average rate of change: essentially the slope between two chosen points, but with the understanding that the rate might be different elsewhere.; Lesson 251 — Average Rate of Change vs Constant Rate Lesson 574 — Average Rate of Change Lesson 1369 — Understanding Average Rate of Change Lesson 1370 — Average Rate of Change Formula Lesson 1372 — The Concept of Instantaneous Rate of Change Lesson 1377 — Average Rate of Change Lesson 1378 — Instantaneous Rate of Change Lesson 1486 — Statement and Geometric Interpretation of the MVT
average speed: is 60 mph—that's your average rate of change of distance with respect to time.; Lesson 1369 — Understanding Average Rate of Change Lesson 1486 — Statement and Geometric Interpretation of the MVT
Average Temperature: If `T(t)` gives temperature at time `t`, find the average temperature over a day by integrating `T(t)` from `t = 0` to `t = 24` and dividing by 24.; Lesson 1698 — Applications of Average Value Lesson 1931 — Average Value of a Function in Three Dimensions
average value: of a function *f* on the interval [*a*, *b*] is given by:; Lesson 1564 — Average Value of a Function Lesson 1565 — The Mean Value Theorem for Integrals Lesson 1575 — Average Value of a Function Lesson 1697 — Average Value of a Function
average velocity: .; Lesson 1371 — Interpreting Average Rate of Change Lesson 1493 — Real-World Interpretation: Speed and Distance Lesson 1698 — Applications of Average Value
Avoiding approximation: Keeping radicals exact preserves precision until the final calculation; Lesson 514 — Why Rationalize Denominators?
Avoiding impossible integrals: Sometimes one order produces an integral with no elementary antiderivative (like `e^(x²)`), while the other order works fine; Lesson 1900 — Choosing Integration Order Strategically
Aw: = μ**w**, then **v** · **w** = 0.; Lesson 2272 — The Spectral Theorem for Symmetric Matrices
Ax: term (for the "slope") and the **B** term (for the "shift") to capture the full behavior.; Lesson 1628 — Irreducible Quadratic Factors: Distinct Lesson 2095 — Rank-Nullity Theorem Lesson 2104 — Matrix Equations and Systems Lesson 2131 — The Left Null Space Lesson 2134 — Orthogonality of Fundamental Subspaces Lesson 2143 — Testing Invertibility Lesson 2234 — The Least Squares Problem Lesson 2236 — Least Squares via Orthogonal Projection (+2 more)
Ax̂: in Col(A) that is *closest* to **b**.; Lesson 2234 — The Least Squares Problem Lesson 2236 — Least Squares via Orthogonal Projection
Ax + B: (not just a constant!; Lesson 1628 — Irreducible Quadratic Factors: Distinct
Ax = 0: .; Lesson 2069 — The Matrix Test for Linear Independence Lesson 2081 — Column Space and Null Space Lesson 2092 — Dimension of Subspaces Lesson 2130 — The Null Space Lesson 2131 — The Left Null Space Lesson 2216 — Properties of Orthogonal Complements
AX = B: by using the inverse of matrix A, transforming the problem into simple matrix multiplication.; Lesson 1307 — Using Inverses to Solve Matrix Equations Lesson 1308 — Applications of Inverse Matrices Lesson 1316 — Solving with Matrix Inverses Lesson 1317 — Cramer's Rule Lesson 1318 — Applications: Real-World Matrix Systems Lesson 2074 — Applications to Solutions of Linear Systems Lesson 2081 — Column Space and Null Space Lesson 2104 — Matrix Equations and Systems (+10 more)
Ax = b₁: and **Ax = b₂**, instead of working with [A | b₁] and then [A | b₂], you work with:; Lesson 2114 — Gaussian Elimination with Multiple Right-Hand Sides
Ax = b₂: , instead of working with [A | b₁] and then [A | b₂], you work with:; Lesson 2114 — Gaussian Elimination with Multiple Right-Hand Sides
axis of symmetry: .; Lesson 425 — The Axis of Symmetry Lesson 433 — Key Features Summary and Complete Graphing Lesson 824 — Kites: Properties and Diagonal Relationships Lesson 1146 — Standard Form: Vertex at Origin, Horizontal Axis Lesson 1148 — Parabolas with Vertex at (h, k): Vertical Axis Lesson 1149 — Parabolas with Vertex at (h, k): Horizontal Axis
Azimuthal direction (θ): At radius ρ and angle φ, moving through angle `dθ` traces an arc of length `ρ sin φ dθ` (since you're on a circle of radius `ρ sin φ`); Lesson 1940 — Volume Elements in Spherical Coordinates

B

b: } is a basis for a vector space *V*, then any vector **v** in *V* can be written as:; Lesson 2094 — Coordinates Relative to a Basis Lesson 2249 — Coordinates Relative to a Basis Lesson 2250 — The Change of Basis Problem Lesson 2252 — Computing Transition Matrices Lesson 2422 — Determining Coefficients from Initial Conditions
B (inside the parentheses): changes the **period** — how often the pattern repeats; Lesson 1024 — Transformations of Tangent: Period and Vertical Stretch
B (Period Change): The period becomes **2π/|B|** instead of 2π.; Lesson 1029 — Transformations of Secant
B = C: Lesson 2139 — Properties of Invertible Matrices
B = P ⁻¹AP: Lesson 2177 — Similar Matrices and Eigenvalue Invariance Lesson 2187 — Similar Matrices Have the Same Characteristic Polynomial Lesson 2255 — Change of Basis for Linear Transformations
B > A: , the major axis is vertical with length `2√B`, and the minor axis is horizontal with length `2√A`; Lesson 1163 — The Major and Minor Axes
b^(log_b(x)) = x: Lesson 618 — Definition of Logarithm as Inverse of Exponential Lesson 627 — Inverse Relationship: b^(log_b(x)) and log_b(b^x)
b₀: (the intercept): where the line crosses the y-axis; Lesson 3063 — Calculating the Regression Coefficients Lesson 3067 — Making Predictions with the Regression Line
B₁: Lesson 1099 — Two Solutions Case in SSA Lesson 2094 — Coordinates Relative to a Basis Lesson 2114 — Gaussian Elimination with Multiple Right-Hand Sides Lesson 2249 — Coordinates Relative to a Basis Lesson 2250 — The Change of Basis Problem Lesson 3063 — Calculating the Regression Coefficients Lesson 3067 — Making Predictions with the Regression Line
b₂: , .; Lesson 2094 — Coordinates Relative to a Basis Lesson 2114 — Gaussian Elimination with Multiple Right- Hand Sides Lesson 2249 — Coordinates Relative to a Basis Lesson 2250 — The Change of Basis Problem
b² - 4ac: that appears under the square root is called the **discriminant**.; Lesson 416 — The Discriminant: b² - 4ac Lesson 417 — Interpreting Discriminant Values Lesson 418 — Complex Solutions from the Quadratic Formula Lesson 1132 — Using the Discriminant to Classify Conics
BA: does not, or they may both exist but produce different-sized results!; Lesson 1281 — Matrix Multiplication Definition and Compatibility Lesson 1303 — Verifying an Inverse by Multiplication
BA = I: (the identity matrix); Lesson 1303 — Verifying an Inverse by Multiplication
back substitution: possible: you can solve for the last variable immediately, then work backward.; Lesson 2108 — Forward Elimination Process Lesson 2110 — Back Substitution Lesson 2121 — Converting REF to RREF Lesson 2246 — Solving Linear Systems Using QR
Back-substitute: to find the third variable using any original equation; Lesson 294 — Solving Three-Variable Systems by Elimination Lesson 638 — Exponential and Logarithmic Equations with Quadratic Form Lesson 1619 — Combining Trig Substitution with Other Techniques
back-substitution: .; Lesson 296 — Back-Substitution in Three Variables Lesson 2105 — Introduction to Gaussian Elimination
Backtrack: when a vertex has no unvisited neighbors; Lesson 2673 — Finding Spanning Trees: Depth-First Search
Backward (Extended): Lesson 2715 — Extended Euclidean Algorithm
Bad (dependence): Lesson 3090 — Checking for Independence: Residual Sequence Plots
Balance: Equal sample sizes across groups maximize statistical power; Lesson 3102 — ANOVA Applications and Experimental Design
bar charts: when comparing counts across many categories.; Lesson 2995 — Categorical Data: Bar Charts and Pie Charts Lesson 2997 — Choosing Appropriate Summaries and Displays
base: is `5` (the number being multiplied); Lesson 132 — What is an Exponent?Lesson 137 — Working with Fractional Bases Lesson 140 — Integer Exponents in Context Lesson 600 — What is an Exponential Function?Lesson 603 — Growth vs. Decay: The Role of the Base Lesson 734 — Base Angles of Isosceles Triangles Lesson 1112 — Area Formula Using Base and Height Lesson 2504 — Proving Divisibility Properties (+3 more)
Base (b): Any side of the parallelogram can be the base; Lesson 858 — Area of Parallelograms Lesson 859 — Area of Triangles
Base 10: `log_b(x) = log(x) / log(b)`; Lesson 625 — Change of Base Formula
Base area: = 6 × 6 = 36 cm²; Lesson 889 — Surface Area of Pyramids Lesson 890 — Surface Area of Cones Lesson 899 — Cavalieri's Principle with Cylinders
Base case: When *n* = 1, left side = 1, right side = 1(2)/2 = 1.; Lesson 2501 — The Principle of Mathematical Induction Lesson 2504 — Proving Divisibility Properties Lesson 2505 — Proving Inequalities by Induction Lesson 2508 — Induction Starting at Arbitrary Indices Lesson 2510 — Induction on Recursively Defined Sequences Lesson 2669 — Counting Vertices and Edges in Trees
Base case (n=1): Left side = 1.; Lesson 2503 — Proving Summation Formulas
Base case(s): Prove `P(n₀)` (and possibly a few more initial cases if needed); Lesson 2513 — The Strong Induction Principle Lesson 2514 — Proving Divisibility Results with Strong Induction Lesson 2515 — Strong Induction for Sequences and Recurrences Lesson 2516 — Structural Induction on Trees and Graphs
Base e: `log_b(x) = ln(x) / ln(b)`; Lesson 625 — Change of Base Formula
bases: of the trapezoid.; Lesson 821 — Trapezoids: Definitions and Basic Properties Lesson 860 — Area of Trapezoids Lesson 2063 — Spanning Sets and Redundancy
Basic rule: Adjust the boundary of your discrete value by **±0.; Lesson 2914 — Continuity Correction
basis: is a set of vectors that is both **linearly independent** and **spans** the entire vector space.; Lesson 2086 — What is a Basis?Lesson 2089 — Finding a Basis from a Spanning Set Lesson 2093 — The Basis Theorem Lesson 2094 — Coordinates Relative to a Basis
Basis transformations preserve structure: When changing between orthonormal bases, the transformation matrix is orthogonal (or unitary in ℂⁿ), meaning it preserves lengths and angles.; Lesson 2212 — Orthonormal Bases
Bayes factor: is the key tool here.; Lesson 3119 — Bayesian Hypothesis Testing and Bayes Factors
Bayesian updating: .; Lesson 2820 — Sequential Updating with Bayes' Theorem
Be precise: Use proper notation ( ≅ for congruence, proper triangle names); Lesson 748 — Writing Two-Column Congruence Proofs
bearing: describes a direction relative to north.; Lesson 979 — Navigation and Bearing Problems Lesson 1101 — Law of Sines with Bearing and Direction
Before calculators existed: , division was *much* harder than multiplication.; Lesson 514 — Why Rationalize Denominators?
Before data collection: Determine how many groups, sample sizes needed (power analysis), and whether blocking factors exist; Lesson 3102 — ANOVA Applications and Experimental Design
Before test: P(Disease) = 0.; Lesson 2801 — Updating Probabilities with New Information
Before we sample: , the sample mean is random and will fall within a predictable range around μ about 95% of the time (say, μ ± 1.; Lesson 3010 — The Logic Behind Confidence Intervals
Behavior patterns: emerge: convergence, divergence, oscillation; Lesson 2305 — Direction Fields
Bell-shaped and symmetric: around zero, just like the standard normal; Lesson 2923 — The Student's t-Distribution: Definition Lesson 3033 — The t-Distribution: Properties and Degrees of Freedom
Berge's Theorem: makes this precise:; Lesson 2701 — Augmenting Paths and Berge's Theorem Lesson 2702 — The Hungarian Algorithm for Maximum Bipartite Matching
Bernoulli: Lesson 2336 — Strategy Guide: Choosing the Right Solution Method
Bernoulli distribution: describes a random variable *X* that takes value 1 with probability *p* and value 0 with probability *1 - p*.; Lesson 2872 — The Bernoulli Trial and Bernoulli Distribution
Bernoulli equation: is a first-order differential equation with a specific structure:; Lesson 2332 — Bernoulli Equations: Form and Recognition Lesson 2334 — Special Cases: n=0 and n=1 in Bernoulli Equations
Bernoulli random variable: ) is the simplest type of random variable you can define.; Lesson 2830 — Indicator Random Variables Lesson 2873 — Bernoulli Distribution: PMF, Expectation, and Variance
Bernoulli trial: is any experiment or random process that has exactly two possible outcomes.; Lesson 2872 — The Bernoulli Trial and Bernoulli Distribution Lesson 2874 — Introduction to the Binomial Distribution
Bernoulli(p): For a single success/failure trial, the MGF is:; Lesson 2962 — MGFs of Common Distributions
Bessel functions: special functions that appear naturally in circular geometry; Lesson 2433 — The Two-Dimensional Wave Equation
Bessel modes: with fascinating nodal patterns—circles and radial lines where the membrane doesn't move.; Lesson 2433 — The Two-Dimensional Wave Equation
Bessel's equation: for the radial part R(r); Lesson 2454 — Separation in Other Coordinate Systems
Best for: Series with exponential terms, factorials in denominators, or repeated multiplication; Lesson 1735 — Choosing Comparison Series Effectively
beta distribution: is a continuous probability distribution defined on the interval **[0, 1]**, making it perfect for modeling proportions, probabilities, and percentages.; Lesson 2919 — The Beta Function and Beta Distribution Lesson 2920 — Properties and Applications of the Beta Distribution Lesson 3114 — Conjugate Priors: Beta-Binomial Model
Better for certain calculations: Useful in systems of equations and advanced applications; Lesson 260 — Standard Form: Ax + By = C
Better for small samples: than Spearman in some cases; Lesson 3110 — Kendall's Tau
between: the two lines being cut by the transversal; Lesson 717 — Same-Side Interior Angles Lesson 742 — SAS (Side-Angle-Side) Congruence Postulate Lesson 1114 — Choosing the Correct Angle in SAS Area Formula
between 0 and 1: causes a **horizontal stretch** (the graph gets wider); Lesson 581 — Horizontal Stretches and Compressions Lesson 603 — Growth vs. Decay: The Role of the Base
Between two parabolas: Region between `y = x²` and `y = 2 - x²` requires finding intersection points, then integrating with those curves as bounds.; Lesson 1906 — Regions Bounded by Curves
Between-group variation: How much do the group means differ from each other?; Lesson 3049 — One-Way ANOVA F Test Introduction Lesson 3093 — Introduction to Analysis of Variance (ANOVA)
Between-group variation (SSB): How much do the *group means* differ from each other?; Lesson 3095 — Between-Group and Within-Group Variation
Bézout coefficients: come in.; Lesson 2756 — Computing Solutions Using Bézout Coefficients
Bézout's identity: .; Lesson 2715 — Extended Euclidean Algorithm Lesson 2736 — Finding Modular Inverses
BF₁₂ < 0.1: Strong evidence for H₂; Lesson 3119 — Bayesian Hypothesis Testing and Bayes Factors
BF₁₂ > 10: Strong evidence for H₁; Lesson 3119 — Bayesian Hypothesis Testing and Bayes Factors
BF₁₂ ≈ 1: Data provide no clear preference; Lesson 3119 — Bayesian Hypothesis Testing and Bayes Factors
BFS tree: or **breadth-first spanning tree**.; Lesson 2674 — Finding Spanning Trees: Breadth-First Search
bijection: ) is a function that is both **injective** (one-to-one) and **surjective** (onto).; Lesson 2564 — Bijective Functions (One-to-One Correspondence)Lesson 2569 — Applications to Infinite Sets
bijective: if it's both injective *and* surjective—a perfect pairing between domain and codomain elements.; Lesson 2559 — Introduction to Function Types Lesson 2568 — Examples with Finite Sets
bijective function: (or **bijection**) is a function that is both **injective** (one-to-one) and **surjective** (onto).; Lesson 2564 — Bijective Functions (One-to-One Correspondence)Lesson 2565 — Inverse Functions and Bijections
binomial coefficient: is simply a standardized way to write "the number of ways to choose *k* objects from *n* objects without regard to order.; Lesson 2599 — Binomial Coefficients: Definition and Notation Lesson 2600 — The Binomial Theorem: Statement and Expansion
binomial distribution: models exactly this situation.; Lesson 2874 — Introduction to the Binomial Distribution Lesson 2875 — Binomial Probability Mass Function Lesson 3104 — Sign Test for Median
Binomial Theorem: is one of the most elegant results in combinatorics.; Lesson 2600 — The Binomial Theorem: Statement and Expansion Lesson 2610 — The Multinomial Theorem
Binomial(n, p): Since it's the sum of *n* independent Bernoulli trials, the MGF is the *product* of *n* identical Bernoulli MGFs:; Lesson 2962 — MGFs of Common Distributions
bipartite graph: is a graph whose vertices can be divided into two disjoint sets—let's call them *U* and *V*—such that every edge connects a vertex in *U* to a vertex in *V*.; Lesson 2653 — Bipartite Graphs Lesson 2696 — Coloring Special Graphs Lesson 2700 — Hall's Marriage Theorem
Birthdays in a room: If 13 people are in a room, at least two share a birth month.; Lesson 2570 — The Pigeonhole Principle: Statement and Intuition
Block diagonal: (when the space decomposes into invariant subspaces); Lesson 2257 — Finding Convenient Bases
BODMAS: (common in the UK) reminds us:; Lesson 31 — Introduction to Order of Operations (PEMDAS/BODMAS)
Body: Present your logical steps with reasons embedded naturally; Lesson 764 — Paragraph Proof Format
Boole's Inequality: or the **union bound**.; Lesson 2789 — Probability Bounds and Boole's Inequality
Boolean algebra: a complete system for manipulating logical expressions.; Lesson 2477 — Other Logical Equivalences and Laws
Borrow: from the next place value to the left.; Lesson 16 — Subtraction with Regrouping (Borrowing)
Borrow from the tens: The 5 tens becomes 4 tens, and the 2 ones becomes 12 ones (because we borrowed 10); Lesson 16 — Subtraction with Regrouping (Borrowing)
both: the original denominator and the expression you're dividing by (which becomes a denominator after flipping).; Lesson 454 — Dividing Rational Expressions: Multiply by Reciprocal Lesson 515 — Rationalizing with Square Roots Lesson 560 — Domain of Composite Functions Lesson 564 — Verifying Inverse Functions Lesson 639 — Mixed Exponential-Logarithmic Equations Lesson 752 — Definition of Similar Triangles Lesson 1273 — Matrix Equality Lesson 1407 — Common Errors and Problem-Solving Strategies (+4 more)
Both powers even: Use power-reducing formulas (half-angle identities); Lesson 1610 — Strategy Selection for Trigonometric Integrals
both sides: of an equation by the LCD of all fractions present, you use the **multiplication property of equality** to transform the equation into an equivalent one without fractions.; Lesson 209 — Equations with Fractions: Clearing Denominators Lesson 1321 — Estimating Limits from Tables of Values
bound: it.; Lesson 1366 — ε-δ Proofs for Quadratic Functions Lesson 1785 — The Lagrange Form of the Remainder Lesson 2487 — Binding Variables and Scope
Boundary conditions: describe what's happening at the spatial boundaries of your domain for all time.; Lesson 2411 — Initial and Boundary Conditions Lesson 2417 — Initial and Boundary Conditions Lesson 2418 — Steady-State Solutions Lesson 2423 — Heat Equation on a Finite Rod
boundary value problem: specifies conditions at two distinct points, say `x = a` and `x = b`:; Lesson 2345 — Boundary Value Problems: Introduction Lesson 2420 — Eigenvalue Problems from Boundary Conditions
Boundary value problems (BVPs): flip this idea: you're given information at *two different locations* (the "boundaries"), and you must find the function that connects them while satisfying the differential equation everywhere in between.; Lesson 2345 — Boundary Value Problems: Introduction Lesson 2404 — Boundary Value Problems: Introduction and Types
bounded: .; Lesson 303 — Bounded vs Unbounded Feasible Regions Lesson 572 — Bounded Functions Lesson 1710 — Monotonic Sequences and Boundedness
bounded above: if there's a ceiling: all outputs are less than or equal to some number *M*.; Lesson 572 — Bounded Functions Lesson 1710 — Monotonic Sequences and Boundedness
bounded below: if there's a floor: all outputs are greater than or equal to some number *m*.; Lesson 572 — Bounded Functions Lesson 1710 — Monotonic Sequences and Boundedness
box plots: (box-and-whisker diagrams), which visualize these five values graphically.; Lesson 2990 — Percentiles and Quartiles Lesson 2997 — Choosing Appropriate Summaries and Displays
Boxes: contain statements (facts, givens, or conclusions); Lesson 765 — Flow Chart Proof Format
Branches: are how we manage this.; Lesson 1234 — Applications: Fractional Powers and Euler's Formula
Break down composite shapes: into familiar pieces (rectangles + semicircles, trapezoids minus triangles); Lesson 866 — Perimeter and Area Problem Solving
Break it down: into shapes you already know (rectangles, triangles, circles, trapezoids, etc.; Lesson 868 — Finding Area of Composite Figures by Addition Lesson 873 — Composite Figures in Real- World Contexts
Breaking apart: √(36) = √(4 · 9) = √4 · √9 = 2 · 3 = 6; Lesson 497 — Product Property of Radicals
Breaking into Steps: Tackle complicated problems one piece at a time rather than all at once.; Lesson 195 — Checking Solutions and Problem-Solving Strategies
Bridges and structures: use parabolic arches because they distribute weight evenly along the curve.; Lesson 1187 — Real-World Conic Applications Review
Bring Down: Bring down the next digit from the dividend and repeat the process.; Lesson 29 — Long Division Algorithm Lesson 350 — Performing Synthetic Division
Build equivalent expressions: by multiplying each rational by the appropriate form of 1; Lesson 464 — Subtracting Rationals with Unlike Denominators
Build equivalent rationals: rewrite each expression with the LCD as its new denominator; Lesson 463 — Adding Rationals with Unlike Denominators
Build intuition: Seeing how different rules produce different outputs deepens your understanding; Lesson 542 — Working with Multiple Functions
Build it: [3/(x+2)] · [(x-1)/(x-1)] = [3(x-1)]/[(x+2)(x-1)]; Lesson 462 — Building Equivalent Rationals
Build number sense: See how numbers relate to each other; Lesson 13 — The Relationship Between Addition and Subtraction Lesson 24 — The Relationship Between Multiplication and Division
Build strategic complements: sometimes counting what you *don't* want is easier than direct counting; Lesson 2626 — Complex Applications and Problem-Solving Strategies
Build the general solution: u(x,t) = Σ [A_n cos(ω t) + B_n sin(ω t)]sin(nπx/L); Lesson 2431 — Fourier Series Solutions for Bounded Domains
Build the transition matrix: P = [v₁ | v₂ | .; Lesson 2258 — Applications: Diagonalization via Change of Basis
Build truth tables: for both propositions, listing all possible combinations of truth values for the atomic propositions involved; Lesson 2475 — Logical Equivalence
Build your design matrix: $A$ where each row contains powers of an $x_i$ value: $[1, x_i, x_i^2, \ldots, x_i^n]$; Lesson 2239 — Applications: Data Fitting and Approximation
Bundle: the objects that must be adjacent into one "block"; Lesson 2583 — Permutations with Restrictions: Adjacent Objects
Business Applications: If `R(x)` is revenue and `C(x)` is cost, the area between them over an interval represents total profit:; Lesson 1652 — Applied Problems: Area Between Curves
Business Revenue: If R(x) represents revenue from selling x units:; Lesson 1504 — Applications: Real-World Monotonicity and Concavity
Bx + C: Lesson 1628 — Irreducible Quadratic Factors: Distinct
By definition: of even: n = 2k for some integer k; Lesson 2491 — Direct Proof: Basic Structure

C

c: , then the column space consists of every vector you can build by taking:; Lesson 2128 — The Column Space Lesson 2250 — The Change of Basis Problem Lesson 2252 — Computing Transition Matrices Lesson 2421 — The General Series Solution Lesson 2463 — Complex Exponential Form of Fourier Series
C (Phase Shift): Shifts the graph horizontally C units to the right (or left if C is negative).; Lesson 1029 — Transformations of Secant
C < 0: shift **left** by |C| units; Lesson 1025 — Transformations of Tangent: Phase Shift and Vertical Shift
C = 2πr: (using the radius); Lesson 861 — Circumference of Circles Lesson 1477 — Geometric Related Rates: Expanding Circles
C = AB: Lesson 1282 — Computing the (i,j)-Entry of a Product
C = EM: (encoded message), you recover the original with **M = E ¹C**.; Lesson 1308 — Applications of Inverse Matrices
C = πd: (using the diameter); Lesson 861 — Circumference of Circles
C > 0: shift **right** by C units; Lesson 1025 — Transformations of Tangent: Phase Shift and Vertical Shift
C cos(ωx): , use:; Lesson 2352 — Particular Solutions for Sine and Cosine Functions
C sin(ωx): or **C cos(ωx)**, use:; Lesson 2352 — Particular Solutions for Sine and Cosine Functions
C value: in `sin(B(x - C))` slides the entire sine (or cosine) graph horizontally left or right.; Lesson 1037 — Phase Shift: Horizontal Translation
C(6,2): = 6!; Lesson 2590 — Computing Basic Combinations
C(7,3): the number of ways to choose 3 items from 7:; Lesson 2590 — Computing Basic Combinations
C(a, b): = combinations ("a choose b"); Lesson 2885 — Hypergeometric Probability Formula
C(AB) = CI: Lesson 2139 — Properties of Invertible Matrices
C(K, k): Ways to choose k successes from K available successes; Lesson 2885 — Hypergeometric Probability Formula
C(n-1, k-1): ways.; Lesson 2604 — Pascal's Identity and Recursive Properties
C(n-1, k): ways.; Lesson 2604 — Pascal's Identity and Recursive Properties
C(N, n): Total ways to choose n items from N; Lesson 2885 — Hypergeometric Probability Formula
C(n,0): = n!; Lesson 2592 — Combinations with All or None: C(n,0) and C(n,n)
C(n,0) – Choosing Nothing: Lesson 2592 — Combinations with All or None: C(n,0) and C(n,n)
C(n,0) = 1: .; Lesson 2592 — Combinations with All or None: C(n,0) and C(n,n)
C(n,k): where k is the term's index; Lesson 2602 — Finding Specific Terms in Binomial Expansions Lesson 2875 — Binomial Probability Mass Function
C(n,n): = n!; Lesson 2592 — Combinations with All or None: C(n,0) and C(n,n)
C(n,n) – Choosing Everything: Lesson 2592 — Combinations with All or None: C(n,0) and C(n,n)
C(n,n) = 1: .; Lesson 2592 — Combinations with All or None: C(n,0) and C(n,n)
C(n,r) = C(n,n-r): Lesson 2591 — Symmetry Property: C(n,r) = C(n,n-r)
c₁: , **c₂**, .; Lesson 2128 — The Column Space Lesson 2250 — The Change of Basis Problem Lesson 2343 — The Principle of Superposition
c₂: , .; Lesson 2128 — The Column Space Lesson 2250 — The Change of Basis Problem Lesson 2343 — The Principle of Superposition
CA = I: Lesson 2139 — Properties of Invertible Matrices
CAH: stands for:; Lesson 950 — The Cosine Ratio (CAH)
calculate: their values.; Lesson 133 — Evaluating Positive Integer Exponents Lesson 136 — Evaluating Negative Exponents Lesson 248 — Finding Slope from a Graph Lesson 873 — Composite Figures in Real-World Contexts Lesson 883 — Volume of Composite Solids Lesson 1965 — Evaluating Vector Fields at Points
Calculate average rates: for each interval using the formula you know:; Lesson 1374 — Estimating Instantaneous Rates Numerically
Calculate differences: For each pair (subject), compute d = measurement₁ - measurement₂; Lesson 3039 — Paired t-Test: Comparing Dependent Samples
Calculate each area: Lesson 868 — Finding Area of Composite Figures by Addition
Calculate each area separately: using formulas you know (rectangle: *lw*, circle: *πr²*, semicircle: *½πr²*); Lesson 871 — Composite Figures with Circles and Semicircles
Calculate expected counts: assuming all populations share the same distribution; Lesson 3045 — Chi-Squared Test for Homogeneity
Calculate expected frequencies: from your theoretical model; Lesson 3043 — Chi-Squared Goodness-of-Fit Test
Calculate standard deviation: of differences: sᴅ; Lesson 3039 — Paired t-Test: Comparing Dependent Samples
Calculate the 2×2 determinant: of that minor.; Lesson 2150 — Determinant of a 3×3 Matrix: Cofactor Expansion
Calculate the area: of each individual face (using formulas you already know: rectangles, triangles, circles, etc.; Lesson 893 — Nets and Surface Area Lesson 1548 — Right Riemann Sums
Calculate the difference: `d` between paired ranks.; Lesson 3109 — Spearman's Rank Correlation
Calculate the imaginary part: Multiply r by sin θ; Lesson 1219 — Converting from Polar to Rectangular Form
Calculate the perpendicular slope: by taking the negative reciprocal; Lesson 268 — Writing Equations of Perpendicular Lines
Calculate the ratio: Horizontal asymptote is y = 3/5; Lesson 489 — Finding Horizontal Asymptotes by Leading Coefficients
Calculate the real part: Multiply r by cos θ; Lesson 1219 — Converting from Polar to Rectangular Form
Calculate the slope: Rise/Run = 4/3; Lesson 244 — Rise Over Run: Understanding Slope Geometrically
Calculate the surface area: of each solid separately (using formulas you've learned); Lesson 892 — Surface Area of Composite Solids
Calculate the t-statistic: `t = (x̄ - μ₀) / (s/√n)` where x̄ is the sample mean, s is the sample standard deviation, and n is the sample size; Lesson 3034 — One-Sample t-Test: Testing a Population Mean
Calculate the test statistic: You compute the **z-statistic**:; Lesson 3025 — One-Sample z-Test for Means Lesson 3048 — F Test for Equality of Variances
Calculate U statistics: for each group; Lesson 3106 — Mann-Whitney U Test (Wilcoxon Rank-Sum)
Calculate x-coordinate: x = -(-8)/(2·2) = 8/4 = 2; Lesson 429 — Finding the Vertex Using x = -b/(2a)
Calculator-based contexts: – Engineering, physics labs, and applied sciences often work directly with decimal values.; Lesson 521 — Applications and When Not to Rationalize
Calendar problems: "What day will it be 100 days from now?; Lesson 2752 — Systems of Linear Congruences
Can get messy: Complicated equations with fractions or decimals require extra steps; Lesson 281 — Comparing Graphing and Substitution Methods
Can I avoid fractions: If one path requires multiplying by 7 and another by 2, choose the simpler one.; Lesson 295 — Choosing Strategic Pairs for Elimination
Can I integrate easily: Lesson 1741 — Mixed Practice: Integral and Comparison Tests
Can I use u-substitution: Look for a function and its derivative present together (or with a constant multiple).; Lesson 1632 — Applications and Strategy Selection
Cancel: those common factors (divide them out); Lesson 445 — Simplifying by Canceling Common Factors Lesson 452 — Simplifying Before Multiplying Lesson 455 — Dividing and Simplifying in One Step Lesson 487 — Holes in Rational Functions Lesson 2437 — Basic Solutions in Cartesian Coordinates
Cancel common factors: across numerators and denominators; Lesson 456 — Mixed Operations: Multiplication and Division Together
cancel out: , giving you an incorrect answer.; Lesson 1647 — Area with Multiple Regions Lesson 2032 — Stokes' Theorem on Piecewise Smooth Surfaces
Cancel the 4: C(7,3) = (7 × 6 × 5) / 3!; Lesson 2590 — Computing Basic Combinations
Cancel the common factors: (remembering they equal 1 when divided); Lesson 448 — Simplifying with Trinomial Factors
Cancelation works: If a·c ≡ b·c (mod n) and GCD(c, n) = 1, you can safely cancel c from both sides.; Lesson 2718 — Relatively Prime Integers
cannot: be parallel to a vertical line; Lesson 270 — Special Cases: Horizontal and Vertical Lines Lesson 392 — Zero Product Property Lesson 549 — Finding Domain from Equations Lesson 870 — Perimeter of Composite Figures Lesson 892 — Surface Area of Composite Solids Lesson 1399 — The Product Rule Formula Lesson 1813 — Computing Limits Using Direct Substitution Lesson 2098 — Matrix Addition and Subtraction (+1 more)
Cannot combine (unlike radicals): Lesson 158 — Adding and Subtracting Radicals
Capacity c(u,v): The maximum amount of flow that can travel along edge (u,v); Lesson 2704 — Networks and Flows: Definitions and Notation
Capacity comparisons: "Which holds more: a sphere with radius 5cm or a cube with side 8cm?; Lesson 884 — Volume Word Problems and Applications
Capacity constraint: For every edge (u,v), we must have 0 ≤ f(u,v) ≤ c(u,v).; Lesson 2704 — Networks and Flows: Definitions and Notation
Car loans: Compare financing options; Lesson 689 — Loan Amortization
cardinality: of a set is the number of elements it contains, denoted |A|.; Lesson 2521 — Sets: Definition and Notation Lesson 2566 — Cardinality and Bijections
Cardioids: (like r = 1 + cos θ): Heart-shaped curves.; Lesson 1919 — Computing Areas Using Polar Integrals
carrying capacity: (denoted *L* or *K*).; Lesson 645 — Logistic Growth Models Lesson 2375 — Logistic Growth Model
Cartesian product: does.; Lesson 2534 — Introduction to Cartesian Products
Case 1: If *f(n)* grows **slower** than *n^(log_b a)*, then **T(n) = Θ(n^(log_b a))**; Lesson 2636 — Divide-and-Conquer Recurrences Lesson 2722 — Infinitude of Primes: Euclid's Proof
Case 1: No Solution: Lesson 1125 — The Ambiguous Case with Obtuse Angles
Case 2: If *f(n)* grows at **exactly the same rate** as *n^(log_b a)*, then **T(n) = Θ(n^(log_b a) · log n)**; Lesson 2636 — Divide-and-Conquer Recurrences Lesson 2722 — Infinitude of Primes: Euclid's Proof
Case 2: One Solution: Lesson 1125 — The Ambiguous Case with Obtuse Angles
Case 3: If *f(n)* grows **faster** than *n^(log_b a)*, then **T(n) = Θ(f(n))**; Lesson 2636 — Divide-and-Conquer Recurrences
Case-by-case addition: Count each valid scenario separately and add them; Lesson 2595 — Combinations with Restrictions
Cauchy form: of the remainder states:; Lesson 1788 — The Cauchy and Integral Forms of the Remainder
Cauchy product: .; Lesson 1769 — Operations on Power Series
Cauchy-Schwarz Inequality: states that for any two vectors **u** and **v** in an inner product space:; Lesson 2204 — The Cauchy-Schwarz Inequality
Cavalieri's Principle: if two solids have the same height and their cross-sections at every level have equal areas, they have equal volumes.; Lesson 886 — Volume of Oblique Solids Lesson 897 — Introduction to Cavalieri's Principle Lesson 899 — Cavalieri's Principle with Cylinders
Cavalieri's Principle states: If two solids have equal heights, and every horizontal cross-section taken at the same height has equal area, then the two solids have equal volumes.; Lesson 900 — Cavalieri's Principle for Pyramids and Cones
CDF method: is a systematic approach:; Lesson 2869 — Functions of Random Variables: Continuous Case
center: is the fixed point in the middle of the circle.; Lesson 827 — Parts of a Circle: Center, Radius, Diameter, and Chord Lesson 991 — Defining the Unit Circle Lesson 1141 — Finding Circle Properties from General Form Lesson 1156 — Parts of an Ellipse: Vertices, Co- vertices, and Foci Lesson 1168 — Components of a Hyperbola: Center, Vertices, and Foci Lesson 1174 — Hyperbolas Centered at (h, k)Lesson 1793 — Equations of Spheres Lesson 2905 — Parameters: Mean and Standard Deviation
Center at the origin: The conic is centered at the point (0, 0); Lesson 1133 — Standard Position and Orientation
Center differences: Which group has a higher median?; Lesson 2994 — Comparing Distributions
center of mass: is the point where an object would perfectly balance—it's the "average position" of all the mass.; Lesson 1700 — Center of Mass for One-Dimensional Systems Lesson 1921 — Applications: Mass and Center of Mass in Polar Lesson 1957 — Moments and First Moments Lesson 1958 — Center of Mass in Two Dimensions
Center the data: Subtract the mean from each column to get **X̃**; Lesson 2289 — Principal Component Analysis (PCA) and SVD
central angle: is an angle whose vertex sits at the center of a circle.; Lesson 833 — Central Angles and Their Measures Lesson 838 — Inscribed Angles Lesson 845 — Congruent Arcs and Central Angles Lesson 864 — Area of Sectors and Arc Length
Central Limit Theorem: , we know the sample mean follows approximately a normal distribution centered at the true population mean μ, with standard error σ/√n.; Lesson 3010 — The Logic Behind Confidence Intervals
Central Limit Theorem (CLT): is one of the most powerful results in probability theory.; Lesson 2977 — Introduction to the Central Limit Theorem
Central tendency: Compare means and medians across groups; Lesson 2994 — Comparing Distributions
centroid: is where the three **medians** meet (remember: medians connect vertices to midpoints of opposite sides).; Lesson 739 — Centroid, Orthocenter, Incenter, and Circumcenter Lesson 1701 — Moments and Centroids of Planar Regions
chain rule: tells us:; Lesson 1410 — Chain Rule with Power Functions Lesson 1412 — Chain Rule with Radicals Lesson 1413 — Chain Rule with Rational Functions Lesson 1433 — Chain Rule with Exponential Functions Lesson 1444 — Chain Rule with Inverse Trigonometric Functions Lesson 1847 — The Gradient is Perpendicular to Level Curves and Surfaces
Chain rule needed: Lesson 1417 — Recognizing When to Use the Chain Rule
Chain rule NOT needed: Lesson 1417 — Recognizing When to Use the Chain Rule
Change: subtraction to addition AND flip the sign of the second number (find its opposite); Lesson 44 — Subtracting Positive and Negative Numbers
Change coordinates: Let **y** = P^T**x** (rotating to align with eigenvectors); Lesson 2277 — Principal Axes and Quadratic Forms
change of basis problem: given a vector's description in one coordinate system (basis), find its description in another coordinate system (basis).; Lesson 2250 — The Change of Basis Problem Lesson 2258 — Applications: Diagonalization via Change of Basis
Change of Variables Formula: states:; Lesson 1947 — Change of Variables Formula for Double Integrals
Change the lower limit: Calculate u(a) = g(a); Lesson 1586 — Definite Integrals and Changing Limits
Change the upper limit: Calculate u(b) = g(b); Lesson 1586 — Definite Integrals and Changing Limits
Change variables: by setting `x = Py`, where `y` is a new coordinate system; Lesson 2296 — Diagonalization of Quadratic Forms
changes: from trial to trial; Lesson 2887 — Hypergeometric vs. Binomial: When to Use Each Lesson 3082 — Categorical Predictors and Interaction Terms
Changes shape: based on degrees of freedom; Lesson 3033 — The t-Distribution: Properties and Degrees of Freedom
changes sign: .; Lesson 1499 — Finding Intervals of Concavity Lesson 2162 — Effect of Row Operations on Determinants
characteristic equation: **det(A - λI) = 0**; Lesson 2169 — Finding Eigenvalues by Hand for 2×2 Matrices Lesson 2338 — The Characteristic Equation for Constant Coefficients Lesson 2339 — Real Distinct Roots: General Solution Lesson 2629 — Linear Homogeneous Recurrence Relations Lesson 2630 — Solving Second-Order Linear Homogeneous Recurrences
Characteristic functions: solve this problem elegantly.; Lesson 2967 — Characteristic Functions (Optional)
Charge distribution: For electric charge density on a surface; Lesson 2018 — Surface Integrals of Scalar Functions
Check: that your answer's units make sense; Lesson 113 — Solving Rate Problems Lesson 123 — Real-World Proportion Problems Lesson 192 — Age Problems and Consecutive Integer Problems Lesson 202 — Combining Addition/Subtraction with Multiplication/Division Lesson 299 — Applications of Three-Variable Systems Lesson 403 — Solving Quadratics by Completing the Square (a = 1)Lesson 443 — Evaluating Rational Expressions Lesson 475 — Applications and Word Problems with Complex Fractions (+7 more)
Check all quadrants: where the trig function takes that value; Lesson 1082 — Finding All Solutions in an Interval
Check all solutions: in the original equation — this step is critical!; Lesson 526 — Equations with Radicals on Both Sides Lesson 527 — Equations Requiring Multiple Squaring Steps Lesson 530 — Mixed Radical and Rational Equations
Check both solutions: against the restricted values to eliminate extraneous solutions; Lesson 482 — Rational Equations Leading to Quadratics
Check boundaries: – If the domain is closed and bounded, evaluate f on the boundary using techniques like parametrization or Lagrange multipliers (coming later).; Lesson 1872 — Applications: Optimization Problems in Two Variables
Check boundary points: where pieces meet — these often create special range values; Lesson 556 — Domain and Range of Piecewise Functions
Check conditions: The approximation works well when both *np* ≥ 10 and *n*(1−*p*) ≥ 10; Lesson 2985 — Approximating Binomial Distributions
Check dimensions: After differentiation, every term should have the same "order" — if you started with x³, your derivative terms shouldn't mix x² with x⁴.; Lesson 1407 — Common Errors and Problem-Solving Strategies
Check each solution: by substituting it back into the original equation; Lesson 481 — Checking for Extraneous Solutions Lesson 486 — Vertical Asymptotes
Check endpoints: if the domain is restricted; Lesson 1510 — Geometric Optimization: Minimizing Perimeter Lesson 1513 — Cost and Revenue Optimization Lesson 1878 — Solving Two-Variable Constrained Problems
Check f: We need g(x) ≥ 0, so x - 3 ≥ 0, which means x ≥ 3; Lesson 560 — Domain of Composite Functions
Check for additional restrictions: within each piece (like square roots or denominators); Lesson 556 — Domain and Range of Piecewise Functions
Check for extraneous solutions: Always verify your answers satisfy domain restrictions (logs need positive arguments, etc.; Lesson 638 — Exponential and Logarithmic Equations with Quadratic Form
Check for overlap: – Does any part of your proposed trial appear in `y_c`?; Lesson 2358 — Common Mistakes and Problem-Solving Strategies
Check for restricted values: – note any values that make denominators zero; Lesson 478 — Solving by Cross-Multiplication
Check for sign change: If f(a) and f(b) have opposite signs (one positive, one negative), then by the IVT, there must be some c in [a, b] where f(c) = 0; Lesson 1357 — Using the IVT to Show Roots Exist
Check for special solutions: (identity or contradiction); Lesson 216 — Mixed Practice: Complex Linear Equations
Check g: x can be any real number; Lesson 560 — Domain of Composite Functions
Check if they span: the entire space.; Lesson 2090 — Extending to a Basis
Check it: Use histograms, normal probability plots (Q-Q plots), or formal tests like Shapiro-Wilk.; Lesson 3040 — Assumptions and Diagnostics for t-Tests
Check linear dependence: See if one constraint's gradient is a scalar multiple of another's; Lesson 1889 — Dealing with Redundant Constraints
Check mutual independence: Lesson 2944 — Pairwise vs Mutual Independence
Check pairwise independence: Lesson 2944 — Pairwise vs Mutual Independence
Check the degrees: Both numerator and denominator have degree 2; Lesson 489 — Finding Horizontal Asymptotes by Leading Coefficients
Check the denominators: – Make sure they're identical; Lesson 69 — Adding Fractions with Like Denominators
Check the exponents: – Are they the same?; Lesson 167 — Adding and Subtracting in Scientific Notation
Check the pattern: Lesson 1106 — Choosing Between Law of Sines and Law of Cosines
Check the result: If the left side becomes I, the right side is now A ¹.; Lesson 2142 — The Gauss-Jordan Method for Finding Inverses
Check the right side: If it's negative, you'll use imaginary numbers.; Lesson 409 — Complex Solutions via Completing the Square
Check the solution: Lesson 2069 — The Matrix Test for Linear Independence
Check the support region: The region where f(x,y) > 0 must be a *rectangle* in the xy-plane (or the entire plane); Lesson 2941 — Checking Independence for Continuous Random Variables
Check validity: Does B₂ make sense?; Lesson 1124 — Solving for Two Possible Triangles
Check where you land: (does your solution satisfy the boundary condition?; Lesson 2406 — Solving Boundary Value Problems: Shooting Method Concept
Check x = -2: log(-2) is undefined!; Lesson 637 — Checking for Extraneous Solutions
Check x = 5: log(5) + log(2) = log(10) = 1 (and both arguments are positive); Lesson 637 — Checking for Extraneous Solutions
Check your answer: by substituting into both original equations; Lesson 274 — Solving Systems by Graphing: Basic Technique Lesson 282 — Application Problems Using Systems and Substitution Lesson 299 — Applications of Three-Variable Systems Lesson 400 — Applications and Word Problems
Check your input: against the conditions for each piece; Lesson 591 — Evaluating Piecewise Functions
Check your solution: by substituting back; Lesson 211 — Multi-Step Equations Requiring Multiple Operations Lesson 478 — Solving by Cross- Multiplication Lesson 524 — Solving Simple Radical Equations Lesson 528 — Radical Equations with Higher Index Roots Lesson 537 — Solving Equations with Fractional Exponents
Check your solutions: in the original equation (critical!; Lesson 636 — Solving Logarithmic Equations with Multiple Logs
Check your work: If 6 + 3 = 9, then 9 - 3 must equal 6; Lesson 13 — The Relationship Between Addition and Subtraction Lesson 24 — The Relationship Between Multiplication and Division Lesson 381 — Applications and Mixed Practice Lesson 421 — Verifying Solutions by Substitution Lesson 712 — Solving Multi-Step Angle Problems
Checking the Range: Lesson 2563 — Testing for Surjectivity
Chemical reactions: How fast is a reactant being consumed at this instant?; Lesson 1376 — Applications: Velocity and Other Rates
chi-squared distribution: (written as χ²) is not a completely new distribution—it's actually a specific version of the gamma distribution you just learned.; Lesson 2921 — The Chi-Squared Distribution: Definition Lesson 3042 — Chi-Squared Distribution Basics
Chi-squared goodness-of-fit test: Tests whether observed categorical data follows an expected distribution; Lesson 2922 — Properties and Applications of Chi-Squared Lesson 3043 — Chi-Squared Goodness-of-Fit Test
Chi-Squared independence test: "We reject the null hypothesis (p = 0.; Lesson 3050 — Interpreting Chi-Squared and F Test Results
chi-squared test for independence: helps you decide whether observed patterns in your data represent real associations or just random chance.; Lesson 3044 — Chi-Squared Test for Independence Lesson 3046 — Expected Frequencies and Assumptions
Chi-squared test of independence: Determines if two categorical variables are independent in contingency tables; Lesson 2922 — Properties and Applications of Chi-Squared
Chi-squared tests: excel when you're analyzing categorical data—like testing whether customer preferences differ across regions (independence test) or whether observed survey responses match theoretical predictions (goodness-of-fit).; Lesson 3051 — Applications and Limitations
Chinese Remainder Theorem: If m₁, m₂, .; Lesson 2753 — The Chinese Remainder Theorem Statement
Cholesky decomposition: is a specialized factorization available only for positive definite matrices.; Lesson 2295 — The Cholesky Decomposition Lesson 2298 — Applications to Optimization and Hessian Matrices
Choose: the right formula (explicit: a = a₁ + (n-1)d); Lesson 659 — Applications of Arithmetic Sequences
Choose a coordinate system: with the origin at the fluid surface, y-axis pointing downward; Lesson 1699 — Fluid Pressure and Force on Submerged Surfaces
Choose a grid: Select evenly-spaced points (like a 5×5 grid) that cover the region of interest; Lesson 1968 — Sketching Vector Fields by Hand
Choose a projection plane: (xy, xz, or yz) that simplifies the shadow region; Lesson 1929 — Triple Integrals over General 3D Regions
Choose a side: for your variable terms (usually the side with the larger coefficient); Lesson 221 — Inequalities with Variables on Both Sides
Choose a variable: pick a letter (like *x*, *n*, or *t*) to represent it; Lesson 189 — Writing Equations from Word Problems
Choose contrapositive proof when: Lesson 2495 — Proof by Contrapositive: Method and Strategy
Choose coordinates: with y measuring vertical height; Lesson 1695 — Pumping Liquids from Tanks Lesson 1956 — Computing Total Mass with Triple Integrals
Choose familiar formulas: If you can decompose using rectangles and triangles rather than trapezoids, you might save time (fewer variables in the formulas).; Lesson 874 — Optimizing Composite Figure Problems
Choose Geometric Mean when: Lesson 805 — Comparing Pythagorean Theorem and Geometric Mean Methods
Choose intervals: around that point: [a, a + h] where h gets smaller; Lesson 1374 — Estimating Instantaneous Rates Numerically
Choose k < r: (the desired rank); Lesson 2288 — Low-Rank Approximation and Data Compression
Choose minimum: Select the edge with the smallest weight among those crossing the boundary.; Lesson 2676 — Minimum Spanning Trees: Prim's Algorithm
Choose Path 1: Often try approaching along a simple line like $y = 0$ (the x-axis) or $x = 0$ (the y-axis); Lesson 1812 — Path-Dependent Limits and Nonexistence
Choose Path 2: Try a different line like $y = x$, or a curve like $y = x^2$; Lesson 1812 — Path-Dependent Limits and Nonexistence
Choose Pythagorean Theorem when: Lesson 805 — Comparing Pythagorean Theorem and Geometric Mean Methods
Choose sample points: in the xy-plane (the "floor" of your 3D space); Lesson 1806 — Graphing Surfaces in 3D
Choose slicing direction: Which makes the setup simpler?; Lesson 1652 — Applied Problems: Area Between Curves
Choose strategically: Sometimes simplifying means making an expression shorter.; Lesson 1079 — Simplifying Expressions with Product and Sum Formulas
Choose the appropriate bounds: based on which segment you're rotating; Lesson 1656 — Finding Limits of Integration for Disk Method
Choose the appropriate formula: Use the surface area formula that matches your axis of rotation.; Lesson 1691 — Applications: Real-World Surfaces
Choose the first pivot: (top-left entry); Lesson 2108 — Forward Elimination Process
Choose the maximum: The corner point giving the highest value is your answer; Lesson 308 — Applications: Maximization Problems
Choose the new basis: Let B = {v₁, v₂, .; Lesson 2258 — Applications: Diagonalization via Change of Basis
Choose the right formula: – Perimeter or area?; Lesson 193 — Perimeter and Area Word Problems Lesson 804 — Finding Missing Lengths with Geometric Means
Choose the right properties: identify which overlapping conditions to track (the "bad" events you want to exclude); Lesson 2626 — Complex Applications and Problem-Solving Strategies
Choose u and dv: using LIATE (just like before); Lesson 1599 — Definite Integrals and Integration by Parts
Choose your expansion: Pick a row or column for cofactor expansion (choose one with zeros if possible!; Lesson 2182 — Computing Characteristic Polynomials for 3×3 Matrices
Choose Your Method: Lesson 1673 — Applications and Problem-Solving Strategies
Choose Your Tool: Lesson 982 — Mixed Application Problems and Problem-Solving Strategies
Choose your weapon: cofactor expansion or row reduction; Lesson 2158 — Practice: Mixed Determinant Computation
chord: is any line segment connecting two points on the circle.; Lesson 827 — Parts of a Circle: Center, Radius, Diameter, and Chord Lesson 845 — Congruent Arcs and Central Angles Lesson 850 — Chord Properties and Definitions
Chords and the Center: A perpendicular line from the center of a circle to a chord bisects (cuts in half) that chord.; Lesson 850 — Chord Properties and Definitions
circle: has an equation like **(x - h)² + (y - k)² = r²**, while a **line** has an equation like **y = mx + b** or **Ax + By = C**.; Lesson 920 — Intersection of Lines and Circles Lesson 942 — Line Symmetry (Reflectional Symmetry)Lesson 1130 — Geometric Definitions of Conics Lesson 1131 — The General Second-Degree Equation Lesson 1132 — Using the Discriminant to Classify Conics Lesson 1133 — Standard Position and Orientation Lesson 1136 — Standard Form of a Circle Lesson 1179 — Eccentricity: Measuring the Shape of Conics (+1 more)
Circles: appear in wheels, gears, and any rotating machinery.; Lesson 1135 — Overview of Conic Applications
Circuit (Closed Walk): Lesson 2658 — Circuits and Cycles
Circuit design: Minimize power loss with voltage and current restrictions; Lesson 1890 — Application: Optimization in Physics and Engineering
Circular patterns: often suggest peaks or valleys; Lesson 1800 — Level Curves and Contour Maps
circular segment: is the region between a chord and the arc it cuts off.; Lesson 843 — Segment Area Lesson 865 — Area of Segments
circulation: patterns where vectors curve around a point; Lesson 1968 — Sketching Vector Fields by Hand Lesson 1981 — Applications: Work and Circulation Lesson 2034 — Physical Interpretations: Circulation and Rotation
Circulation ≠ 0: indicates rotational behavior (like vortices in fluid flow); Lesson 1981 — Applications: Work and Circulation
Circulation Density: Lesson 2009 — Interpreting Curl Physically
circumcenter: is where the **perpendicular bisectors** of the three sides intersect.; Lesson 739 — Centroid, Orthocenter, Incenter, and Circumcenter Lesson 832 — Inscribed and Circumscribed Circles
circumference: is the distance around a circle—like measuring the perimeter of a polygon, but for a curved shape.; Lesson 828 — Circumference of a Circle Lesson 830 — Circles and Pi (π)Lesson 861 — Circumference of Circles Lesson 1477 — Geometric Related Rates: Expanding Circles Lesson 1664 — Introduction to the Shell Method
circumscribed circle: (circumcircle)—the circle passing through all three vertices.; Lesson 739 — Centroid, Orthocenter, Incenter, and Circumcenter Lesson 832 — Inscribed and Circumscribed Circles
Circumscribed circles: wrap *around* a polygon and pass through all its vertices (corners).; Lesson 832 — Inscribed and Circumscribed Circles
Claim: Vertical angles are congruent.; Lesson 770 — Proving Theorems About Lines and Angles Lesson 2497 — Proof by Contradiction: Basic Idea Lesson 2503 — Proving Summation Formulas Lesson 2504 — Proving Divisibility Properties Lesson 2514 — Proving Divisibility Results with Strong Induction Lesson 2573 — Pigeonhole Principle in Number Theory Lesson 2743 — Proof of Fermat's Little Theorem
Clairaut's Theorem: states that if the mixed partial derivatives are **continuous** in a neighborhood of a point, then they are equal at that point:; Lesson 1827 — Mixed Partial Derivatives and Clairaut's Theorem
Class intervals: the ranges (e.; Lesson 2992 — Histograms and Frequency Distributions
Classic example: Proving the Fundamental Theorem of Arithmetic (every integer > 1 has a unique prime factorization).; Lesson 2512 — Review: Ordinary vs. Strong Induction
Classify: using the test conditions; Lesson 1867 — Applying the Second Derivative Test Lesson 2335 — Applications of Exact and Bernoulli Equations
Classify critical points: – Use the Second Derivative Test (discriminant D) to identify local maxima, minima, or saddle points.; Lesson 1872 — Applications: Optimization Problems in Two Variables
Classify each interval: Lesson 1499 — Finding Intervals of Concavity
Classify the Given Information: Lesson 982 — Mixed Application Problems and Problem-Solving Strategies
Cleaner pattern recognition: – Rationalized forms reveal mathematical structure better.; Lesson 521 — Applications and When Not to Rationalize
Cleaner patterns: Many common functions have elegant Maclaurin series; Lesson 1774 — Maclaurin Series: Taylor Series at Zero
Clear denominators: by multiplying both sides by the LCD; Lesson 482 — Rational Equations Leading to Quadratics Lesson 1623 — Linear Factors: Distinct Denominators
Clear fractions: Multiply everything by 4: `6(x - 4) + 20 = x + 6 - 4`; Lesson 216 — Mixed Practice: Complex Linear Equations Lesson 530 — Mixed Radical and Rational Equations
Clear fractions or decimals: if needed (multiply both sides by LCD or power of 10); Lesson 211 — Multi-Step Equations Requiring Multiple Operations Lesson 216 — Mixed Practice: Complex Linear Equations
Clear the fraction: Multiply both sides by $x$: $x\sqrt{x} = 6$; Lesson 530 — Mixed Radical and Rational Equations
Clearer integer relationships: Some real-world problems naturally produce integer coefficients; Lesson 260 — Standard Form: Ax + By = C
closed: (includes both endpoints with brackets: [*a*, *b*]); Lesson 1506 — The Extreme Value Theorem Lesson 1981 — Applications: Work and Circulation
closed and bounded region: , the Second Derivative Test helps you classify critical points *inside* the region (the interior).; Lesson 1869 — Boundary Considerations for Optimization Lesson 1870 — Optimization on Closed and Bounded Regions
Closed circle (●): Use when the endpoint *is* included (≤ or ≥); Lesson 217 — Inequality Notation and the Number Line
Closed endpoint: (≤ or ≥): Use a **solid dot** • to show the point is included; Lesson 592 — Graphing Piecewise Functions
Closed Interval Method: works exactly the same way.; Lesson 1508 — The Closed Interval Method
Closed Intervals [a, b]: Lesson 1353 — Continuity on an Interval
Closed under addition: If **u** and **v** are in H, then **u + v** is also in H; Lesson 2078 — Subspaces: Definition and Subspace Test Lesson 2083 — Intersection of Subspaces Lesson 2084 — Sum of Subspaces Lesson 2215 — Orthogonal Complements
Closed under scalar multiplication: If **u** is in H and *c* is any scalar, then *c***u** is in H; Lesson 2078 — Subspaces: Definition and Subspace Test Lesson 2083 — Intersection of Subspaces Lesson 2084 — Sum of Subspaces Lesson 2215 — Orthogonal Complements
Closed-form results: No computational burden.; Lesson 3114 — Conjugate Priors: Beta-Binomial Model
Closely spaced curves: indicate steep terrain (rapid change in z); Lesson 1800 — Level Curves and Contour Maps
Clustering: might indicate dependence issues; Lesson 3101 — Checking ANOVA Assumptions
Clusters or groupings: May suggest missing categorical variables or interaction effects.; Lesson 3084 — Residual Plots: Fitted Values vs. Residuals
co-vertices: are the endpoints of the minor axis—the closest points from the center.; Lesson 1156 — Parts of an Ellipse: Vertices, Co-vertices, and Foci Lesson 1168 — Components of a Hyperbola: Center, Vertices, and Foci
Cobb-Douglas production function: models how inputs create output:; Lesson 1810 — Applications: Modeling with Multivariable Functions
Codomain: The entire set B that the relation maps *into*.; Lesson 2540 — Domain, Codomain, and Range of Relations
Codomain of R: = {a, b, c, d, e} — the entire set B by definition; Lesson 2540 — Domain, Codomain, and Range of Relations
coefficient: is the number multiplied by a variable within a term.; Lesson 171 — Terms, Coefficients, and Constants Lesson 310 — What is a Polynomial?Lesson 312 — Terms, Coefficients, and Constants Lesson 2055 — What is a Linear Combination?
Coefficient before f: `−2` means a vertical stretch by factor 2 AND a reflection across the x-axis (negative sign); Lesson 586 — Identifying Transformations from Equations
Coefficient matrix determinant D: Lesson 1299 — Cramer's Rule for Solving Systems
Coefficient with x: If you saw `f(2x)`, that's a horizontal compression by factor ½; Lesson 586 — Identifying Transformations from Equations
coefficients: (the numbers in front) while keeping the radical part unchanged.; Lesson 158 — Adding and Subtracting Radicals Lesson 171 — Terms, Coefficients, and Constants Lesson 288 — Choosing the Easier Variable to Eliminate Lesson 508 — Multiplying Radicals with the Same Index
cofactor expansion: (or expansion by minors).; Lesson 1292 — Determinant of a 3×3 Matrix: Expansion by Minors Lesson 2150 — Determinant of a 3×3 Matrix: Cofactor Expansion
Cofunction identities: sin(90° - θ) = cos θ, etc.; Lesson 1050 — Simplifying Trigonometric Expressions
Cohen's d: measures effect size by expressing the difference between two means in standard deviation units:; Lesson 3041 — Effect Size and Practical Significance in t-Tests
Coincident: Same slope, same intercept (or equivalent equations) → Infinite solutions; Lesson 276 — Graphing Systems with Parallel and Coincident Lines
Collect all distinct values: (removing duplicates); Lesson 2824 — The Range of a Discrete Random Variable
Collect evidence: and see what happens under that assumption; Lesson 3024 — The Logic of Hypothesis Testing
Collect samples: from each population (e.; Lesson 3045 — Chi-Squared Test for Homogeneity
Collecting a large sample: of observations; Lesson 2975 — Applications: Estimating Probabilities and Means
Collinear: All points on the *same line* (1-dimensional constraint); Lesson 698 — Collinear and Coplanar Points
Collinear points: are points that all lie on the same straight line.; Lesson 698 — Collinear and Coplanar Points
Colon notation: 3:4; Lesson 106 — Writing Ratios in Different Forms
Colon notation (3:4): is compact and clear—it literally means "3 compared to 4.; Lesson 106 — Writing Ratios in Different Forms
column space: (also called the *range* or *image*) of a matrix **A** is the set of all possible linear combinations of its column vectors.; Lesson 2081 — Column Space and Null Space Lesson 2096 — Bases for Important Subspaces Lesson 2128 — The Column Space Lesson 2134 — Orthogonality of Fundamental Subspaces Lesson 2262 — The Range (Image) of a Linear Transformation Lesson 2283 — Left and Right Singular Vectors
Column space view: The rank equals the dimension of the span of the matrix's columns.; Lesson 2126 — Definition of Matrix Rank
Column Space, C(A): Lesson 2132 — The Four Fundamental Subspaces
column vector: (or column matrix) has exactly **one column** but can have multiple rows.; Lesson 1272 — Types of Matrices: Square, Row, Column, and Zero Lesson 2097 — Matrix Notation and Dimensions
columns: (like tiny boxes) stacked from the xy-plane up to the surface.; Lesson 1892 — Introduction to Double Integrals Lesson 2069 — The Matrix Test for Linear Independence
combination: a selection of objects where *order doesn't matter*.; Lesson 2588 — Introduction to Combinations Lesson 3100 — Two-Way ANOVA With Interaction
Combinations: Use when the **order doesn't matter**.; Lesson 2594 — Distinguishing When to Use Combinations vs Permutations Lesson 2599 — Binomial Coefficients: Definition and Notation Lesson 2615 — Relationship to Combinations and Stars-and-Bars
Combinations (order doesn't matter): {A,B}, {A,C}, {B,C} → 3 selections; Lesson 2588 — Introduction to Combinations
Combinations of continuous functions: (sums, products, compositions); Lesson 1813 — Computing Limits Using Direct Substitution
combine: separate radicals into one: √a / √b = √(a/b).; Lesson 500 — Quotient Property of Radicals Lesson 507 — Simplifying Before Adding or Subtracting Lesson 883 — Volume of Composite Solids Lesson 2274 — Eigenspaces and Multiplicity for Symmetric Matrices Lesson 2400 — Finding Power Series Solutions Near Ordinary Points
Combine all allowed x-values: the domain is typically the union of all conditions *unless* there are gaps; Lesson 556 — Domain and Range of Piecewise Functions
Combine all possible y-values: , watching for overlaps or gaps; Lesson 556 — Domain and Range of Piecewise Functions
Combine both samples: and rank all values together (smallest = 1); Lesson 3106 — Mann-Whitney U Test (Wilcoxon Rank-Sum)
Combine each group separately: Lesson 1197 — Adding and Subtracting Complex Numbers
Combine everything: and add your constant of integration; Lesson 1542 — Antiderivatives of Polynomial Functions
combine like terms: .; Lesson 181 — Simplifying Expressions with Multiple Steps Lesson 203 — Equations with Variables on One Side Lesson 208 — Equations with Parentheses and the Distributive Property Lesson 211 — Multi-Step Equations Requiring Multiple Operations Lesson 216 — Mixed Practice: Complex Linear Equations Lesson 328 — Multiplying a Binomial by a Trinomial Lesson 329 — Multiplying Two Trinomials Lesson 1202 — Simplifying Complex Expressions
Combine numerators: using addition and subtraction from left to right; Lesson 467 — Adding and Subtracting Three or More Rationals
Combine the results: working back outward; Lesson 1338 — Combining Multiple Limit Laws
Combine them: The general solution is **y = y_h + y_p**; Lesson 2356 — Second-Order Examples with Undetermined Coefficients
Combine using the formula: Add up the contributions from each pathway; Lesson 1854 — Chain Rule for Independent Variables
Combined: `1/√(1 - (3x)²) · 3 = 3/√(1 - 9x²)`; Lesson 1444 — Chain Rule with Inverse Trigonometric Functions Lesson 1559 — Properties of Definite Integrals: Linearity Lesson 2859 — Properties of Variance: Scaling and Translation
Combined rate: = sum of individual rates; Lesson 468 — Applications of Adding and Subtracting Rationals
Combined variation: happens when a relationship involves *both* direct and inverse variation together.; Lesson 127 — Combined Variation Lesson 128 — Finding the Constant of Variation
combining: groups of objects.; Lesson 10 — Addition Facts to 10 Lesson 497 — Product Property of Radicals
Combining logs: If you see `log₂(x) + log₂(5) = 3`, use the product rule to combine:; Lesson 629 — Applying Logarithm Properties to Solve Equations
Commands: "Solve for x.; Lesson 2468 — Propositions and Truth Values
Common applications: Lesson 1359 — Applications of Continuity Theorems
Common conversion tools: Lesson 1090 — Equations Involving Multiple Functions
Common denominator method: Combine fractions over a common denominator; Lesson 1532 — Indeterminate Differences: ∞ - ∞
common difference: , usually denoted by *d*.; Lesson 651 — Definition of Arithmetic Sequences Lesson 652 — Finding Terms Using the Common Difference Lesson 653 — The Explicit Formula for Arithmetic Sequences Lesson 654 — Finding the Common Difference from Two Terms Lesson 656 — Determining if a Sequence is Arithmetic Lesson 657 — Finding the Number of Terms in a Finite Sequence Lesson 693 — Stacking and Construction Problems
common divisors: 1, 2, 3, and 6.; Lesson 2711 — Greatest Common Divisor (GCD): Definition Lesson 2712 — Finding GCD by Listing Divisors
common factor: is a number that divides evenly into two or more numbers without leaving a remainder.; Lesson 51 — Common Factors Lesson 487 — Holes in Rational Functions
common factors: 1, 2, 3, and 6.; Lesson 51 — Common Factors Lesson 520 — Simplifying After Rationalizing
Common forms: Lesson 3022 — Test Statistics and P-values
Common labeling patterns: Lesson 1114 — Choosing the Correct Angle in SAS Area Formula
Common mistake: Don't do addition first just because you "like" it better!; Lesson 35 — Addition and Subtraction (Left to Right)
Common modulus attack: Never encrypt the same message to multiple recipients using the same `n`.; Lesson 2770 — Security and Practical Considerations
common multiples: .; Lesson 53 — Common Multiples Lesson 54 — Least Common Multiple (LCM)
common ratio: , often denoted by *r*.; Lesson 660 — Introduction to Geometric Sequences Lesson 662 — Finding the Common Ratio Lesson 663 — Writing Formulas for Geometric Sequences Lesson 666 — Growth and Decay Applications Lesson 667 — Convergence and Divergence of Geometric Sequences Lesson 1722 — Geometric Series: Definition and Formula Lesson 1728 — Comparing Geometric and p-Series
Common scenario types: Lesson 973 — Combined Problems with Distance and Height
Common scenarios: Lesson 1079 — Simplifying Expressions with Product and Sum Formulas
Common vulnerabilities: Lesson 2770 — Security and Practical Considerations
Communication: A central server (cut vertex) whose failure splits a network into isolated clusters.; Lesson 2661 — Cut Vertices and Cut Edges
commutative: .; Lesson 929 — Composition of Translations and Reflections Lesson 1275 — Properties of Matrix Addition Lesson 2048 — Vector Addition Lesson 2098 — Matrix Addition and Subtraction Lesson 2524 — Union of Sets
commutative property: .; Lesson 22 — The Commutative and Associative Properties of Multiplication Lesson 36 — The Commutative Property
Compare: If both paths head toward the same y-value, that's your limit!; Lesson 1322 — Estimating Limits from Graphs Lesson 1507 — Critical Points in Optimization Lesson 1515 — Optimization on Open Intervals Lesson 1812 — Path-Dependent Limits and Nonexistence Lesson 3034 — One-Sample t-Test: Testing a Population Mean Lesson 3043 — Chi-Squared Goodness-of-Fit Test Lesson 3045 — Chi-Squared Test for Homogeneity
Compare all function values: from steps 1 and 2; Lesson 1871 — Global Extrema: Finding Absolute Maximum and Minimum
Compare all these values: Lesson 1508 — The Closed Interval Method
Compare apples to oranges: Compare values from different normal distributions on a common scale; Lesson 2907 — Standardization and Z-Scores
Compare candidates: Evaluate the objective function at all critical points (including boundary points if applicable).; Lesson 1891 — Verifying and Interpreting Solutions
Compare denominators: Determine what factor you need to multiply the current denominator by to get the desired denominator; Lesson 462 — Building Equivalent Rationals
Compare hundreds first: (if both numbers have hundreds); Lesson 6 — Comparing and Ordering Whole Numbers
Compare the result columns: if they're identical in every row, the propositions are logically equivalent; Lesson 2475 — Logical Equivalence
Compare to chi-squared distribution: Under the null hypothesis (all groups come from identical distributions), H approximately follows a chi-squared distribution with *k* − 1 degrees of freedom (*k* = number of groups).; Lesson 3107 — Kruskal-Wallis Test
Compare to t-distribution: with (n-1) degrees of freedom; Lesson 3039 — Paired t-Test: Comparing Dependent Samples
Compare to the original: Divide the change by the original value; Lesson 101 — Percent Increase and Decrease
Compare values: – The global extremum is the best value among all critical points and boundary values.; Lesson 1872 — Applications: Optimization Problems in Two Variables
Comparing rates: helps you make smart decisions by putting everything on equal footing.; Lesson 111 — Comparing Rates to Make Decisions
Comparing Two Variances: To test if σ₁² = σ₂² from independent samples, compute F = s₁²/s₂².; Lesson 2926 — Properties and Applications of the F-Distribution
Comparison: "Sam has 30 stickers.; Lesson 18 — Word Problems with Addition and Subtraction
Comparison Property: works the same way: If one function stays at or above another function throughout an interval, then its integral (the "accumulated total") must be at least as large.; Lesson 1563 — Comparison Properties of Definite Integrals
Comparison Test: lets you determine convergence without finding the exact value.; Lesson 1640 — Comparison Test for Improper Integrals
Comparison Tests: (compare to p-series).; Lesson 1750 — Strategy: Choosing the Right Convergence Test Lesson 1759 — Choosing the Right Test for Alternating Series
Compass bearings: Measured from north or south toward east or west (e.; Lesson 979 — Navigation and Bearing Problems
Compatible numbers: shine in division when you want clean, easy calculations; Lesson 30 — Estimating Products and Quotients
complement: of the other; Lesson 706 — Complementary Angles Lesson 2526 — Set Difference and Complement Lesson 2778 — Complement of an Event
Complementary angles: are two angles that together form a right angle.; Lesson 706 — Complementary Angles Lesson 1049 — Cofunction Identities
Complementary approach: Lesson 2597 — Complementary Counting with Combinations
complementary counting: Lesson 2583 — Permutations with Restrictions: Adjacent Objects Lesson 2595 — Combinations with Restrictions Lesson 2614 — Distribution Problems with Constraints
Complete answer: x³ - (5x²)/2 + 7x + C; Lesson 1542 — Antiderivatives of Polynomial Functions
Complete bipartite graph K₃,₃: Three vertices in each set, all connected across sets.; Lesson 2689 — Planar Graphs and the Definition of Planarity
Complete graph K₄: Four vertices where every vertex connects to every other.; Lesson 2689 — Planar Graphs and the Definition of Planarity
Complete graph K₅: Five vertices all connected to each other.; Lesson 2689 — Planar Graphs and the Definition of Planarity
Complete orthonormal basis: Using the Gram-Schmidt process within each eigenspace (for repeated eigenvalues), we can construct an orthonormal basis of eigenvectors for ℝⁿ.; Lesson 2272 — The Spectral Theorem for Symmetric Matrices
Complete the parallelogram: using the two vectors as adjacent sides; Lesson 1243 — Adding Vectors Geometrically
Complete the square: as usual (factor out leading coefficient if needed, move constant, add (b/2)² to both sides).; Lesson 409 — Complex Solutions via Completing the Square Lesson 1174 — Hyperbolas Centered at (h, k)Lesson 1175 — Converting General Form to Standard Form Lesson 1793 — Equations of Spheres
Completing the square: reverses this process.; Lesson 402 — Completing the Square with Leading Coefficient 1 Lesson 406 — Converting to Vertex Form Lesson 428 — Converting Standard Form to Vertex Form Lesson 1140 — Converting Between Circle Forms Lesson 1150 — Converting to Standard Form by Completing the Square Lesson 1680 — Simplifying Arc Length Integrals
complex conjugate: of a complex number is formed by changing the sign of its imaginary part while keeping the real part the same.; Lesson 1196 — Complex Conjugates Lesson 1199 — Complex Conjugates
Complex domains: (pentagons, domains with holes); Lesson 2455 — Limitations and Extensions of the Method
Complex eigenvalues: oscillatory behavior with frequency determined by imaginary part; Lesson 2278 — Applications to Differential Equations and Dynamics
Complex exponentials: For \(\sum \frac{n^n}{(n+1)^n}\), the nth root yields \(\frac{n}{n+1} \to 1\) (inconclusive, need another test).; Lesson 1746 — Applying the Root Test
complex fraction: is simply a fraction where the numerator, the denominator, or both contain fractions themselves.; Lesson 80 — Complex Fractions Lesson 469 — What is a Complex Fraction?Lesson 471 — Simplifying Complex Fractions by Simplifying Numerator and Denominator Lesson 1384 — Computing Derivatives from the Definition: Rational Functions
complex numbers: they have a real part (–3) and an imaginary part (±4i).; Lesson 409 — Complex Solutions via Completing the Square Lesson 418 — Complex Solutions from the Quadratic Formula Lesson 1189 — The Imaginary Unit i
complex plane: (or Argand diagram).; Lesson 1192 — Plotting Complex Numbers on the Complex Plane Lesson 1207 — Complex Numbers as Points in the Plane
Complicated Products: Lesson 1458 — When to Use Logarithmic Differentiation
Complicated Quotients: Lesson 1458 — When to Use Logarithmic Differentiation
component form: .; Lesson 1237 — Component Form of Vectors in 2D Lesson 1252 — Resolving Vectors into Components
Component form in 3D: uses an **ordered triple** to represent a vector:; Lesson 1238 — Component Form of Vectors in 3D
component functions: .; Lesson 1963 — Introduction to Vector Fields Lesson 1964 — Component Functions of Vector Fields
composite number: is any whole number greater than 1 that has *more than two factors*.; Lesson 56 — Composite Numbers Lesson 2720 — Prime Numbers: Definition and Basic Properties
Composite numbers: have *more than two factors* (like 4, 6, 8, 9, 10…).; Lesson 56 — Composite Numbers
composition: .; Lesson 940 — Composition of Rigid Motions Lesson 1355 — Continuity of Combinations of Functions Lesson 1410 — Chain Rule with Power Functions Lesson 2548 — Composition of Relations
composition of transformations: means applying one transformation after another.; Lesson 929 — Composition of Translations and Reflections Lesson 938 — Combining Rotations and Dilations with Other Transformations
Compound events: Lesson 2775 — Simple Events vs Compound Events Lesson 2834 — Computing Probabilities from PMFs
compound interest: means you earn interest not just on your initial amount, but also on the interest you've already earned.; Lesson 642 — Compound Interest and Continuous Compounding Lesson 687 — Compound Interest and Financial Growth
compression: of the basic wave pattern.; Lesson 1033 — Amplitude: Vertical Stretch and Compression Lesson 2290 — Applications of SVD: Image Compression and Noise Reduction
Computational complexity: measures how the workload grows as the problem size increases.; Lesson 2115 — Computational Complexity and Efficiency
Computational simplicity: You only need to compute one inner product to test orthogonality; Lesson 2206 — Angle Between Vectors and Orthogonality
Compute: f(a) (the function value at the nice point); Lesson 1470 — Estimating Function Values with Linear Approximation Lesson 1849 — Directional Derivatives in Non-Unit Directions Lesson 1977 — Computing Vector Line Integrals via Parametrization Lesson 1982 — Practice with Line Integrals in 2D and 3D Lesson 2040 — Applying the Theorem to a Sphere
Compute a: by integrating f(x)cos(nx) over one period; Lesson 2459 — Computing Fourier Series for Simple Functions
Compute a₀: by integrating f(x) over one full period; Lesson 2459 — Computing Fourier Series for Simple Functions
Compute and interpret: Don't forget units and context!; Lesson 1652 — Applied Problems: Area Between Curves
Compute and Verify: Lesson 1673 — Applications and Problem-Solving Strategies
Compute b: by integrating f(x)sin(nx) over one period; Lesson 2459 — Computing Fourier Series for Simple Functions
Compute both partial derivatives: Find ∂f/∂x and ∂f/∂y; Lesson 1862 — Critical Points in Two Variables
Compute denominator: 3!; Lesson 2590 — Computing Basic Combinations
Compute det(A - λI): Form the characteristic polynomial.; Lesson 2188 — Using the Characteristic Polynomial in Practice
Compute du and v: by differentiating and integrating; Lesson 1599 — Definite Integrals and Integration by Parts
Compute dx/dt and dy/dt: from your parametric equations; Lesson 1853 — Computing Derivatives along Parametric Curves
Compute each integral: Apply your standard line integral techniques to each segment; Lesson 1979 — Line Integrals Over Piecewise Smooth Curves
Compute f(x + h): f(x + h) = m(x + h) + b = mx + mh + b; Lesson 1381 — Computing Derivatives from the Definition: Linear Functions
Compute M: = m₁ · m₂ · .; Lesson 2755 — The CRT Construction Algorithm
Compute mean difference: d̄ (average of all differences); Lesson 3039 — Paired t-Test: Comparing Dependent Samples
Compute sample moments: from your data: m₁ = x̄, m₂ = (1/n)Σx ᵢ², etc.; Lesson 3003 — Method of Moments for Multiple Parameters
Compute SVD: **X̃ = UΣV ᵀ**; Lesson 2289 — Principal Component Analysis (PCA) and SVD
Compute the antiderivative: as usual; Lesson 1637 — Convergence and Divergence of Improper Integrals
Compute the chi-squared statistic: χ² = Σ[(Observed - Expected)²/Expected]; Lesson 3045 — Chi-Squared Test for Homogeneity
Compute the derivative: f'(a) to find the rate of change; Lesson 1474 — Tangent Line Approximation in Applied Problems
Compute the determinant: using cofactor expansion (typically along the row or column with the most zeros); Lesson 2178 — Computing Eigenvalues for 3×3 Matrices Lesson 2181 — Computing Characteristic Polynomials for 2×2 Matrices
Compute the discriminant: D = f_xx · f_yy - (f_xy)²; Lesson 1867 — Applying the Second Derivative Test
Compute the gradient: at the point: **∇F(x₀, y₀, z₀)** = ⟨Fₓ, Fᵧ, Fᵤ⟩ evaluated at *(x₀, y₀, z₀)*; Lesson 1834 — Finding Tangent Planes to Level Surfaces
Compute the H-statistic: A formula (similar in spirit to the F-statistic) measures whether rank sums differ more than chance would predict.; Lesson 3107 — Kruskal-Wallis Test
Compute the Jacobian: Find the derivative dx/dy (how x changes with respect to y); Lesson 2870 — The Jacobian Method for Transformations
Compute the necessary derivatives: of your proposed solution; Lesson 2307 — Verifying Solutions by Substitution
Compute the probability: that this alternative distribution falls in the rejection region; Lesson 3058 — Calculating Power for Simple Tests
Compute the projection matrix: P = A(A ᵀA) ¹A ᵀ; Lesson 2231 — The Projection Matrix
Compute the small power: Calculate a^r mod p, which is manageable since *r* < *p*-1; Lesson 2744 — Computing Large Powers Using Fermat's Little Theorem
Compute the t-statistic: t = d̄ / (sᴅ/√n), where n is the number of pairs; Lesson 3039 — Paired t-Test: Comparing Dependent Samples
Compute the χ² statistic: using the formula above; Lesson 3043 — Chi-Squared Goodness-of-Fit Test
Compute z: for each (x,y) pair using your function; Lesson 1806 — Graphing Surfaces in 3D
Computing expectations: E[g(X)h(Y)] = E[g(X)] · E[h(Y)]; Lesson 2946 — Independence and Functions of Random Variables
Computing powers: A¹⁰⁰ = PD¹⁰⁰P ¹, and D¹⁰⁰ is trivial (just raise diagonal entries to the 100th power); Lesson 2258 — Applications: Diagonalization via Change of Basis
Computing the sample average: Lesson 2975 — Applications: Estimating Probabilities and Means
concave down: on an interval when its graph bends downward, like a frown or an upside-down cup.; Lesson 1497 — Introduction to Concavity Lesson 1498 — The Second Derivative and Concavity Lesson 1499 — Finding Intervals of Concavity Lesson 1501 — The Second Derivative Test for Local Extrema Lesson 1521 — Second Derivative and Concavity Lesson 1525 — Comprehensive Curve Sketching Algorithm Lesson 1550 — Comparing Left, Right, and Midpoint Approximations
Concave down intervals: Where f''(x) < 0 (graph curves downward like ∩); Lesson 1503 — Analyzing Function Behavior with Derivatives
concave up: on an interval when its graph bends upward, like a cup or smile.; Lesson 1497 — Introduction to Concavity Lesson 1498 — The Second Derivative and Concavity Lesson 1499 — Finding Intervals of Concavity Lesson 1501 — The Second Derivative Test for Local Extrema Lesson 1521 — Second Derivative and Concavity Lesson 1525 — Comprehensive Curve Sketching Algorithm Lesson 1550 — Comparing Left, Right, and Midpoint Approximations
Concave up intervals: Where f''(x) > 0 (graph curves upward like ∪); Lesson 1503 — Analyzing Function Behavior with Derivatives
concavity: describes *how* the curve bends as you move along it.; Lesson 1497 — Introduction to Concavity Lesson 1498 — The Second Derivative and Concavity Lesson 1504 — Applications: Real-World Monotonicity and Concavity Lesson 1521 — Second Derivative and Concavity
Concentration: = A(t)/V(t) (volume might change!; Lesson 2377 — Mixing Problems and Compartment Models
Conclude: The triangles are congruent by SSS; Lesson 741 — SSS (Side-Side-Side) Congruence Postulate Lesson 767 — Proof by Contradiction Lesson 771 — Proving Properties of Parallel Lines Lesson 1344 — The Squeeze Theorem Lesson 1757 — The Absolute Convergence Test Lesson 2493 — Direct Proof Examples and Practice Lesson 2495 — Proof by Contrapositive: Method and Strategy Lesson 2497 — Proof by Contradiction: Basic Idea
Conclude congruence: Apply SSS, SAS, ASA, AAS, or HL; Lesson 748 — Writing Two-Column Congruence Proofs
Conclusion: Since all consecutive differences equal 6, this sequence **is arithmetic** with common difference d = 6.; Lesson 656 — Determining if a Sequence is Arithmetic Lesson 764 — Paragraph Proof Format Lesson 789 — Acute and Obtuse Triangle Tests Lesson 1071 — Verifying Identities with Double and Half Angle Formulas Lesson 1488 — Verifying the Hypotheses of the MVT Lesson 1494 — Using the First Derivative to Determine Monotonicity Lesson 1732 — Applying the Integral Test to p-Series and Beyond Lesson 2306 — Existence and Uniqueness Theorems (+3 more)
Conclusion (Q): The result that follows; Lesson 2489 — Structure of Mathematical Statements
Conclusion reached: Lesson 768 — Direct Proof Techniques
condense: means combining separate logarithms into one:; Lesson 624 — Combining Logarithm Properties Lesson 629 — Applying Logarithm Properties to Solve Equations
Conditional probabilities: Use p(x|y) = p(x, y) / p_Y(y); Lesson 2927 — Joint Probability Distributions: Discrete Case
Conditional probability test: Does P(A|B) = P(A)?; Lesson 2804 — Testing for Independence
conditional variance: measures spread within the conditioned world:; Lesson 2932 — Properties of Conditional Distributions Lesson 2934 — Conditional Expectation and Conditional Variance
conditionally convergent: ).; Lesson 1755 — Absolute Convergence Lesson 1756 — Conditional Convergence Lesson 1758 — Rearrangements and the Riemann Rearrangement Theorem
Conditions needed: Initial temperature distribution `u(x,0) = f(x)` and boundary conditions (insulated ends, fixed temperatures, etc.; Lesson 2415 — Physical Interpretation and Modeling
Cones: The equation **φ = c** describes a cone opening from the origin.; Lesson 1939 — Graphing Surfaces in Spherical Coordinates
confidence interval: is a range of values constructed from sample data that aims to capture an unknown population parameter.; Lesson 3009 — Introduction to Confidence Intervals Lesson 3015 — Confidence Interval for a Population Proportion Lesson 3018 — Confidence Interval for the Difference of Two Proportions Lesson 3069 — Inference for the Slope Lesson 3116 — Credible Intervals vs. Confidence Intervals
Confidence intervals: What's a plausible range for the true slope value?; Lesson 3069 — Inference for the Slope
confidence level: (commonly 90%, 95%, or 99%) tells us how often our method would succeed *in the long run*.; Lesson 3009 — Introduction to Confidence Intervals Lesson 3013 — Choosing Sample Size for Desired Margin of Error
Confirm: that both sides equal zero (or are equal to each other); Lesson 421 — Verifying Solutions by Substitution
Confirm the sign change: an inflection point exists only if f''(x) switches from positive to negative or negative to positive; Lesson 1522 — Inflection Points
Confirmation of extrema: Using the Second Derivative Test; Lesson 1503 — Analyzing Function Behavior with Derivatives
Confusing f with f(x): Lesson 547 — Common Function Notation Mistakes
Congruence: means two figures are *exactly* the same in both shape and size.; Lesson 729 — Distinguishing Congruence from Similarity Lesson 941 — Congruence Through Rigid Motions
Congruence modulo n: is a relation on the integers that captures when two numbers have the same remainder after division by n.; Lesson 2557 — Congruence Modulo n
congruent: (equal in measure).; Lesson 715 — Alternate Interior Angles Lesson 716 — Alternate Exterior Angles Lesson 722 — Understanding Congruence Lesson 723 — Congruent Segments and Angles Lesson 724 — Congruence Transformations Lesson 729 — Distinguishing Congruence from Similarity Lesson 740 — Congruent Triangles: Definition and Notation Lesson 835 — Congruent Circles and Circle Relationships (+3 more)
Congruent Chords: In the same circle or in congruent circles, chords that are equal in length are equidistant from the center.; Lesson 850 — Chord Properties and Definitions
Congruent figures: (≅): same shape AND same size; Lesson 726 — Introduction to Similarity
Congruent hypotenuses: (the sides opposite the right angles); Lesson 745 — HL (Hypotenuse-Leg) Congruence Theorem
congruent modulo n: (written *a ≡ b (mod n)*) if their difference is divisible by n.; Lesson 2557 — Congruence Modulo n Lesson 2730 — Introduction to Congruence Notation
Congruent Sides: Lesson 824 — Kites: Properties and Diagonal Relationships
conjugate: .; Lesson 517 — Conjugates and Binomial Denominators Lesson 518 — Rationalizing with Sum or Difference Conjugates Lesson 1177 — Conjugate Hyperbolas Lesson 1342 — Limits Involving Conjugates Lesson 2201 — Standard Inner Products on ℝⁿ and ℂⁿ Lesson 3114 — Conjugate Priors: Beta-Binomial Model
conjugate axis: (perpendicular to the opening direction).; Lesson 1168 — Components of a Hyperbola: Center, Vertices, and Foci Lesson 1169 — The Transverse and Conjugate Axes
Connect the dots: Sometimes you'll need to use angles from *one* transversal to find angles on *another*, building a chain of reasoning; Lesson 720 — Parallel Lines Cut by Multiple Transversals Lesson 1806 — Graphing Surfaces in 3D
connected: if there exists a path between every pair of vertices.; Lesson 2659 — Connected Graphs Lesson 2667 — Applications of Connectivity
Connected and acyclic: – The graph is connected and has no cycles (the definition above); Lesson 2668 — Definition and Properties of Trees
Connection to factorials: For positive integers n, Γ(n) = (n − 1)!; Lesson 2916 — Introduction to the Gamma Function
Connectivity and vulnerability: Cut vertices and cut edges reveal structural weaknesses.; Lesson 2661 — Cut Vertices and Cut Edges
Consecutive angles are supplementary: (they add up to 180°); Lesson 816 — Parallelograms: Definition and Properties
Consecutive even/odd integers: "Find two consecutive even integers whose squares differ by 68.; Lesson 438 — Number and Consecutive Integer Problems
Consecutive integers: "Find two consecutive integers whose product is 132.; Lesson 438 — Number and Consecutive Integer Problems
Consider absolute values: when dealing with even roots and variables; Lesson 504 — Combining Simplification Techniques
Consider the context: (Do you paint all faces?; Lesson 895 — Word Problems Involving Surface Area
Consider the problem context: For rotations, use a basis aligned with the rotation axis.; Lesson 2257 — Finding Convenient Bases
Consistency: It's a standard form, making it easier to compare answers; Lesson 514 — Why Rationalize Denominators?Lesson 2999 — Properties of Good Estimators
Consistent: If no such contradictory row exists, the system is consistent and has at least one solution.; Lesson 2111 — Consistent and Inconsistent Systems Lesson 2999 — Properties of Good Estimators Lesson 3000 — Sample Mean and Variance as Estimators
constant: is a term with no variable—just a plain number that never changes.; Lesson 171 — Terms, Coefficients, and Constants Lesson 312 — Terms, Coefficients, and Constants Lesson 1692 — Work Done by a Constant Force
Constant base: 2^x → use d/dx[a^x] = a^x ln(a); Lesson 1462 — Variable Bases and Variable Exponents
Constant coefficients: `P(x)` and `Q(x)` are both constants (like `y'' + 3y' - 2y = 0`); Lesson 2337 — Form and Classification of Second-Order Linear ODEs Lesson 2349 — The Method of Undetermined Coefficients Overview
Constant density: If ρ is constant, mass = ρ × (area or volume), and the integral simplifies nicely.; Lesson 1954 — Mass and Density Functions
Constant exponent: x³ → use the power rule; Lesson 1462 — Variable Bases and Variable Exponents
Constant Factor Rule: Lesson 672 — Properties of Summation: Linearity
Constant Multiple Law: works the same way with limits:; Lesson 1337 — The Constant Multiple Law
Constant Multiple Rule: and the **Sum/Difference Rules** you learned earlier for polynomials.; Lesson 1424 — Derivatives of Trig Functions with Coefficients and Sums Lesson 1559 — Properties of Definite Integrals: Linearity Lesson 1708 — Limit Laws for Sequences
constant of variation: (a fixed number that never changes); Lesson 124 — Direct Variation Lesson 125 — Inverse Variation Lesson 128 — Finding the Constant of Variation
Constraint: The perimeter must equal 200 meters; Lesson 1505 — Introduction to Optimization
Constraint Boundary Analysis: Lesson 1880 — Identifying Maxima and Minima from Critical Points
Constraints: A system of linear inequalities that represent your limitations or requirements.; Lesson 304 — Introduction to Linear Programming
Construct a set: Define *S* as the set of positive integers for which the statement fails; Lesson 2518 — Proofs Using Well-Ordering
Construct geometric figures: Given three vertices of a rectangle, find the fourth by writing parallel and perpendicular line equations; Lesson 272 — Systems and Geometric Reasoning with Parallel/Perpendicular Lines
Construct the solution: x ≡ a₁M₁y₁ + a₂M₂y₂ + .; Lesson 2755 — The CRT Construction Algorithm
Construction and Carpentry: Builders use the 3-4-5 triple (or its multiples like 6-8-10) to ensure corners are perfectly square.; Lesson 790 — Applications of Triples and the Converse
Contains all vertices: from the original graph; Lesson 2672 — Spanning Trees of a Graph
Contains the zero vector: Since U and V are both subspaces, each contains **0**.; Lesson 2083 — Intersection of Subspaces
Contains zero vector: Since ⟨**0**, w ⟩ = 0 for all w ∈ W, we have **0** ∈ W ⊥.; Lesson 2215 — Orthogonal Complements
Context matters: ρ = 0.; Lesson 2956 — Examples and Applications of Correlation
contingency table: ) displays counts or frequencies for combinations of two events.; Lesson 2793 — Computing Conditional Probabilities from Tables Lesson 3044 — Chi-Squared Test for Independence
Continue: until you have a linearly independent set that still spans the space; Lesson 2089 — Finding a Basis from a Spanning Set Lesson 2119 — Converting to Row Echelon Form Lesson 2673 — Finding Spanning Trees: Depth-First Search Lesson 2674 — Finding Spanning Trees: Breadth-First Search
Continue multiplying: by the derivative of each successive inner layer until you reach the innermost basic function; Lesson 1414 — Multiple Compositions
Continue step-by-step: – Each new statement follows logically from earlier ones; Lesson 768 — Direct Proof Techniques
Continue the chain: until you reach the final answer; Lesson 781 — Multi-Step Problems with Multiple Triangles
Continue up: Does 3 work?; Lesson 48 — Understanding Factors
continuity: , though we haven't named it yet.; Lesson 1324 — Limits at Points Where Functions Are Defined Lesson 1488 — Verifying the Hypotheses of the MVT Lesson 2486 — Nested Quantifiers in Mathematics
continuity correction: fixes this mismatch by treating each discrete value as if it occupies an interval.; Lesson 2914 — Continuity Correction Lesson 2983 — The Continuity Correction Lesson 2986 — Rule of Thumb for CLT Applicability
Continuity holds: within an interval containing your initial condition; Lesson 2312 — Handling Absolute Values and Domains
Continuity is essential: A function with a jump or break might not have a maximum or minimum (think of a graph with a hole right where the peak should be).; Lesson 1358 — The Extreme Value Theorem
Continuity of ∂f/∂y: (called the *Lipschitz condition* in stronger forms) prevents solution curves from crossing or branching.; Lesson 2306 — Existence and Uniqueness Theorems
Continuity of f: ensures the direction field doesn't have wild jumps—the solution can "flow" smoothly.; Lesson 2306 — Existence and Uniqueness Theorems
continuous: at that point.; Lesson 576 — Continuity of Functions Lesson 593 — Continuity of Piecewise Functions Lesson 1356 — The Intermediate Value Theorem Lesson 1358 — The Extreme Value Theorem Lesson 1486 — Statement and Geometric Interpretation of the MVT Lesson 1488 — Verifying the Hypotheses of the MVT Lesson 1506 — The Extreme Value Theorem Lesson 1731 — The Integral Test: Statement and Conditions (+9 more)
Continuous case: Lesson 2943 — Independence of More Than Two Random Variables
Continuous curves: If a parabola or line extends infinitely in both directions, the domain is "all real numbers" (−∞, ∞).; Lesson 552 — Domain from Graphs
Continuous portions: appear as **smooth, increasing segments** between the jumps; Lesson 2850 — Mixed Random Variables and Their CDFs
Continuous probability density: across intervals (like a PDF); Lesson 2840 — Mixed Random Variables
continuous random variable: is one that can take on *any value* within an interval or collection of intervals on the real number line.; Lesson 2825 — Continuous Random Variables: Definition and Examples Lesson 2827 — Distinguishing Discrete from Continuous Lesson 2843 — Definition and Interpretation of the CDF Lesson 2848 — Recovering PMF and PDF from the CDF
contour map: is a collection of several level curves for different k values, displayed together on the xy-plane.; Lesson 1800 — Level Curves and Contour Maps Lesson 1805 — Level Curves and Contour Maps
contradiction: .; Lesson 206 — Recognizing Identity and Contradiction Equations Lesson 212 — Equations with No Solution Lesson 297 — Inconsistent Three-Variable Systems Lesson 2478 — Tautologies, Contradictions, and Contingencies Lesson 2499 — Classic Contradiction Proofs
Contradiction (no solution): When simplifying leads to a statement that's *never true*, like `5 = 7` or `0 = 3`, you have a **contradiction**.; Lesson 206 — Recognizing Identity and Contradiction Equations
Contradiction approach: Lesson 2497 — Proof by Contradiction: Basic Idea
contrapositive: is "if not Q then not P" (symbolically: if ¬Q then ¬P).; Lesson 2494 — Contrapositive: Logical Equivalence Lesson 2496 — Contrapositive Proof Examples
Contrapositive approach: Prove "If n is not even (odd), then n² is not even (odd).; Lesson 2495 — Proof by Contrapositive: Method and Strategy Lesson 2496 — Contrapositive Proof Examples
control: how precise your estimate is.; Lesson 3013 — Choosing Sample Size for Desired Margin of Error Lesson 3102 — ANOVA Applications and Experimental Design
Convergence: means the limit exists and equals a finite value.; Lesson 1637 — Convergence and Divergence of Improper Integrals Lesson 1704 — Visualizing Sequences: Terms and Graphs Lesson 1715 — Convergence and Divergence of Series Lesson 2922 — Properties and Applications of Chi-Squared
Convergence in distribution: (also called *convergence in law*) means that as n grows, the cumulative distribution function (CDF) of X gets closer and closer to the CDF of some limiting random variable X.; Lesson 2980 — Convergence in Distribution
Convergence in probability: formalizes this intuition: a sequence of random variables X₁, X₂, X₃, .; Lesson 2971 — Convergence in Probability Lesson 2974 — Almost Sure Convergence vs. Convergence in Probability
Convergence rule: The new series converges at least on the **smaller** of the two original intervals.; Lesson 1769 — Operations on Power Series
Convergence Test: Geometric series test the ratio; p-series test the exponent.; Lesson 1728 — Comparing Geometric and p-Series
Convergent: (like a geometric series with |r| < 1, a p-series with p > 1, or another known convergent series); Lesson 1736 — Direct Comparison Test: Proving Convergence
converges: (settles toward a specific number) or **diverges** (grows without limit or oscillates forever).; Lesson 667 — Convergence and Divergence of Geometric Sequences Lesson 679 — Infinite Series: Convergence vs Divergence Lesson 683 — Infinite Geometric Series: Convergence Condition Lesson 1634 — Type 1 Improper Integrals: Infinite Upper Limit Lesson 1635 — Type 1 Improper Integrals: Infinite Lower Limit Lesson 1638 — Type 2 Improper Integrals: Discontinuity at an Endpoint Lesson 1639 — Type 2 Improper Integrals: Interior Discontinuity Lesson 1714 — Sequence of Partial Sums (+5 more)
converges absolutely: Lesson 1742 — The Ratio Test: Statement and Logic Lesson 1760 — Examples and Applications of Alternating Series
converges in probability: to μ, written X̄ →ᵖ μ.; Lesson 2970 — Weak Law of Large Numbers: Statement Lesson 2971 — Convergence in Probability
Converges to 0: when `-1 < r < 1` (equivalently, `|r| < 1`); Lesson 667 — Convergence and Divergence of Geometric Sequences
converges to L: if:; Lesson 1706 — Formal Definition: Convergent Sequences Lesson 1715 — Convergence and Divergence of Series
converging: (coming together) — there's a sink at that point; Lesson 2004 — The Divergence of a Vector Field Lesson 2006 — Interpreting Divergence Physically
converse: flips this logic: if you *start* with two congruent angles in any triangle, you can prove the triangle *must be* isosceles.; Lesson 735 — Converse of Isosceles Triangle Theorem Lesson 787 — The Converse of the Pythagorean Theorem
Conversions and Travel: Currency exchange, speed calculations, or distance on maps.; Lesson 123 — Real-World Proportion Problems
Convert: each mixed number to an improper fraction; Lesson 76 — Multiplying Mixed Numbers Lesson 99 — Finding What Percent One Number Is of Another
Convert back: to a mixed number if needed; Lesson 76 — Multiplying Mixed Numbers Lesson 1613 — Substitution for √(a²-x²): Using x = a sin θ
Convert to equivalent fractions: Lesson 73 — Subtracting Fractions with Unlike Denominators
Convert to exponential form: using the definition of logarithm; Lesson 636 — Solving Logarithmic Equations with Multiple Logs
Convert to percent: Multiply by 100 (or move the decimal two places right); Lesson 101 — Percent Increase and Decrease
Convert to polar form: Write z = r(cos θ + i sin θ); Lesson 1226 — Using De Moivre's Theorem to Compute Powers
Convert to unit rates: (miles per hour):; Lesson 111 — Comparing Rates to Make Decisions
convolution: of $f(t)$ and $g(t)$:; Lesson 2395 — Transforms of Integrals and Convolution Lesson 2641 — Operations on Generating Functions: Multiplication Lesson 2957 — Sums of Independent Random Variables Lesson 2959 — Convolution Formula for Continuous RVs
convolution formula: gives you the answer.; Lesson 2958 — Convolution Formula for Discrete RVs Lesson 2959 — Convolution Formula for Continuous RVs
Cook's Distance: quantifies this influence by measuring how much all fitted values would shift if you deleted a single observation and re-ran the regression.; Lesson 3088 — Cook's Distance and Influence Measures Lesson 3089 — Outliers in Regression
Cooling towers: use hyperbolic shapes because the curve provides maximum strength with minimum material.; Lesson 1187 — Real-World Conic Applications Review
Coordinate calculations become trivial: If **B** = {**u**₁, **u**₂, .; Lesson 2212 — Orthonormal Bases
Coordinate systems: Points on a torus can be viewed as ` ℝ²` modulo a lattice relation; Lesson 2558 — Applications and Quotient Sets
Coordinates: (√3/2, 1/2); Lesson 994 — Reference Angles and the First Quadrant Lesson 1004 — Trigonometric Functions in Each Quadrant Lesson 2046 — Vectors as Ordered Lists of Numbers Lesson 2094 — Coordinates Relative to a Basis
Coordinates at 45°: (√2/2, √2/2); Lesson 994 — Reference Angles and the First Quadrant
Coplanar: All points on the *same plane* (2-dimensional constraint); Lesson 698 — Collinear and Coplanar Points Lesson 699 — Planes: Flat Surfaces Extending Infinitely
Coplanar points: are points that all lie on the same flat plane (like a flat sheet of paper extending infinitely).; Lesson 698 — Collinear and Coplanar Points
Coprimality: It must be relatively prime to φ(n), meaning gcd(e, φ(n)) = 1; Lesson 2764 — Choosing the Public Exponent e
coprime: Lesson 2711 — Greatest Common Divisor (GCD): Definition Lesson 2718 — Relatively Prime Integers
Core idea: Before eliminating entries in a column, scan downward from your current pivot position.; Lesson 2113 — Partial Pivoting Strategy
Core strategy: If someone is *x* years old now, they were *(x - 5)* five years ago and will be *(x + 5)* five years from now.; Lesson 192 — Age Problems and Consecutive Integer Problems
Corollary: All inscribed angles that intercept the same arc are **congruent** to each other—they all equal half that arc's measure, no matter where the vertex sits on the circle.; Lesson 838 — Inscribed Angles
Correct orientation: when bounds appear in the "wrong" order; Lesson 1560 — Properties of Definite Integrals: Intervals
Correlation: is a single number (typically Pearson's *r*, ranging from -1 to +1) that quantifies *how strongly* two variables move together.; Lesson 3066 — Correlation vs Regression
Corresponding angles: are pairs of angles that sit in the **same relative position** at each intersection.; Lesson 714 — Corresponding Angles Lesson 718 — Using Angle Relationships to Find Measures Lesson 721 — Applications of Parallel Line Properties
cos(x): Lesson 1014 — Period of Sine and Cosine Functions Lesson 1421 — Derivatives of sin(x) and cos(x): Pattern Recognition
Cosecant: Asymptotes wherever sine equals zero: **x = nπ**; Lesson 1032 — Comparing and Analyzing All Six Trig Functions
Cosecant (csc): is the reciprocal of sine: `csc(θ) = 1/sin(θ)`; Lesson 1009 — Reciprocal Trigonometric Functions: Secant, Cosecant, Cotangent
Cosine: = adjacent side ÷ hypotenuse; Lesson 947 — Introduction to Trigonometric Ratios Lesson 950 — The Cosine Ratio (CAH)Lesson 997 — Fourth Quadrant Points Lesson 1427 — Higher-Order Derivatives of Trigonometric Functions
Cosine (CAH): Use when working with **adjacent** and **hypotenuse**; Lesson 956 — Choosing the Appropriate Trig Ratio
Cosine is Chill: " (even—no sign change), while "**Sine is Sneaky**" (odd—it flips sign).; Lesson 1048 — Even-Odd Identities
Cosine series: Extend evenly, compute only `a ` coefficients.; Lesson 2462 — Half-Range Expansions
Cost Analysis: If C(x) is the cost of producing x items:; Lesson 1504 — Applications: Real-World Monotonicity and Concavity
Cost function: If `C(n) = 15 + 3n` represents the cost in dollars to produce n items:; Lesson 546 — Function Evaluation in Context
Cost rate: How much does producing one more unit cost at this exact production level?; Lesson 1376 — Applications: Velocity and Other Rates
Cotangent: Vertical asymptotes at **x = nπ** (multiples of π); Lesson 1032 — Comparing and Analyzing All Six Trig Functions
Cotangent (cot): is the reciprocal of tangent: `cot(θ) = 1/tan(θ)`; Lesson 1009 — Reciprocal Trigonometric Functions: Secant, Cosecant, Cotangent
Coterminal angles: are angles that end up pointing in the same direction—they share the same terminal side.; Lesson 990 — Coterminal Angles in Radians Lesson 998 — Coterminal Angles on the Unit Circle Lesson 1006 — Coterminal Angles and Periodicity
Count: arrangements of this block with the remaining objects; Lesson 2583 — Permutations with Restrictions: Adjacent Objects
Count carefully: Show that you have more pigeons than holes; Lesson 2575 — Pigeonhole Principle in Geometry
Count combinatorial objects: by manipulating series; Lesson 2638 — What Is a Generating Function?
Count full squares: Tally all unit squares that lie *completely* inside the shape.; Lesson 872 — Irregular Composite Figures on Grids
Count how many places: you need to move the decimal point to make it a whole number; Lesson 91 — Dividing by Decimals
Count partial squares: Identify squares that are *partially* covered by the shape.; Lesson 872 — Irregular Composite Figures on Grids
Count the remaining positions: and objects; Lesson 2582 — Permutations with Restrictions: Fixed Positions
Count the rise: Starting from the first point, count how many units you move vertically to reach the height of the second point; Lesson 248 — Finding Slope from a Graph
Count the run: Count how many units you move horizontally (left to right) between the points; Lesson 248 — Finding Slope from a Graph
Count the signs: How many differences are positive?; Lesson 3104 — Sign Test for Median
Count the terms: – Four terms often suggest grouping; Lesson 381 — Applications and Mixed Practice
Count the transversals: crossing them; Lesson 720 — Parallel Lines Cut by Multiple Transversals
Count them: if you can gather **n** linearly independent eigenvectors total across all eigenspaces, you're done —the matrix is diagonalizable; Lesson 2190 — The Diagonalization Theorem
Count your negatives: An even number of negatives makes a positive result; an odd number keeps it negative.; Lesson 457 — Multiplying and Dividing with Negative Signs
Count your terms: Product rule should give you exactly 2 terms.; Lesson 1407 — Common Errors and Problem-Solving Strategies
Countable outcomes: The set of possible values is finite or countably infinite (like {1, 2, 3, .; Lesson 2823 — Discrete Random Variables: Definition and Examples
Counter-example: Can sides 2, 3, and 6 form a triangle?; Lesson 737 — Triangle Inequality Theorem Lesson 2804 — Testing for Independence
Counter-rotation: Subtract the arguments; Lesson 1224 — Geometric Interpretation of Polar Operations
counterclockwise: starting from the upper-right:; Lesson 238 — The Four Quadrants Lesson 1007 — Negative Angles and Trigonometric Values Lesson 1210 — Argument of a Complex Number Lesson 1994 — Statement of Green's Theorem Lesson 2027 — Orientation of Surfaces and Boundary Curves
Counterexample: Suppose you have sides of length 5 and 3, with a 30° angle opposite the side of length 3.; Lesson 746 — Why AAA and SSA Don't Work
Counterexample for Non-Surjectivity: Lesson 2563 — Testing for Surjectivity
Counting: Summing indicators counts how many events occur in a collection.; Lesson 2830 — Indicator Random Variables
counting problems: .; Lesson 2607 — Applications: Probability and Counting Lesson 2790 — Uniform Probability on Finite Sample Spaces
Course: (or **ground velocity**, **actual path**): The *resultant* vector—where the vehicle actually ends up going; Lesson 1266 — Navigation Problems: Heading and Course
Cov(X, c) = 0: for any constant *c*; Lesson 2949 — Properties of Covariance
Cov(X, X) = Var(X): Lesson 2949 — Properties of Covariance
covariance matrix: describing correlations between variables—this matrix is symmetric.; Lesson 2279 — Spectral Theorem in Quantum Mechanics and Statistics Lesson 2299 — Covariance Matrices and Applications Lesson 2955 — Covariance Matrix
Covering a cylindrical tank: Find the surface area of the cylinder—both circular bases plus the curved lateral surface—to determine how much sheet metal you need.; Lesson 895 — Word Problems Involving Surface Area
CPCTC: (Corresponding Parts of Congruent Triangles are Congruent), ∠B ≅ ∠C.; Lesson 750 — Congruence in Isosceles and Equilateral Triangles
Cramer's Rule: gives you a direct formula using determinants.; Lesson 1299 — Cramer's Rule for Solving Systems
Create a grid: of (x, y) points in your region of interest; Lesson 2305 — Direction Fields
Create zeros below: that leading entry using row operations (add multiples of the pivot row to rows below); Lesson 2119 — Converting to Row Echelon Form
Credit cards: Understand minimum payment traps; Lesson 689 — Loan Amortization
Critical: Always translate your decision back to the actual question.; Lesson 3050 — Interpreting Chi-Squared and F Test Results
Critical condition: *a* has a multiplicative inverse modulo *n* **if and only if gcd(a, n) = 1** (they're relatively prime).; Lesson 2735 — Division and Modular Inverses
Critical detail: Parentheses matter tremendously!; Lesson 138 — Exponents with Negative Bases
Critical mistake to avoid: Never say "we accept H₀" or "we proved H₀ is true.; Lesson 3030 — Interpreting Hypothesis Test Results
critical point: occurs where the derivative equals zero (horizontal tangent) or doesn't exist.; Lesson 1507 — Critical Points in Optimization Lesson 1520 — Critical Points and Local Extrema Lesson 1862 — Critical Points in Two Variables
critical points: potential peaks, valleys, or flat spots where the function's behavior might change.; Lesson 1495 — Finding Intervals of Increase and Decrease Lesson 1503 — Analyzing Function Behavior with Derivatives Lesson 1506 — The Extreme Value Theorem Lesson 1507 — Critical Points in Optimization Lesson 1519 — First Derivative and Intervals of Increase/Decrease Lesson 2298 — Applications to Optimization and Hessian Matrices
Critical reminder: Be careful with the subtraction sign!; Lesson 464 — Subtracting Rationals with Unlike Denominators
Critical requirement: Cramer's Rule only works when det(A) ≠ 0, meaning the system has exactly one unique solution.; Lesson 1317 — Cramer's Rule
critical value: is a specific percentile used in statistical inference—often the boundary values for decision regions.; Lesson 2910 — Finding Percentiles and Critical Values Lesson 3035 — Calculating the t-Statistic and Finding Critical Values
Critical value approach: Reject H₀ if z > z_α (the value where P(Z > z_α) = α); Lesson 3027 — One-Tailed z-Tests
Critically Damped: (Δ = 0, repeated real root); Lesson 2347 — Applications: Free Oscillations and Damped Motion
Cross through twice: (two different triangles!; Lesson 1096 — The SSA Configuration Introduction
Cross twice: intersect the opposite side at two different points (two triangles!; Lesson 1121 — Introduction to the Ambiguous Case (SSA)
cross-multiplication: to create a linear equation.; Lesson 215 — Ratios and Proportions Leading to Linear Equations Lesson 1885 — Solving Systems with Multiple Lagrange Multipliers
Cross-sections: slicing the surface with planes gives you curves (like the level curves you learned, but vertical slices too); Lesson 1806 — Graphing Surfaces in 3D Lesson 1929 — Triple Integrals over General 3D Regions
CRT Construction Algorithm: gives us the explicit formula.; Lesson 2755 — The CRT Construction Algorithm
Crust length: (arc length): How much crust that slice has; Lesson 846 — Applications of Arcs and Sectors
Cryptography: Many encryption methods require finding numbers with specific remainders under different moduli; Lesson 2752 — Systems of Linear Congruences
Cube both sides: to eliminate the cube root:; Lesson 528 — Radical Equations with Higher Index Roots
Cube roots: can handle negative numbers: ∛(−8) = −2 because (−2) × (−2) × (−2) = −8; Lesson 153 — Cube Roots and Higher-Order Roots Lesson 157 — Simplifying Higher-Order Radicals
Cube roots (n=3): Three points forming an equilateral triangle, 120° apart; Lesson 1230 — Geometric Interpretation of nth Roots
cumulative distribution function: (or CDF) of a random variable X is a function, typically denoted F(x) or F_X(x), that gives the probability that X takes on a value less than or equal to x:; Lesson 2843 — Definition and Interpretation of the CDF Lesson 2849 — Percentiles and Quantiles from the CDF
curl: F** = ⟨R_y - Q_z, P_z - R_x, Q_x - P_y ⟩**; Lesson 1987 — Testing for Conservative Fields: The Curl Test in ℝ³ Lesson 2007 — The Curl of a Vector Field
curl F: = **i**(∂R/∂y - ∂Q/∂z) - **j**(∂R/∂x - ∂P/∂z) + **k**(∂Q/∂x - ∂P/∂y); Lesson 2008 — Computing Curl Using the Determinant Formula Lesson 2009 — Interpreting Curl Physically Lesson 2011 — Irrotational Fields and Conservative Vector Fields
Curl product rule: Lesson 2010 — Properties of Divergence and Curl
current value: , not the original—successive changes don't simply add up!; Lesson 104 — Multi-Step Percentage Problems Lesson 613 — The Derivative Property: Why e is Special
Curved pattern: (like a U-shape or inverted-U): Suggests the relationship isn't linear.; Lesson 3084 — Residual Plots: Fitted Values vs. Residuals Lesson 3086 — Normal Probability Plots (Q-Q Plots)
Curves with asymptotes: may exclude certain y-values; Lesson 554 — Range from Graphs
Customer arrivals: at a bank, store, or website (e.; Lesson 2893 — Applications of the Poisson Distribution
cut edges: the critical points and links whose removal disconnects a graph—and understand why they matter for graph robustness.; Lesson 2661 — Cut Vertices and Cut Edges Lesson 2667 — Applications of Connectivity
cut vertices: and **cut edges**—the critical points and links whose removal disconnects a graph—and understand why they matter for graph robustness.; Lesson 2661 — Cut Vertices and Cut Edges Lesson 2667 — Applications of Connectivity
Cycle (Simple Circuit): Lesson 2658 — Circuits and Cycles
Cycles and trees: Any cycle graph (C_n) and any tree are planar because their simple structures allow flat layouts.; Lesson 2689 — Planar Graphs and the Definition of Planarity
Cycles or waves: Residuals oscillate in regular patterns; Lesson 3090 — Checking for Independence: Residual Sequence Plots
cylindrical: (r, θ, z), and **spherical** (ρ, θ, φ).; Lesson 1943 — Choosing the Best Coordinate System Lesson 1961 — Computing Moments of Inertia Lesson 2454 — Separation in Other Coordinate Systems

D

D (Vertical Shift): Moves the entire graph up or down D units, shifting the midline from y = 0 to y = D.; Lesson 1029 — Transformations of Secant
D < 0: shift **down** by |D| units; Lesson 1025 — Transformations of Tangent: Phase Shift and Vertical Shift Lesson 1868 — Inconclusive Cases and Further Analysis
D = 0: ?; Lesson 1868 — Inconclusive Cases and Further Analysis
D > 0: shift **up** by D units; Lesson 1025 — Transformations of Tangent: Phase Shift and Vertical Shift Lesson 1868 — Inconclusive Cases and Further Analysis
D ₓ: = determinant when you replace the x-column with the constants (e, f); Lesson 1299 — Cramer's Rule for Solving Systems
D ᵧ: = determinant when you replace the y-column with the constants; Lesson 1299 — Cramer's Rule for Solving Systems
D_u f: is the directional derivative in direction **u**; Lesson 1844 — Computing Directional Derivatives Using the Gradient
D, E: Control the linear terms (x and y); Lesson 1131 — The General Second-Degree Equation
D(0) = 1: (by convention); Lesson 2586 — Derangements
D(1) = 0: (one object must stay in place); Lesson 2586 — Derangements
D(n): or **!; Lesson 2586 — Derangements Lesson 2622 — Derangements and the Derangement Formula
D^n: is just diagonal entries raised to the nth power (λ ᵢ^n), you immediately see:; Lesson 2199 — Applications of Diagonalization
Data analysis: Extract dominant patterns from noisy datasets; Lesson 2288 — Low-Rank Approximation and Data Compression
Data structures: A binary search tree with 100 nodes contains exactly 99 parent-child links.; Lesson 2669 — Counting Vertices and Edges in Trees
Data violates parametric assumptions: even after transformations; Lesson 3111 — Choosing and Interpreting Nonparametric Tests
Days of the week: The pattern Monday-Tuesday-.; Lesson 570 — Periodic Functions
De Moivre's Theorem states: For any real number θ and any positive integer n; Lesson 1225 — De Moivre's Theorem: Statement and Proof
De Morgan's Law: Is ¬(p ∧ q) equivalent to (¬p) ∨ (¬q)?; Lesson 2475 — Logical Equivalence
decay: (things getting smaller).; Lesson 666 — Growth and Decay Applications Lesson 690 — Population Growth and Decay Models Lesson 1437 — Applications: Exponential Growth and Decay Rates Lesson 2316 — Applications: Exponential Growth and Decay
Decay rate: `N'(t) = N₀ · (-k) · e^(-kt) = -k · N(t)`; Lesson 1437 — Applications: Exponential Growth and Decay Rates
Decide: reject or fail to reject the null based on your significance level; Lesson 3034 — One-Sample t-Test: Testing a Population Mean Lesson 3043 — Chi-Squared Goodness-of- Fit Test
Decide your approach: Lesson 1514 — Optimization with Implicit Constraints
Decision: Reject H₀ if F exceeds the critical value (or if p-value < α); Lesson 3048 — F Test for Equality of Variances
Decomposing problems: In electromagnetism, Maxwell's equations involve both divergence (Gauss's law) and curl (Faraday's and Ampère's laws).; Lesson 2045 — Combining with Stokes' Theorem and Applications
Decouple the system: Change coordinates using **y = P ¹x**, transforming **dx/dt = Ax** into **dy/dt = Dy**, where **D** is diagonal; Lesson 2278 — Applications to Differential Equations and Dynamics
decreasing: by 5 degrees each hour.; Lesson 247 — Slope as Rate of Change Lesson 571 — Monotonicity: Increasing and Decreasing Functions Lesson 1494 — Using the First Derivative to Determine Monotonicity Lesson 1495 — Finding Intervals of Increase and Decrease Lesson 1519 — First Derivative and Intervals of Increase/Decrease Lesson 1525 — Comprehensive Curve Sketching Algorithm Lesson 1731 — The Integral Test: Statement and Conditions Lesson 2869 — Functions of Random Variables: Continuous Case
Decreasing (or non-increasing): Each term is less than or equal to the previous one: a₁ ≥ a₂ ≥ a₃ ≥ .; Lesson 1710 — Monotonic Sequences and Boundedness
Decreasing intervals: Where f'(x) < 0; Lesson 1503 — Analyzing Function Behavior with Derivatives
Decreasing terms: The sequence {b } is eventually decreasing; Lesson 1753 — Verifying Conditions for the Alternating Series Test
Decrypt: Compute *(m^e)^d = m^(ed)*; Lesson 2768 — Why RSA Works: Euler's Theorem Lesson 2769 — RSA Example with Small Primes
Define "bad" properties: (things you want to exclude); Lesson 2623 — The Sieve Method in Practice
Define the generating function: G(x) = a₀ + a₁x + a₂x² + .; Lesson 2643 — Solving Recurrence Relations with Generating Functions
Define the pigeonholes: These might be regions of space, distance categories, or angle ranges; Lesson 2575 — Pigeonhole Principle in Geometry
Define variables: Let *x* = number of cookies, *y* = number of brownies; Lesson 308 — Applications: Maximization Problems Lesson 1516 — Applied Optimization Strategy
definite integral: .; Lesson 1553 — Riemann Sums as n Approaches Infinity Lesson 1555 — The Limit Definition of the Definite Integral Lesson 1558 — Notation and Components of the Definite Integral Lesson 1586 — Definite Integrals and Changing Limits Lesson 1599 — Definite Integrals and Integration by Parts
Definiteness: Lesson 2200 — Definition of Inner Product Spaces Lesson 2202 — Inner Products on Function Spaces Lesson 2292 — Positive Definite, Negative Definite, and Indefinite Matrices
Definition: Lesson 1634 — Type 1 Improper Integrals: Infinite Upper Limit Lesson 2939 — Independence via Joint and Marginal Distributions
Definitions: Lesson 766 — Common Reasons in Proofs
degree: tells you the algebraic power to which that highest derivative is raised (once the equation is polynomial in derivatives).; Lesson 2301 — Order and Degree of ODEs Lesson 2647 — What is a Graph? Vertices and Edges Lesson 2650 — Degree of a Vertex and the Handshaking Lemma Lesson 2651 — Adjacency and Incidence
Degree 0: Constant: Example: `5` or `-12`; Lesson 315 — Classifying Polynomials by Degree
Degree 1: Linear: Example: `3x + 7`; Lesson 315 — Classifying Polynomials by Degree
Degree 2: Quadratic: Example: `x² - 4x + 1`; Lesson 315 — Classifying Polynomials by Degree
Degree 3: Cubic: Example: `2x³ + x - 5`; Lesson 315 — Classifying Polynomials by Degree
Degree 4: Quartic: Example: `x⁴ - 3x² + 2`; Lesson 315 — Classifying Polynomials by Degree
Degree 5: Quintic: Example: `x⁵ + 2x³ - x + 9`; Lesson 315 — Classifying Polynomials by Degree
degrees: (symbolized as °).; Lesson 703 — Measuring Angles and Angle Notation Lesson 833 — Central Angles and Their Measures Lesson 837 — Arc Length Formula Lesson 842 — Sector Area Formula Lesson 983 — What is a Radian?
degrees of freedom: , denoted by ν (the Greek letter "nu").; Lesson 2921 — The Chi-Squared Distribution: Definition Lesson 2923 — The Student's t-Distribution: Definition Lesson 3033 — The t-Distribution: Properties and Degrees of Freedom
Degrees of freedom (df): df = n - 1; Lesson 3035 — Calculating the t-Statistic and Finding Critical Values
Denoising: Drop singular values below a threshold that captures noise level; Lesson 2290 — Applications of SVD: Image Compression and Noise Reduction
denominator: (bottom number) tells you *how many equal parts the whole is divided into*; Lesson 59 — What is a Fraction?Lesson 442 — What is a Rational Expression?Lesson 531 — Converting Between Radical and Exponential Form Lesson 532 — Evaluating Expressions with Fractional Exponents Lesson 1222 — Dividing Complex Numbers in Polar Form Lesson 1338 — Combining Multiple Limit Laws Lesson 2793 — Computing Conditional Probabilities from Tables Lesson 3096 — The F-Statistic and F- Distribution
Denominator expressions: 1/(x² + 1) → u = x² + 1; Lesson 1580 — Identifying the Inner Function u
Denominators: 2, 3, 6; Lesson 209 — Equations with Fractions: Clearing Denominators Lesson 480 — Solving Equations with Multiple Terms
dependent: .; Lesson 2067 — Testing Linear Independence in R² Lesson 2068 — Testing Linear Independence in R³ Lesson 2069 — The Matrix Test for Linear Independence
Depreciation: A machine worth $10,000 loses $500 in value each year.; Lesson 659 — Applications of Arithmetic Sequences Lesson 666 — Growth and Decay Applications Lesson 691 — Depreciation and Asset Value
Depth-first search: is a natural algorithm for constructing one: it explores as far as possible along each branch before backtracking, adding edges as it discovers new vertices.; Lesson 2673 — Finding Spanning Trees: Depth-First Search
derangement: is a permutation of objects where no object appears in its original position.; Lesson 2586 — Derangements Lesson 2622 — Derangements and the Derangement Formula
Derivation: For a ≤ x ≤ b, integrate the PDF from a to x:; Lesson 2895 — PDF and CDF of the Uniform Distribution
derivative: of a function `f` at a point `x` is defined as:; Lesson 1380 — The Limit Definition of the Derivative Lesson 1473 — Applications of Differentials in Measurement Lesson 2438 — Boundary Value Problems for Laplace's Equation Lesson 2848 — Recovering PMF and PDF from the CDF
Derivative of inside: 6x; Lesson 1411 — The Outside-Inside Method
Derivative of outside: (using Power Rule): 7u⁶, but keep u = 3x² + 5 → 7(3x² + 5)⁶; Lesson 1411 — The Outside-Inside Method
Derive a contradiction: Use properties of *m* being minimal to construct a smaller element in *S*, contradicting minimality; Lesson 2518 — Proofs Using Well-Ordering
Deriving distributions: You can analyze transformations without worrying about induced dependencies; Lesson 2946 — Independence and Functions of Random Variables
Describe the projection: using Type I or Type II region limits (from double integrals); Lesson 1929 — Triple Integrals over General 3D Regions
Description: "Take f(x) and shift it left 4 units, compress it vertically by 1/2, then shift down 3.; Lesson 585 — Writing Transformed Functions from Descriptions
Design: A logo might combine a square and a triangle; Lesson 873 — Composite Figures in Real-World Contexts
det(A): or **|A|** and is calculated as:; Lesson 1290 — What is a Determinant?Lesson 2175 — Properties of Eigenvalues: Trace and Determinant
det(A) < 0: , the transformation flips orientation (like a mirror reflection); Lesson 1290 — What is a Determinant?Lesson 2149 — Determinant of a 2×2 Matrix
det(A) = 0: , the matrix squashes space flat—vectors become dependent, and the matrix has no inverse; Lesson 1290 — What is a Determinant?Lesson 1297 — Using Determinants to Test Invertibility Lesson 1301 — When Does an Inverse Exist?Lesson 2149 — Determinant of a 2×2 Matrix Lesson 2163 — Determinants and Linear Independence
det(A) ≠ 0: , then A is invertible.; Lesson 1297 — Using Determinants to Test Invertibility Lesson 1301 — When Does an Inverse Exist?Lesson 2163 — Determinants and Linear Independence
determinant: is a single scalar value that summarizes key information about the matrix.; Lesson 1290 — What is a Determinant?Lesson 2008 — Computing Curl Using the Determinant Formula Lesson 2140 — Computing 2×2 Matrix Inverses Lesson 2143 — Testing Invertibility Lesson 2148 — What is a Determinant?Lesson 2165 — Determinant and Matrix Rank Lesson 2188 — Using the Characteristic Polynomial in Practice
Determine: the output formula in each region; Lesson 598 — Writing Functions as Piecewise Definitions Lesson 1698 — Applications of Average Value
Determine d(y): , how far that slice must be lifted; Lesson 1695 — Pumping Liquids from Tanks
Determine direction: – Are they looking above (elevation) or below (depression) the horizontal?; Lesson 967 — Identifying Angles in Real-World Scenarios
Determine how many terms: we need for a desired accuracy; Lesson 1790 — Applications to Common Functions
Determine intersection: Before solving algebraically, predict whether two lines will meet by comparing slopes; Lesson 272 — Systems and Geometric Reasoning with Parallel/Perpendicular Lines
Determine limits: by finding the bounds of your region; Lesson 1665 — Setting Up a Shell Integral Lesson 1673 — Applications and Problem-Solving Strategies
Determine orientation: Check which variable is squared.; Lesson 1151 — Graphing Parabolas from Standard Form
Determine the density function: ρ(x, y); Lesson 1955 — Computing Total Mass with Double Integrals
Determine the domain: – Is it an open region, or does it have boundaries you must check?; Lesson 1872 — Applications: Optimization Problems in Two Variables
Determine the sign: of f''(x) on different intervals:; Lesson 1521 — Second Derivative and Concavity
Determine the solving order: which triangle can you solve first?; Lesson 781 — Multi-Step Problems with Multiple Triangles
Determine the tolerance: (maximum allowed deviation); Lesson 235 — Applications and Mixed Absolute Value Problems
Diagonal: (best case—eigenvalue basis when the operator is diagonalizable); Lesson 2257 — Finding Convenient Bases
diagonal matrix: has non-zero entries only along its main diagonal (top-left to bottom-right), with zeros everywhere else.; Lesson 1295 — Determinants of Special Matrices Lesson 2157 — Determinant of Special Matrices
Diagonals: The outermost diagonals are all 1's.; Lesson 2598 — Pascal's Triangle: Construction and Patterns
Diagonals are congruent: This is unique to rectangles!; Lesson 818 — Rectangles: Properties and Characteristics
Diagonals bisect each other: (cut each other exactly in half); Lesson 816 — Parallelograms: Definition and Properties Lesson 817 — Proving Quadrilaterals Are Parallelograms Lesson 818 — Rectangles: Properties and Characteristics
diameter: is a line segment that passes through the center and connects two points on the circle.; Lesson 827 — Parts of a Circle: Center, Radius, Diameter, and Chord Lesson 829 — Area of a Circle Lesson 830 — Circles and Pi (π)Lesson 839 — Angles Inscribed in a Semicircle Lesson 850 — Chord Properties and Definitions Lesson 2660 — Distance and Diameter Lesson 2667 — Applications of Connectivity
Diameter Is Special: The diameter is the longest chord, and it divides the circle into two equal halves (semicircles).; Lesson 850 — Chord Properties and Definitions
difference: of two separate logarithms.; Lesson 622 — The Quotient Rule for Logarithms Lesson 1819 — Properties and Composition of Continuous Functions Lesson 2912 — Linear Combinations of Normal Variables
Difference of Cubes: Lesson 339 — Sum and Difference of Cubes Lesson 387 — Difference of Cubes Pattern Lesson 389 — Applying Cube Formulas with Coefficients
difference of squares: is one of the most elegant patterns in algebra.; Lesson 382 — Difference of Squares Pattern Lesson 386 — Sum of Squares (Why It Doesn't Factor)
difference of squares pattern: called that because you're subtracting (difference) one perfect square from another.; Lesson 336 — Difference of Squares Pattern Lesson 518 — Rationalizing with Sum or Difference Conjugates
difference quotient: Lesson 574 — Average Rate of Change Lesson 1379 — The Difference Quotient
Difference Rule: lim(a - b ) = L - M; Lesson 1708 — Limit Laws for Sequences
Differentiability: on the open interval (a, b); Lesson 1488 — Verifying the Hypotheses of the MVT
Differentiable: on the open interval (a, b); Lesson 1486 — Statement and Geometric Interpretation of the MVT Lesson 1488 — Verifying the Hypotheses of the MVT
differential equation: is an equation that involves one or more **derivatives** of an unknown function.; Lesson 2300 — What is a Differential Equation?Lesson 2304 — Initial Value Problems
Differential equations: Systems like dx/dt = Ax become decoupled when you change to eigenvector coordinates; Lesson 2258 — Applications: Diagonalization via Change of Basis
Differentials: give us a precise notation for the tiny changes involved in that approximation.; Lesson 1471 — Differentials and Notation
differentiate: .; Lesson 1566 — Definite Integrals with Variable Limits Lesson 1780 — Differentiation and Integration of Power Series
Differentiate and Solve: Lesson 1516 — Applied Optimization Strategy
Differentiate both sides: of the equation with respect to x; Lesson 1449 — The Basic Technique of Implicit Differentiation Lesson 1455 — Second Derivatives by Implicit Differentiation Lesson 1463 — Logarithmic Differentiation with Radicals Lesson 1478 — Related Rates with Rectangles and Squares Lesson 1479 — Related Rates with Triangles Lesson 1481 — Ladder and Shadow Problems Lesson 1482 — Distance and Position Related Rates Lesson 1484 — Water Tank and Fluid Flow Problems (+1 more)
Differentiate both sides implicitly: with respect to x:; Lesson 1435 — Derivatives of Exponential Expressions with Variable Bases Lesson 1459 — Logarithmic Differentiation: Basic Technique
Differentiate each term: in `∂z/∂x` with respect to `x` again; Lesson 1859 — Second-Order Derivatives via the Chain Rule
differentiate implicitly: with respect to *x*.; Lesson 1459 — Logarithmic Differentiation: Basic Technique Lesson 1462 — Variable Bases and Variable Exponents Lesson 1480 — Volume Related Rates: Spheres and Cones Lesson 1483 — Angle Related Rates
Differentiate term-by-term: y' = Σ(n=1 to ∞) na xⁿ ¹, y'' = Σ(n=2 to ∞) n(n-1)a xⁿ ²; Lesson 2400 — Finding Power Series Solutions Near Ordinary Points
Differentiate the outer layer: while keeping everything inside intact; Lesson 1414 — Multiple Compositions
Differentiate the outside: (leave the inside alone for now); Lesson 1411 — The Outside-Inside Method Lesson 1425 — Chain Rule with Trigonometric Functions Lesson 1444 — Chain Rule with Inverse Trigonometric Functions
Differentiate x-terms normally: (they're already in terms of x); Lesson 1452 — Implicit Differentiation with Circles and Ellipses
Differentiation: `f'(x) = Σ n·c (x - a)ⁿ ¹` (same radius of convergence); Lesson 1780 — Differentiation and Integration of Power Series
Dilate: about the origin using your scale factor; Lesson 936 — Dilations Centered at Points Other Than the Origin Lesson 938 — Combining Rotations and Dilations with Other Transformations
dilation: is a transformation that changes the size of a figure without changing its shape.; Lesson 934 — Introduction to Dilations and Scale Factor Lesson 935 — Dilations Centered at the Origin
dimension: .; Lesson 2046 — Vectors as Ordered Lists of Numbers Lesson 2086 — What is a Basis?Lesson 2091 — Dimension of a Vector Space Lesson 2092 — Dimension of Subspaces Lesson 2172 — Eigenspaces and Their Dimension
Dimension of image: `det(A) = 0` means the transformation represented by A reduces dimensions; Lesson 2165 — Determinant and Matrix Rank
dimensions: (or **order** or **size**) of a matrix tell us how many rows and columns it has.; Lesson 1271 — What is a Matrix? Dimensions and Notation Lesson 2097 — Matrix Notation and Dimensions
Dirac's Theorem: , which requires *every* vertex to have degree at least n/2, you might wonder if there's a more flexible condition.; Lesson 2686 — Ore's Theorem
Direct all existing edges: from U to V (capacity 1 each); Lesson 2707 — Applications of Network Flows to Matching Problems
Direct approach: Messy—count committees with just Alice, just Bob, or both.; Lesson 2597 — Complementary Counting with Combinations
Direct Comparison: = Quick and clean when the relationship is obvious; Lesson 1740 — Strategic Selection Between Comparison Tests Lesson 1750 — Strategy: Choosing the Right Convergence Test
Direct Comparison Test: works similarly with infinite series.; Lesson 1734 — The Direct Comparison Test: Basic Principle Lesson 1740 — Strategic Selection Between Comparison Tests Lesson 1741 — Mixed Practice: Integral and Comparison Tests
Direct Construction (Algebraic): Lesson 2563 — Testing for Surjectivity
Direct form: "If P, then Q"; Lesson 2489 — Structure of Mathematical Statements
Direct method: Parametrize the sphere (painful), compute normals, evaluate the surface integral.; Lesson 2041 — Simplifying Flux Calculations with Divergence
Direct projections: onto subspaces; Lesson 2211 — Orthogonal Bases
direct proof: starts with given information (facts you know are true) and uses logical steps—one after another— to reach the conclusion you're trying to prove.; Lesson 768 — Direct Proof Techniques Lesson 2491 — Direct Proof: Basic Structure
Direct proof attempt: Starting from "n² is even" and trying to extract information about n directly is awkward.; Lesson 2496 — Contrapositive Proof Examples
Direct solution methods: like Gaussian elimination (which you know!; Lesson 2147 — Computational Considerations and Singular Matrices
direct substitution: when the function is defined at that point.; Lesson 1347 — Exponential and Logarithmic Limits Lesson 1527 — Indeterminate Forms 0/0 and ∞/∞
Direct variation: describes a relationship between two variables where they change together in a predictable way: when one doubles, the other doubles; when one triples, the other triples.; Lesson 124 — Direct Variation Lesson 125 — Inverse Variation Lesson 127 — Combined Variation Lesson 128 — Finding the Constant of Variation Lesson 129 — Real-World Direct Variation Problems
Directed: The connection has a direction (like following someone on social media); Lesson 2647 — What is a Graph? Vertices and Edges
Directed edges: Pipes or channels connecting vertices; Lesson 2704 — Networks and Flows: Definitions and Notation
Directed edges (arrows): from a to b whenever (a,b) ∈ R; Lesson 2541 — Representing Relations: Matrices and Digraphs
Directed graphs: (or **digraphs**) change this by assigning a direction to each edge, creating one-way relationships.; Lesson 2649 — Directed Graphs (Digraphs)
direction: (positive = right, negative = left) and **distance** (the number itself) to move the decimal point.; Lesson 164 — Converting from Scientific Notation to Standard Form Lesson 423 — Vertex Form of a Quadratic Function Lesson 433 — Key Features Summary and Complete Graphing Lesson 930 — Introduction to Rotations in the Coordinate Plane Lesson 1235 — Introduction to Vectors: Magnitude and Direction Lesson 1965 — Evaluating Vector Fields at Points Lesson 2007 — The Curl of a Vector Field Lesson 2049 — Scalar Multiplication of Vectors (+1 more)
direction angle: is the angle θ (theta) that a vector makes with the positive x-axis, measured counterclockwise.; Lesson 1250 — Direction Angles and Components Lesson 1252 — Resolving Vectors into Components
Direction of maximum decrease: = opposite direction of ∇f (i.; Lesson 1846 — Maximum and Minimum Rates of Change
Direction of maximum increase: = direction of ∇f; Lesson 1846 — Maximum and Minimum Rates of Change
Directional derivatives: generalize this idea: they measure the rate of change of a function as you move in *any* direction from a point.; Lesson 1843 — Directional Derivatives: Motivation and Definition
directrix: ).; Lesson 1130 — Geometric Definitions of Conics Lesson 1146 — Standard Form: Vertex at Origin, Horizontal Axis Lesson 1147 — Finding the Focus and Directrix from Equations Lesson 1148 — Parabolas with Vertex at (h, k): Vertical Axis Lesson 1149 — Parabolas with Vertex at (h, k): Horizontal Axis
Dirichlet: Fixed temperature at the boundary (e.; Lesson 2417 — Initial and Boundary Conditions
Dirichlet (Type I): Specifies the function value itself; Lesson 2404 — Boundary Value Problems: Introduction and Types
Dirichlet conditions: guarantee success.; Lesson 2460 — Convergence of Fourier Series
discontinuous: at that point.; Lesson 1351 — Three Conditions for Continuity Lesson 1817 — Continuity at a Point
Discontinuous integrand: The function has a vertical asymptote or other discontinuity within or at the boundaries of the interval; Lesson 1633 — Introduction to Improper Integrals
Discover patterns: hidden in sequences; Lesson 2638 — What Is a Generating Function?
Discrete: Number of students (0, 1, 2, .; Lesson 2827 — Distinguishing Discrete from Continuous Lesson 2831 — Mixed and Other Types of Random Variables Lesson 2867 — Linear Transformations of Random Variables Lesson 2939 — Independence via Joint and Marginal Distributions Lesson 3113 — Discrete Parameter Spaces and Bayesian Updating
Discrete case: Lesson 2943 — Independence of More Than Two Random Variables
Discrete portions: appear as **jump discontinuities** in the CDF—sudden vertical leaps at specific values where probability mass concentrates; Lesson 2850 — Mixed Random Variables and Their CDFs
Discrete probability masses: to certain specific values (like a PMF); Lesson 2840 — Mixed Random Variables
discrete random variable: is a random variable that can take on only a countable number of distinct values.; Lesson 2823 — Discrete Random Variables: Definition and Examples Lesson 2827 — Distinguishing Discrete from Continuous Lesson 2843 — Definition and Interpretation of the CDF Lesson 2848 — Recovering PMF and PDF from the CDF
discriminant: .; Lesson 416 — The Discriminant: b² - 4ac Lesson 417 — Interpreting Discriminant Values Lesson 432 — Using the Discriminant to Predict Intercepts Lesson 1131 — The General Second-Degree Equation Lesson 1132 — Using the Discriminant to Classify Conics
disjoint: ); Lesson 2554 — Partitions of a Set Lesson 2731 — Equivalence Classes in Modular Arithmetic Lesson 2779 — Mutually Exclusive Events Lesson 2803 — Independence vs. Disjoint Events
Disjoint events: cannot happen at the same time.; Lesson 2803 — Independence vs. Disjoint Events
disk: (like slicing a loaf of bread).; Lesson 1653 — Volume of Revolution: Introduction and Disk Method Concept Lesson 1663 — Choosing Between Disk and Washer Methods
disk method: .; Lesson 1653 — Volume of Revolution: Introduction and Disk Method Concept Lesson 1654 — Disk Method: Rotation About the x-axis Lesson 1661 — Rotation About Horizontal Lines y = k Lesson 1663 — Choosing Between Disk and Washer Methods
Displacement: Moving 8 meters to the right from your starting point; Lesson 1235 — Introduction to Vectors: Magnitude and Direction Lesson 1265 — Velocity and Displacement Vectors Lesson 1267 — Work Done by a Constant Force Lesson 1573 — Area and Signed Area Lesson 1578 — Applications and Problem Solving with FTC
displacement vector: on the complex plane.; Lesson 1212 — Geometric Interpretation of Subtraction Lesson 1241 — Position Vectors and Displacement Lesson 1265 — Velocity and Displacement Vectors
distance: (the number itself) to move the decimal point.; Lesson 164 — Converting from Scientific Notation to Standard Form Lesson 226 — Understanding Absolute Value as Distance Lesson 852 — Equidistant Chords Theorem Lesson 1658 — Washer Method: Identifying Outer and Inner Radii Lesson 1820 — Limits and Continuity in Three or More Variables Lesson 2660 — Distance and Diameter Lesson 2667 — Applications of Connectivity
Distance and Navigation: When finding the shortest path or checking if routes meet perpendicularly, recognizing triples speeds up your work dramatically.; Lesson 790 — Applications of Triples and the Converse
Distance between points: Two towns are located at different positions from a radio tower.; Lesson 1109 — Applications: Distance and Navigation Problems
distance formula: to find AB, BC, and AC; Lesson 907 — Distance and Midpoint with Special Triangles Lesson 919 — Writing Equations of Circles from Given Information Lesson 1045 — The Pythagorean Identity Lesson 1103 — Deriving the Law of Cosines Lesson 1240 — Finding Magnitude from Components
distance from zero: , which is never negative.; Lesson 229 — Absolute Value Equations with No Solution or All Solutions Lesson 233 — Absolute Value Inequalities of the Form |x| < a
Distance to directrix: `|x - a²/c|`; Lesson 1183 — Deriving Conic Equations from Eccentricity
Distance to focus: `√[(x - c)² + y²]`; Lesson 1183 — Deriving Conic Equations from Eccentricity
distinct: (non-repeating) linear factors, you can decompose it into a sum of simpler fractions.; Lesson 1623 — Linear Factors: Distinct Denominators Lesson 2630 — Solving Second-Order Linear Homogeneous Recurrences Lesson 2762 — The RSA Setup: Choosing Primes
distribute: (or apply) that exponent to **each factor individually**.; Lesson 145 — Power of a Product Rule Lesson 208 — Equations with Parentheses and the Distributive Property Lesson 211 — Multi-Step Equations Requiring Multiple Operations Lesson 216 — Mixed Practice: Complex Linear Equations
Distribute the first term: of the binomial (x) to all three terms of the trinomial:; Lesson 328 — Multiplying a Binomial by a Trinomial
Distribute the second term: of the binomial (+2) to all three terms of the trinomial:; Lesson 328 — Multiplying a Binomial by a Trinomial
Distributing Exponents Incorrectly: Lesson 150 — Common Mistakes with Exponent Rules
Distributions are severely skewed: or have heavy outliers; Lesson 3111 — Choosing and Interpreting Nonparametric Tests
Distributive opportunities: – Sometimes factoring out a common term from the numerator helps you cancel with the denominator.; Lesson 520 — Simplifying After Rationalizing
distributive property: lets you multiply a number by a sum by multiplying that number by each addend separately, then adding the results.; Lesson 38 — The Distributive Property Lesson 186 — Simplifying Before Evaluating Lesson 208 — Equations with Parentheses and the Distributive Property Lesson 325 — Multiplying a Monomial by a Polynomial
div F = 0: The field is **incompressible** — no net flow in or out; Lesson 2004 — The Divergence of a Vector Field Lesson 2012 — Incompressible Fields and the Continuity Equation Lesson 2043 — Physical Interpretation: Conservation Laws
Divergence: means the limit is infinite or doesn't exist at all.; Lesson 1637 — Convergence and Divergence of Improper Integrals Lesson 1704 — Visualizing Sequences: Terms and Graphs Lesson 1715 — Convergence and Divergence of Series Lesson 1737 — Direct Comparison Test: Proving Divergence Lesson 1994 — Statement of Green's Theorem Lesson 2002 — Normal Form of Green's Theorem Lesson 2004 — The Divergence of a Vector Field Lesson 2036 — Introduction to the Divergence Theorem
Divergence method: Lesson 2041 — Simplifying Flux Calculations with Divergence
Divergence product rule: Lesson 2010 — Properties of Divergence and Curl
Divergence Test: ) formalizes this intuition:; Lesson 686 — The nth Term Test for Divergence
Divergence Theorem: (also called Gauss's Theorem) states that for a vector field **F** defined on a solid region *E* with closed boundary surface *S*:; Lesson 2038 — Statement of the Divergence Theorem Lesson 2041 — Simplifying Flux Calculations with Divergence
diverges: (grows without limit or oscillates forever).; Lesson 667 — Convergence and Divergence of Geometric Sequences Lesson 679 — Infinite Series: Convergence vs Divergence Lesson 680 — Partial Sums and Series Limits Lesson 686 — The nth Term Test for Divergence Lesson 1634 — Type 1 Improper Integrals: Infinite Upper Limit Lesson 1635 — Type 1 Improper Integrals: Infinite Lower Limit Lesson 1637 — Convergence and Divergence of Improper Integrals Lesson 1714 — Sequence of Partial Sums (+9 more)
Divide: Look at the first digit(s) of the dividend (the number being divided).; Lesson 29 — Long Division Algorithm Lesson 99 — Finding What Percent One Number Is of Another Lesson 443 — Evaluating Rational Expressions Lesson 471 — Simplifying Complex Fractions by Simplifying Numerator and Denominator Lesson 1302 — Finding the Inverse of a 2×2 Matrix Lesson 2930 — Conditional Distributions: Discrete Case
Divide and conquer: Lesson 1636 — Type 1 Improper Integrals: Both Limits Infinite Lesson 1909 — Regions Requiring Multiple Integrals
Divide as usual: , treating the numbers as if the decimal weren't there; Lesson 90 — Dividing Decimals by Whole Numbers Lesson 91 — Dividing by Decimals
Divide both: the numerator and denominator by the GCF; Lesson 65 — Simplifying Fractions
Divide both terms: by the GCF; Lesson 107 — Simplifying Ratios
Divide by 5: `y = 5`; Lesson 208 — Equations with Parentheses and the Distributive Property
Divide by the coefficient: to get x; Lesson 1084 — Equations with Multiple Angles Lesson 1608 — Trigonometric Integrals with Linear Arguments
Divide each component: by that magnitude; Lesson 1249 — Unit Vectors and Normalization
Divide each term: by that GCF; Lesson 358 — Factoring Out the GCF from Polynomials
Divide the coefficients: (the decimal numbers in front); Lesson 166 — Dividing Numbers in Scientific Notation Lesson 510 — Dividing Radicals with the Same Index
Divide the curve: into n small pieces, each with arc length Δs; Lesson 1684 — Deriving the Surface Area Formula
Divide the exponent: Write *n* = *q*(p-1) + *r* where 0 ≤ *r* < *p*-1; Lesson 2744 — Computing Large Powers Using Fermat's Little Theorem
Divide the radicands: (the numbers inside the radicals); Lesson 510 — Dividing Radicals with the Same Index
Divide to get 1: Make the right side equal 1; Lesson 1175 — Converting General Form to Standard Form
Dividend: The expression being divided (the "inside" number); Lesson 344 — Introduction to Polynomial Division
Divides: An integer *a* divides *b* (written *a | b*) if there exists an integer *k* such that *b = ak*.; Lesson 2492 — Direct Proof: Working with Definitions
Dividing: by the total volume of the region; Lesson 1931 — Average Value of a Function in Three Dimensions
Divisibility by 1: 1 divides every integer; Lesson 2709 — Divisibility: Definition and Basic Properties
Divisibility by 11: Lesson 2740 — Applications to Divisibility Tests
Division: Splitting or finding how many groups fit (like "How many ½-cup servings are in 3 cups?; Lesson 82 — Word Problems Involving Fraction Operations Lesson 172 — Writing Expressions from Word Problems Lesson 534 — Multiplying and Dividing with Fractional Exponents Lesson 1224 — Geometric Interpretation of Polar Operations
Division Algorithm for Polynomials: .; Lesson 348 — The Division Algorithm for Polynomials
Division by zero: Exclude points where denominators vanish; Lesson 1803 — Domain and Range of Multivariable Functions
Division equation: Lesson 201 — Solving One-Step Multiplication/Division Equations
Division phrases: Lesson 188 — Translating Words into Mathematical Expressions
Division Property example: Lesson 200 — Multiplication and Division Properties of Equality
Division Property of Equality: says you can divide both sides by the same nonzero number, and equality is preserved.; Lesson 200 — Multiplication and Division Properties of Equality
Division uses fractions: Write expressions as fractions rather than using ÷; Lesson 178 — Reading and Interpreting Algebraic Notation
Divisor: The expression you're dividing by (the "outside" number); Lesson 344 — Introduction to Polynomial Division
Divisors of 24: Lesson 2712 — Finding GCD by Listing Divisors
Divisors of 36: Lesson 2712 — Finding GCD by Listing Divisors
DNA Sequencing: Bioinformatics uses Eulerian paths to reconstruct long DNA sequences from overlapping fragments, treating fragments as edges in a graph.; Lesson 2682 — Applications of Eulerian Paths: The Königsberg Bridge Problem
Do this: Lesson 256 — Writing Equations in Slope-Intercept Form
Does it converge: Yes!; Lesson 1756 — Conditional Convergence
Does it converge absolutely: Check the absolute values:; Lesson 1756 — Conditional Convergence
Does it involve products: Integration by parts handles products of different function types (polynomial × exponential, polynomial × trig, etc.; Lesson 1632 — Applications and Strategy Selection
does not exist: .; Lesson 1332 — When Two-Sided Limits Exist Lesson 1333 — One-Sided Limits at Jump Discontinuities Lesson 1811 — Limits of Functions of Two Variables Lesson 1812 — Path-Dependent Limits and Nonexistence
doesn't change sign: , the critical point is **neither** a max nor min (maybe an inflection point with horizontal tangent); Lesson 1496 — The First Derivative Test for Local Extrema Lesson 1502 — Comparing First and Second Derivative Tests
Domain: Always all real numbers; Lesson 433 — Key Features Summary and Complete Graphing Lesson 485 — Introduction to Rational Functions Lesson 548 — What Are Domain and Range?Lesson 549 — Finding Domain from Equations Lesson 550 — Domain Restrictions from Square Roots Lesson 551 — Domain Restrictions from Rational Expressions Lesson 552 — Domain from Graphs Lesson 556 — Domain and Range of Piecewise Functions (+11 more)
Domain = integers ℤ: TRUE.; Lesson 2482 — Domains of Discourse
Domain of R: = {1, 2, 3} — only these elements from A participate; Lesson 2540 — Domain, Codomain, and Range of Relations
Domain restrictions: .; Lesson 481 — Checking for Extraneous Solutions Lesson 494 — Applications of Rational Functions Lesson 1518 — Vertical and Horizontal Asymptotes
Domain: All real numbers: – You can plug *any* x-value into an exponential function.; Lesson 605 — Domain, Range, and Asymptotes
Don't forget +C: Add your constant of integration at the very end; Lesson 1597 — Cyclic Integration by Parts Lesson 1627 — Integrating Partial Fractions with Linear Terms
Don't forget the r: The area element in polar coordinates is r dr dθ, not just dr dθ.; Lesson 1917 — Integrating over Sectors and Wedges
Done: Your current matching is maximum; Lesson 2702 — The Hungarian Algorithm for Maximum Bipartite Matching
dot product: (also called the scalar product) is a way to multiply two vectors that produces a single number (a scalar), not another vector.; Lesson 1253 — Definition of the Dot Product Lesson 1256 — Finding Angles Between Vectors Lesson 1267 — Work Done by a Constant Force Lesson 1282 — Computing the (i,j)-Entry of a Product
double integral: extends this idea to three dimensions: it calculates the **volume under a surface** z = f(x,y) above a region in the xy-plane.; Lesson 1892 — Introduction to Double Integrals Lesson 1922 — Introduction to Triple Integrals Lesson 1955 — Computing Total Mass with Double Integrals
Doubles: Same number twice (4 + 4 = 8, 5 + 5 = 10); Lesson 10 — Addition Facts to 10
down: , you divide all parts by the same number to make them smaller.; Lesson 118 — Scaling Up and Down Lesson 239 — Plotting Points on the Coordinate Plane Lesson 578 — Vertical Shifts of Functions Lesson 1025 — Transformations of Tangent: Phase Shift and Vertical Shift
downward: (focus below vertex); Lesson 1145 — Standard Form: Vertex at Origin, Vertical Axis Lesson 1148 — Parabolas with Vertex at (h, k): Vertical Axis Lesson 1501 — The Second Derivative Test for Local Extrema
DP ⁻¹: .; Lesson 2192 — Verifying Diagonalization
Draw: the radius to that point of tangency; Lesson 849 — Tangent-Radius Perpendicularity Theorem
Draw a diagram: showing the observer at the top; Lesson 970 — Using SOH-CAH-TOA with Depression Angles Lesson 1516 — Applied Optimization Strategy
Draw a horizontal line: at the observer's height; Lesson 976 — Angles of Depression: Definition and Setup
Draw a north line: at each relevant point; Lesson 1101 — Law of Sines with Bearing and Direction
Draw a perpendicular: from P to line ℓ (this uses perpendicular line concepts); Lesson 927 — Reflections Across Arbitrary Lines
Draw a reference rectangle: with dimensions 2a × 2b; Lesson 1171 — Graphing Hyperbolas Centered at the Origin
Draw a sample rectangle: perpendicular to the axis; Lesson 1665 — Setting Up a Shell Integral
Draw and label: everything clearly; Lesson 712 — Solving Multi-Step Angle Problems Lesson 1479 — Related Rates with Triangles Lesson 1481 — Ladder and Shadow Problems Lesson 1483 — Angle Related Rates
Draw and label everything: you know; Lesson 855 — Applications of Tangents and Chords Lesson 866 — Perimeter and Area Problem Solving
Draw conclusions: Conclude that some hole must contain multiple pigeons, which often implies a geometric property about distances, collinearity, or proximity; Lesson 2575 — Pigeonhole Principle in Geometry
Draw or visualize: the point *(a, b)* on the complex plane; Lesson 1210 — Argument of a Complex Number
Draw the asymptotes: through opposite corners of this rectangle; Lesson 1171 — Graphing Hyperbolas Centered at the Origin
Draw the diagonal: from the common starting point—this is the resultant; Lesson 1243 — Adding Vectors Geometrically
Draw the directrix: A line `p` units from the vertex in the *opposite* direction.; Lesson 1151 — Graphing Parabolas from Standard Form
Draw the first vector: starting from any point; Lesson 1243 — Adding Vectors Geometrically
Draw the ground level: as another horizontal line (parallel to the first); Lesson 976 — Angles of Depression: Definition and Setup
Draw the resultant: from the tail of the first vector to the tip of the second; Lesson 1243 — Adding Vectors Geometrically
Draw vectors: From each point, draw an arrow in the direction ⟨P, Q⟩; Lesson 1968 — Sketching Vector Fields by Hand
Drawing altitudes: Drop a perpendicular from a vertex to the opposite side to create right triangles, which often unlock congruence proofs using HL or create useful angle relationships.; Lesson 769 — Auxiliary Lines and Constructions
Drawing diagonals: In quadrilaterals, a diagonal can split the figure into two triangles, letting you use triangle congruence or similarity theorems.; Lesson 769 — Auxiliary Lines and Constructions
Drawing Diagrams: Visual representations make complex situations clearer.; Lesson 195 — Checking Solutions and Problem-Solving Strategies
Drug elimination: from the bloodstream; Lesson 2327 — Applications: RC Circuits and Decay
ds: a tiny piece of arc length along the curve; Lesson 1972 — Introduction to Line Integrals Lesson 1973 — Parametric Curves and Arc Length Element
du: .; Lesson 1581 — Computing du and Substituting Lesson 1590 — The Integration by Parts Formula
During collection: Maintain randomization and record all relevant factors; Lesson 3102 — ANOVA Applications and Experimental Design
dv: the function you'll integrate to get v; Lesson 1590 — The Integration by Parts Formula Lesson 1609 — Completing Trigonometric Integrals by Parts Lesson 1922 — Introduction to Triple Integrals Lesson 1923 — Setting Up Triple Integrals over Rectangular Boxes Lesson 1956 — Computing Total Mass with Triple Integrals
dx: an independent variable representing a small change in x (you choose this); Lesson 1471 — Differentials and Notation Lesson 1473 — Applications of Differentials in Measurement Lesson 1685 — Surface Area Formula for Revolution Around x-axis Lesson 1837 — The Total Differential
dx/dt = Ax: , where **x** is a vector changing over time, diagonalization lets you "decouple" the system by writing **A = PDP ¹**.; Lesson 2199 — Applications of Diagonalization Lesson 2278 — Applications to Differential Equations and Dynamics
dy: a dependent variable representing the corresponding change in y along the tangent line; Lesson 1471 — Differentials and Notation Lesson 1473 — Applications of Differentials in Measurement Lesson 1660 — Washer Method: Rotation About the y-axis Lesson 1662 — Rotation About Vertical Lines x = h Lesson 1671 — Shell Method with Functions of y Lesson 1837 — The Total Differential
dy/dt = Dy: Lesson 2199 — Applications of Diagonalization Lesson 2278 — Applications to Differential Equations and Dynamics

E

E = 3: edges; Lesson 2691 — Euler's Formula for Planar Graphs Lesson 2764 — Choosing the Public Exponent e
E = 4: edges; Lesson 2691 — Euler's Formula for Planar Graphs
E = 5: (added one edge); Lesson 2691 — Euler's Formula for Planar Graphs
e = c/a: Lesson 1164 — Eccentricity of an Ellipse Lesson 1181 — Calculating Eccentricity for Hyperbolas
E[...]: gives us the average squared distance from μ; Lesson 2856 — Variance: Definition and Interpretation
E[g(X)]: for some function *g* and random variable *X*, but the integral or sum is messy.; Lesson 2976 — Monte Carlo Methods and Simulation
E[h(U)]: where U ~ Uniform(0,1).; Lesson 2976 — Monte Carlo Methods and Simulation
E[X] = 1/p: Lesson 2862 — Expectation and Variance in Common Distributions
E[X] = np: Lesson 2862 — Expectation and Variance in Common Distributions Lesson 2877 — Expectation and Variance of Binomial Distribution
E[X] = p: (the probability of success); Lesson 2872 — The Bernoulli Trial and Bernoulli Distribution
E[X] = α/β: Lesson 2918 — Properties of the Gamma Distribution
E[X] = λ: Lesson 2862 — Expectation and Variance in Common Distributions
E[X^n] = M^(n)(0): Lesson 2961 — Finding Moments from MGFs
E[θ̂] = θ: .; Lesson 2999 — Properties of Good Estimators
e^x: are inverse functions.; Lesson 632 — Solving Equations with Natural Exponentials Lesson 1347 — Exponential and Logarithmic Limits Lesson 1790 — Applications to Common Functions
E⁻¹: .; Lesson 1308 — Applications of Inverse Matrices
e₁: = (1, 0, 0); Lesson 2209 — Orthogonal and Orthonormal Vectors Lesson 2214 — Coordinates with Respect to Orthonormal Bases Lesson 2249 — Coordinates Relative to a Basis
e₂: = (0, 1, 0); Lesson 2209 — Orthogonal and Orthonormal Vectors Lesson 2214 — Coordinates with Respect to Orthonormal Bases Lesson 2249 — Coordinates Relative to a Basis
Each leg: = hypotenuse ÷ √2 (or hypotenuse × √2/2, which simplifies the radical); Lesson 793 — Finding Missing Sides in 45-45-90 Triangles
Each piece has weight: = (weight per unit length) × Δx; Lesson 1696 — Lifting Cables and Chains
Easier evaluation: Computing derivatives at zero is often straightforward; Lesson 1774 — Maclaurin Series: Taylor Series at Zero
Easy to find intercepts: Set x = 0 to find the y-intercept; set y = 0 to find the x-intercept; Lesson 260 — Standard Form: Ax + By = C
eccentricity: measures exactly how "stretched out" or "circular" an ellipse is using a single number.; Lesson 1164 — Eccentricity of an Ellipse Lesson 1179 — Eccentricity: Measuring the Shape of Conics Lesson 1181 — Calculating Eccentricity for Hyperbolas Lesson 1182 — Focus-Directrix Definition of Conics Lesson 1183 — Deriving Conic Equations from Eccentricity
Economics: Analyzing cost functions with multiple variables; Lesson 354 — Applications of Polynomial Division
edge: .; Lesson 2682 — Applications of Eulerian Paths: The Königsberg Bridge Problem Lesson 2698 — Applications of Graph Coloring
edge connectivity: focuses on *edges*.; Lesson 2663 — Edge Connectivity Lesson 2666 — Menger's Theorem Lesson 2667 — Applications of Connectivity
Edge count formula: – It's connected and has exactly *n - 1* edges; Lesson 2668 — Definition and Properties of Trees
Edge Version: Lesson 2666 — Menger's Theorem
edge-disjoint: paths between any two vertices *u* and *v* equals the minimum number of edges whose removal disconnects *u* from *v*.; Lesson 2666 — Menger's Theorem Lesson 2708 — Edge-Disjoint Paths and Menger's Theorem
Education: Assessing three teaching methods across different classrooms; Lesson 3102 — ANOVA Applications and Experimental Design
Efficiency: Computing det(A) and det(B) separately is often faster than finding **AB** first, then computing its determinant; Lesson 2159 — Determinant of a Product Lesson 2999 — Properties of Good Estimators Lesson 3008 — Comparing MLE and Method of Moments
efficient: .; Lesson 50 — Factor Pairs Lesson 2999 — Properties of Good Estimators Lesson 3000 — Sample Mean and Variance as Estimators
Efficient for "solved" equations: When one equation is already in the form y = .; Lesson 281 — Comparing Graphing and Substitution Methods
eigenfunctions: .; Lesson 2405 — Eigenvalue Problems and Sturm-Liouville Theory Lesson 2420 — Eigenvalue Problems from Boundary Conditions
eigenspace: .; Lesson 2170 — Finding Eigenvectors from Eigenvalues Lesson 2172 — Eigenspaces and Their Dimension
eigenvalue: , denoted **λ** (lambda).; Lesson 2168 — Definition of Eigenvectors and Eigenvalues Lesson 2425 — Physical Interpretation and Decay Rates
Eigenvalue criterion: All eigenvalues must be positive; Lesson 2298 — Applications to Optimization and Hessian Matrices
eigenvalue problem: .; Lesson 2430 — Standing Waves and Normal Modes Lesson 2448 — Solving the Spatial ODE: Eigenvalue Problems
Eigenvalues: .; Lesson 2177 — Similar Matrices and Eigenvalue Invariance Lesson 2279 — Spectral Theorem in Quantum Mechanics and Statistics Lesson 2284 — Computing SVD via Eigenvalue Decomposition Lesson 2405 — Eigenvalue Problems and Sturm-Liouville Theory Lesson 2420 — Eigenvalue Problems from Boundary Conditions
Eigenvectors: = quantum states with definite values; Lesson 2279 — Spectral Theorem in Quantum Mechanics and Statistics Lesson 2284 — Computing SVD via Eigenvalue Decomposition
Eighteen: = ten + eight; Lesson 2 — Counting from 11 to 100
Electric fields: Field lines point from positive to negative charges; Lesson 1971 — Field Lines and Flow Lines
Electromagnetics: Light waves in free space; Lesson 2434 — The Three-Dimensional Wave Equation
electromagnetism: , Green's Theorem helps analyze electric and magnetic fields in two-dimensional cross-sections.; Lesson 2003 — Applications and Extensions of Green's Theorem Lesson 2045 — Combining with Stokes' Theorem and Applications
Electrostatics: The electric potential in a charge-free region obeys Laplace's equation.; Lesson 2436 — Introduction to Laplace's Equation
Elementary function combinations: (sums, products, compositions of continuous functions) remain continuous wherever all component functions are defined.; Lesson 1818 — Continuity on a Region
elementary row operations: to create zeros below each leading entry (pivot), working from left to right, top to bottom.; Lesson 1313 — Gaussian Elimination Process Lesson 2107 — Augmented Matrices
Eliminate constants: Express variable dimensions in terms of height using similar triangles or given ratios; Lesson 1484 — Water Tank and Fluid Flow Problems
Eliminate the radical: by squaring (or cubing, etc.; Lesson 530 — Mixed Radical and Rational Equations
Elimination: Lesson 1885 — Solving Systems with Multiple Lagrange Multipliers Lesson 2723 — The Sieve of Eratosthenes
ellipse: is the set of all points where the **sum of distances** to two fixed points (called **foci**) is constant.; Lesson 1130 — Geometric Definitions of Conics Lesson 1131 — The General Second-Degree Equation Lesson 1132 — Using the Discriminant to Classify Conics Lesson 1133 — Standard Position and Orientation Lesson 1155 — Definition and Standard Form of an Ellipse Lesson 1179 — Eccentricity: Measuring the Shape of Conics Lesson 1182 — Focus-Directrix Definition of Conics
Ellipses: dominate astronomy.; Lesson 1135 — Overview of Conic Applications
elliptic paraboloid: looks like:; Lesson 1797 — Quadric Surfaces: Ellipsoids and Paraboloids Lesson 1807 — Common Surfaces: Planes, Paraboloids, and Saddles
Emails: arriving in an inbox; Lesson 2893 — Applications of the Poisson Distribution
Emergency room visits: during a specific time window; Lesson 2893 — Applications of the Poisson Distribution
empirical rule: or 68-95-99.; Lesson 2904 — The Normal Distribution: Shape and Properties Lesson 2911 — The Empirical Rule (68-95- 99.7 Rule)
empty set: that contains *nothing at all*.; Lesson 2522 — Universal Set and Empty Set Lesson 2531 — Introduction to Power Sets Lesson 2774 — Events as Subsets of the Sample Space Lesson 2782 — Probability of the Empty Set and Impossible Events
Encrypt: Send *m^e*; Lesson 2768 — Why RSA Works: Euler's Theorem Lesson 2769 — RSA Example with Small Primes
Engineering: Breaking down complex transfer functions in electronics; Lesson 354 — Applications of Polynomial Division Lesson 1100 — Applications of the Law of Sines
English: "Every student passed the exam.; Lesson 2483 — Translating Between English and Quantifiers
English to logic: , identify:; Lesson 2483 — Translating Between English and Quantifiers
Enlargement: Lesson 119 — Scale Factors
Entry (1,1): (Row 1) · (Column 1) = 2(1) + 3(0) = **2**; Lesson 1283 — Multiplying 2×2 Matrices
Entry (1,2): (Row 1) · (Column 2) = 2(4) + 3(2) = 8 + 6 = **14**; Lesson 1283 — Multiplying 2×2 Matrices
Entry (2,1): (Row 2) · (Column 1) = 5(1) + 1(0) = **5**; Lesson 1283 — Multiplying 2×2 Matrices
Entry (2,2): (Row 2) · (Column 2) = 5(4) + 1(2) = 20 + 2 = **22**; Lesson 1283 — Multiplying 2×2 Matrices
Environmental Science: The area between pollution level curves over time quantifies cumulative environmental impact differences between scenarios.; Lesson 1652 — Applied Problems: Area Between Curves
equal: because of parallel lines.; Lesson 971 — Alternate Interior Angles in Elevation/Depression Lesson 1242 — Equal Vectors and Parallel Vectors Lesson 1273 — Matrix Equality Lesson 2047 — Vector Equality and Zero Vector
Equal variance: ANOVA's F test assumes homogeneity of variance.; Lesson 3051 — Applications and Limitations
Equal Variance (Homoscedasticity): Lesson 3068 — Assumptions of Simple Linear Regression
Equal Variances (Homoscedasticity): Lesson 3094 — One-Way ANOVA: The Model and Assumptions
equals: the angle of elevation from the object back up to you.; Lesson 970 — Using SOH-CAH-TOA with Depression Angles Lesson 1553 — Riemann Sums as n Approaches Infinity
Equals sign (=): Lesson 11 — Understanding Subtraction as Taking Away
Equate coefficients: of like powers:; Lesson 1625 — Solving for Coefficients: Equating Coefficients Lesson 2400 — Finding Power Series Solutions Near Ordinary Points
Equate imaginary parts: `y - 1 = 5` → `y = 6`; Lesson 1205 — Solving Equations with Complex Numbers
Equate real parts: `2x + 3 = 7` → `x = 2`; Lesson 1205 — Solving Equations with Complex Numbers
Equating coefficients: for the remaining unknowns (match powers of x on both sides); Lesson 1631 — Complex Mixed Denominator Cases
Equation: *x* + 7 = 15; Lesson 189 — Writing Equations from Word Problems Lesson 193 — Perimeter and Area Word Problems Lesson 778 — Word Problems Involving Right Triangles Lesson 991 — Defining the Unit Circle Lesson 1173 — Writing Equations from Given Information
equidistant: from the center (the same distance away).; Lesson 852 — Equidistant Chords Theorem Lesson 927 — Reflections Across Arbitrary Lines Lesson 1130 — Geometric Definitions of Conics
Equilateral Triangle: Lesson 730 — Classifying Triangles by Sides Lesson 736 — Properties of Equilateral Triangles Lesson 794 — The 30-60-90 Triangle: Side Ratios Lesson 812 — Finding Individual Exterior Angles in Regular Polygons Lesson 942 — Line Symmetry (Reflectional Symmetry)
equilibrium: , it means the object isn't accelerating—it's either at rest or moving at constant velocity.; Lesson 1264 — Equilibrium: Balancing Multiple Forces Lesson 1269 — Tension in Cables and Ropes Lesson 1270 — Applications in Physics and Engineering
Equilibrium 1: `x = 0, y = 0` (extinction); Lesson 2376 — Predator-Prey Models: The Lotka-Volterra Equations
Equilibrium 2: Solve `αx - βxy = 0` and `-γy + δxy = 0` simultaneously:; Lesson 2376 — Predator-Prey Models: The Lotka-Volterra Equations
Equilibrium solutions: appear where all segments are horizontal (slope = 0); Lesson 2305 — Direction Fields
equivalence class: is the collection of all elements that are equivalent to a particular element.; Lesson 2552 — Equivalence Classes Lesson 2731 — Equivalence Classes in Modular Arithmetic
equivalence relation: is a special kind of relation that behaves like "sameness" or "similarity.; Lesson 2550 — Definition of Equivalence Relations Lesson 2739 — Properties of Congruence Relations
Equivalent condition: All eigenvalues of A are strictly positive.; Lesson 2292 — Positive Definite, Negative Definite, and Indefinite Matrices
Equivalent fractions: are different fractions that represent the same value.; Lesson 64 — Equivalent Fractions
Equivalently: (using the contrapositive): if `x₁ ≠ x₂`, then `f(x₁) ≠ f(x₂)`.; Lesson 2560 — Injective Functions (One-to-One)
Establishing constraints: It helps prove important bounds (like K₅ being non-planar); Lesson 2691 — Euler's Formula for Planar Graphs
estimate: $\mu$ from observed data.; Lesson 2968 — Understanding Sample Means and Averages Lesson 2998 — What is Point Estimation?Lesson 3014 — Confidence Interval for a Population Mean (σ Unknown)
Estimate partials: For each partial square, estimate what fraction is covered (½, ¼, ¾, etc.; Lesson 872 — Irregular Composite Figures on Grids
Estimated area: ≈ 12 + 4 = **16 square units**; Lesson 872 — Irregular Composite Figures on Grids
Euclid's brilliant proof: Assume there are only finitely many primes: p₁, p₂, .; Lesson 2499 — Classic Contradiction Proofs
Euclidean Algorithm: gives you a much faster way: it uses the fact that `gcd(a, b) = gcd(b, r)` where `r` is the remainder when you divide `a` by `b`.; Lesson 2713 — The Euclidean Algorithm
Euler's formula: `e^(iθ) = cos θ + i sin θ`.; Lesson 1234 — Applications: Fractional Powers and Euler's Formula Lesson 2341 — Complex Conjugate Roots: Euler's Formula Approach Lesson 2381 — Complex Eigenvalues in 2×2 Systems Lesson 2693 — K₅ and K₃,₃ are Non-Planar
Euler's identity: states:; Lesson 1220 — Euler's Formula and Exponential Form
Euler's Theorem: works for **any modulus** n (when gcd(a,n) = 1): `a^φ(n) ≡ 1 (mod n)`; Lesson 2751 — Problem Solving with Fermat's and Euler's Theorems
Eulerian: Focus on *edges* (traverse every edge once); Lesson 2683 — Hamiltonian Paths and Cycles: Definition and Examples
Eulerian circuit: (or Eulerian cycle), which is a closed Eulerian path.; Lesson 2678 — Eulerian Paths and Circuits: Definition and Motivation Lesson 2679 — Euler's Theorem for Eulerian Circuits Lesson 2680 — Euler's Theorem for Eulerian Paths
Eulerian path: a walk through a graph that traverses every edge exactly once.; Lesson 2678 — Eulerian Paths and Circuits: Definition and Motivation
Evaluate: numerically when products are simpler than sums; Lesson 1078 — Applying Sum-to-Product Formulas Lesson 1342 — Limits Involving Conjugates Lesson 1344 — The Squeeze Theorem Lesson 1586 — Definite Integrals and Changing Limits Lesson 1698 — Applications of Average Value Lesson 1932 — Applications: Density and Total Mass Lesson 1982 — Practice with Line Integrals in 2D and 3D Lesson 2018 — Surface Integrals of Scalar Functions (+1 more)
Evaluate [uv] ₐᵇ: by substituting b, then a, and subtracting; Lesson 1599 — Definite Integrals and Integration by Parts
Evaluate and compare: to find your optimum; Lesson 1514 — Optimization with Implicit Constraints
Evaluate at the bounds: using the limit; Lesson 1637 — Convergence and Divergence of Improper Integrals
Evaluate endpoints: Calculate f(a) and f(b); Lesson 1357 — Using the IVT to Show Roots Exist
Evaluate f_xx: at each critical point; Lesson 1867 — Applying the Second Derivative Test
Evaluate f(m̄ ᵢ): at each midpoint—this becomes your rectangle height; Lesson 1549 — Midpoint Riemann Sums
Evaluate f(x, y): at each grid point to get the slope; Lesson 2305 — Direction Fields
Evaluate the definite integral: ∫[a,t] f(x)dx using the Fundamental Theorem of Calculus; Lesson 1634 — Type 1 Improper Integrals: Infinite Upper Limit
Evaluate the field: At each point (x₀, y₀), compute **F**(x₀, y₀) to get the vector components; Lesson 1968 — Sketching Vector Fields by Hand
Evaluate the function: at:; Lesson 1508 — The Closed Interval Method Lesson 1548 — Right Riemann Sums
Evaluate the iterated integral: Lesson 1955 — Computing Total Mass with Double Integrals
Evaluate the limit: along this path; Lesson 1812 — Path-Dependent Limits and Nonexistence
Evaluate the objective function: Plug each corner point into *P* = 2*x* + 3*y*; Lesson 308 — Applications: Maximization Problems Lesson 309 — Applications: Minimization Problems
Evaluate the remaining integral: ∫ₐᵇ v du and apply its limits; Lesson 1599 — Definite Integrals and Integration by Parts
Evaluate the vector field: on the surface by substituting the parametrization; Lesson 2021 — Computing Flux for Parametric Surfaces
Evaluating: a rational expression means plugging in a specific value for the variable and calculating the result —just like you would with any algebraic expression.; Lesson 443 — Evaluating Rational Expressions Lesson 546 — Function Evaluation in Context
Evaluating a polynomial: (plugging in a number); Lesson 351 — The Remainder Theorem
Evaluation Theorem: ) gives us a shortcut:; Lesson 1570 — The Fundamental Theorem of Calculus: Part 2
even: or **odd**.; Lesson 138 — Exponents with Negative Bases Lesson 569 — Even and Odd Functions Lesson 1048 — Even- Odd Identities Lesson 1601 — Powers of Sine and Cosine: Even Powers Lesson 1603 — Powers of Tangent and Secant: Even Powers of Secant Lesson 1776 — Maclaurin Series for sin(x) and cos(x)Lesson 2603 — The Binomial Theorem for Negative Binomials
Even cycles: (n even): Alternate two colors around the cycle.; Lesson 2696 — Coloring Special Graphs
Even exponent: → **Positive result**; Lesson 138 — Exponents with Negative Bases
Even extension: Mirror the function symmetrically across the vertical axis.; Lesson 2462 — Half-Range Expansions
Even function: `f(-x) = f(x)` (symmetric across the y-axis); Lesson 594 — The Absolute Value Function Lesson 1517 — Domain, Intercepts, and Symmetry
even functions: have y-axis symmetry (f(-x) = f(x)), like f(x) = x², while **odd functions** have origin symmetry (f(-x) = -f(x)), like f(x) = x³.; Lesson 587 — Even and Odd Functions Under Transformation Lesson 1561 — Definite Integrals of Even and Odd Functions Lesson 2461 — Even and Odd Functions: Cosine and Sine Series
Even integers: An integer *n* is even if there exists an integer *k* such that *n = 2k*.; Lesson 2492 — Direct Proof: Working with Definitions
Even number: `(1 + x² + x⁴ + .; Lesson 2642 — Using Generating Functions to Count Distributions
Even power of secant: Factor out `sec² x` for du/dx and convert remaining secants to tangents; Lesson 1610 — Strategy Selection for Trigonometric Integrals
Even-odd identities: sin(-θ) = -sin θ, cos(-θ) = cos θ; Lesson 1050 — Simplifying Trigonometric Expressions
Event probabilities: Sum p(x, y) over all (x, y) pairs in your event; Lesson 2927 — Joint Probability Distributions: Discrete Case
Events are dependent: Lesson 2804 — Testing for Independence
Events are independent: Lesson 2804 — Testing for Independence
every: term in the trinomial.; Lesson 328 — Multiplying a Binomial by a Trinomial Lesson 2654 — Subgraphs and Induced Subgraphs
exact: area under a curve.; Lesson 1553 — Riemann Sums as n Approaches Infinity Lesson 1556 — Computing Simple Integrals Using Riemann Sum Limits Lesson 2328 — Exact Differential Equations: Definition and Test Lesson 2329 — Solving Exact Equations: Finding the Potential Function Lesson 2336 — Strategy Guide: Choosing the Right Solution Method
Exact answers: Always gives precise numerical solutions, including fractions and decimals; Lesson 281 — Comparing Graphing and Substitution Methods
Exact answers required: – Homework, tests, and theoretical work often demand exact values.; Lesson 521 — Applications and When Not to Rationalize
exactly: the derivative of μ(x)·y:; Lesson 2321 — The Integrating Factor Method Lesson 2876 — Computing Binomial Probabilities
Exactly j objects: `xʲ`; Lesson 2642 — Using Generating Functions to Count Distributions
exactly one triangle: Lesson 1098 — One Solution Case in SSA Lesson 1123 — Using the Height to Determine Triangle Existence
Examine: the integrand's structure; Lesson 1587 — u-Substitution Requiring Algebraic Manipulation
Examine Cross-Sections: Lesson 1868 — Inconclusive Cases and Further Analysis
Examine edges in order: , one at a time; Lesson 2675 — Minimum Spanning Trees: Kruskal's Algorithm
Examine the boundary separately: (parametrize it, substitute constraints, etc.; Lesson 1869 — Boundary Considerations for Optimization
Example (Area): A rectangle's length is 3 meters more than its width.; Lesson 420 — Applications: Projectile Motion and Area Problems
Example (Projectile): A ball is thrown upward with initial velocity 48 ft/s from a 6-foot platform.; Lesson 420 — Applications: Projectile Motion and Area Problems
Example 1: 5 + 2³ × 3; Lesson 33 — Exponents in Order of Operations Lesson 43 — Adding Positive and Negative Numbers Lesson 44 — Subtracting Positive and Negative Numbers Lesson 69 — Adding Fractions with Like Denominators Lesson 84 — Reading and Writing Decimals Lesson 104 — Multi-Step Percentage Problems Lesson 141 — Mixed Practice with Integer Exponents Lesson 142 — Product Rule for Exponents (+50 more)
Example 1 - Simplifying: If you see `2sin(15°)cos(15°)`, recognize it as `sin(2·15°) = sin(30°) = 1/2`.; Lesson 1063 — Double Angle Formula for Sine
Example 1 (Addition): "Maria had some stickers.; Lesson 190 — Solving One-Step Word Problems
Example 1 (Convergent): The geometric series ∑(n=1 to ∞) (1/2ⁿ) = 1/2 + 1/4 + 1/8 + .; Lesson 1715 — Convergence and Divergence of Series
Example 2: 20 ÷ 4 + 3²; Lesson 33 — Exponents in Order of Operations Lesson 43 — Adding Positive and Negative Numbers Lesson 44 — Subtracting Positive and Negative Numbers Lesson 69 — Adding Fractions with Like Denominators Lesson 84 — Reading and Writing Decimals Lesson 104 — Multi-Step Percentage Problems Lesson 141 — Mixed Practice with Integer Exponents Lesson 142 — Product Rule for Exponents (+50 more)
Example 2 - Solving: To solve `sin(2x) = sin(x)`, rewrite it as `2sin(x)cos(x) = sin(x)`, then factor: `sin(x)(2cos(x) - 1) = 0`.; Lesson 1063 — Double Angle Formula for Sine
Example 2 (Multiplication): "A pack of pencils costs $3.; Lesson 190 — Solving One-Step Word Problems
Example 3: 4 - 6; Lesson 44 — Subtracting Positive and Negative Numbers Lesson 69 — Adding Fractions with Like Denominators Lesson 84 — Reading and Writing Decimals Lesson 141 — Mixed Practice with Integer Exponents Lesson 142 — Product Rule for Exponents Lesson 143 — Quotient Rule for Exponents Lesson 144 — Power of a Power Rule Lesson 146 — Power of a Quotient Rule (+27 more)
Example 3 (Subtraction): "Tom had 50 marbles and lost some.; Lesson 190 — Solving One-Step Word Problems
Example 4: 3 - (-8); Lesson 44 — Subtracting Positive and Negative Numbers Lesson 69 — Adding Fractions with Like Denominators Lesson 84 — Reading and Writing Decimals Lesson 141 — Mixed Practice with Integer Exponents Lesson 144 — Power of a Power Rule Lesson 393 — Solving x² = c by Square Roots Lesson 469 — What is a Complex Fraction?Lesson 505 — Adding and Subtracting Like Radicals (+3 more)
Example 4 (Division): "48 students are split equally into groups.; Lesson 190 — Solving One-Step Word Problems
Example 5: (1/2) ¹ = 1 ÷ (1/2) = **2** (negative exponent flips the fraction); Lesson 141 — Mixed Practice with Integer Exponents
Example analogy: Climbing a ladder where each rung might be supported by the previous *two* rungs—you need to know both lower rungs are stable before trusting the next one.; Lesson 2515 — Strong Induction for Sequences and Recurrences
Example approach: Lesson 513 — Mixed Operations with Radicals Lesson 907 — Distance and Midpoint with Special Triangles Lesson 1682 — Applications: Arc Length in Physics and Geometry
Example check: Suppose `p(x) = x/6` for `x ∈ {1, 2, 3}`.; Lesson 2842 — Verifying Valid PMFs and PDFs
Example context: Suppose $X$ counts total claims and $Y$ counts the number of policyholders.; Lesson 2936 — Iterated Expectations and the Tower Property
Example continued: Since r² = 16, we find r = √16 = **4**.; Lesson 917 — Finding Center and Radius from Circle Equations Lesson 1842 — Computing Gradient Vectors
Example flow: In a 3×4 matrix, you'd first create a pivot in row 1, zero out everything below it in that column, then create a pivot in row 2 (in a column to the right), zero out what's below it, and finally handle row 3.; Lesson 2119 — Converting to Row Echelon Form
Example format: Lesson 763 — Two-Column Proof Format
Example idea: If -1 ≤ sin(1/x) ≤ 1 for all x ≠ 0, then multiplying by x² gives: -x² ≤ x²sin(1/x) ≤ x².; Lesson 1344 — The Squeeze Theorem
Example in 2D: Consider **F**(x, y) = ⟨y, x⟩.; Lesson 1963 — Introduction to Vector Fields
Example in 3D: If **u** = ⟨2, 0, -1⟩ and **v** = ⟨1, 3, 5⟩:; Lesson 1253 — Definition of the Dot Product Lesson 1963 — Introduction to Vector Fields
Example in ℂ²: If **u** = (1+i, 2) and **v** = (i, 3), then:; Lesson 2201 — Standard Inner Products on ℝⁿ and ℂⁿ
Example in ℝ³: If **u** = (1, 2, 3) and **v** = (4, −1, 2), then:; Lesson 2201 — Standard Inner Products on ℝⁿ and ℂⁿ
Example mindset: To count passwords with at least one digit, one uppercase, and one symbol, you might define A ᵢ = "passwords missing requirement i," then use inclusion-exclusion to find |A₁ ∪ A₂ ∪ A₃| and subtract from all possible passwords.; Lesson 2626 — Complex Applications and Problem-Solving Strategies
Example of ∞/∞: Lesson 1340 — Indeterminate Forms: 0/0 and ∞/∞
Example of 0/0: Lesson 1340 — Indeterminate Forms: 0/0 and ∞/∞
Example of Infinite Solutions: Lesson 290 — Special Cases: No Solution and Infinite Solutions
Example of No Solution: Lesson 290 — Special Cases: No Solution and Infinite Solutions
Example of the error: Suppose you're proving that the sum 1 + 2 + .; Lesson 2509 — Common Mistakes in Induction Proofs
Example pattern: If someone sends you **C = EM** (encoded message), you recover the original with **M = E ¹C**.; Lesson 1308 — Applications of Inverse Matrices Lesson 2507 — Multiple Base Cases
Example patterns: Lesson 2353 — Products of Basic Functions
Example scenario: If x - 3 is a factor of 2x³ + kx² - 7x + 12, you know dividing by (x - 3) must give remainder 0.; Lesson 354 — Applications of Polynomial Division Lesson 1367 — Common Techniques and Restrictions on δ Lesson 1649 — Choosing Vertical vs Horizontal Slicing
Example situation: You stand 50 feet from a flagpole and look up at a 32° angle of elevation to see the top.; Lesson 969 — Using SOH-CAH-TOA with Elevation Angles
Example situations: Lesson 419 — Choosing Between Factoring and the Formula
Example structure: An observer sees a bird at an angle of elevation of 30°.; Lesson 972 — Multi-Step Elevation and Depression Problems Lesson 2850 — Mixed Random Variables and Their CDFs
Example walkthrough: Lesson 592 — Graphing Piecewise Functions
Example with decimals: Lesson 289 — Elimination with Fractions and Decimals
Example with fractions: Lesson 289 — Elimination with Fractions and Decimals
Example with multiple fractions: Solve `(x/3) + (x/6) = 5`; Lesson 205 — Equations with Fractions or Decimals
Example: 6 × 30: Lesson 26 — Multiplying by Multiples of 10, 100, and 1000
Example: 8 × 500: Lesson 26 — Multiplying by Multiples of 10, 100, and 1000
Example: Shrinking Squares Area: Lesson 692 — Infinite Series in Geometry
Examples of independent stages: Lesson 2808 — Independence in Multi-Stage Experiments
Examples of separable equations: Lesson 2308 — What is a Separable Differential Equation?
excess kurtosis: = Kurt - 3, where 3 is the kurtosis of a normal distribution.; Lesson 2861 — Higher Moments: Skewness and Kurtosis Lesson 2863 — Higher Moments: Skewness and Kurtosis
Exclude those x-values: from your domain; Lesson 549 — Finding Domain from Equations
Execute: the substitution with confidence; Lesson 1587 — u-Substitution Requiring Algebraic Manipulation
Execute carefully: track sign changes and scalar multiplications; Lesson 2158 — Practice: Mixed Determinant Computation
exhaustive: ); Lesson 2554 — Partitions of a Set Lesson 2772 — Sample Spaces: The Set of All Possible Outcomes Lesson 2780 — Partitions of the Sample Space
Existence: If W has an orthonormal basis {u₁, u₂, .; Lesson 2217 — Decomposition into Orthogonal Subspaces Lesson 2412 — Well-Posed Problems: Existence, Uniqueness, Stability Lesson 2488 — Uniqueness Quantifier and Applications Lesson 2724 — Unique Factorization: Statement of the Theorem
Existence and Uniqueness Theorem: answers both.; Lesson 2306 — Existence and Uniqueness Theorems
exists: in some interval around x₀; Lesson 2306 — Existence and Uniqueness Theorems Lesson 2346 — Existence and Uniqueness for Second- Order ODEs
expand: a logarithm means breaking it into simpler parts using the rules:; Lesson 624 — Combining Logarithm Properties Lesson 629 — Applying Logarithm Properties to Solve Equations Lesson 1059 — Sum and Difference Formulas in Equation Solving Lesson 1088 — Equations Requiring Sum and Difference Formulas
Expand and simplify: to get a cubic polynomial: -λ³ + (terms with λ² and λ) + constant; Lesson 2178 — Computing Eigenvalues for 3×3 Matrices
Expand factorials strategically: Instead of computing full factorials, cancel common terms:; Lesson 2590 — Computing Basic Combinations
Expand G(x): back into a power series to read off the coefficient formula; Lesson 2643 — Solving Recurrence Relations with Generating Functions
Expand the determinant: This creates three 2×2 determinants; Lesson 2182 — Computing Characteristic Polynomials for 3×3 Matrices
Expand to standard form: Lesson 2181 — Computing Characteristic Polynomials for 2×2 Matrices
Expanded form: means writing the number as the *sum of these place values*:; Lesson 5 — Expanded Form and Standard Form
Expanding and Regrouping: Lesson 2315 — Separable Equations with Algebraic Manipulation
Expanding/contracting shapes: A balloon's radius grows at 2 cm/s.; Lesson 1418 — Chain Rule Applications and Problem-Solving
expectation: or **mean** of a discrete random variable tells you the "average" value you'd expect if you could repeat the random experiment infinitely many times.; Lesson 2852 — Definition of Expectation (Mean) for Discrete Random Variables Lesson 2855 — Definition of Expectation for Continuous Random Variables Lesson 2861 — Higher Moments: Skewness and Kurtosis
expected value: (or mean) of a discrete random variable by taking a weighted average of all possible outcomes.; Lesson 2852 — Definition of Expectation (Mean) for Discrete Random Variables Lesson 2968 — Understanding Sample Means and Averages
explicit formula: Lesson 657 — Finding the Number of Terms in a Finite Sequence Lesson 661 — Finding Terms in a Geometric Sequence
Exploit orthogonality: If working in an inner product space, an orthonormal basis often simplifies computations (projection formulas become trivial, norms are preserved).; Lesson 2257 — Finding Convenient Bases
Explore: an unvisited neighbor *u* of *v*; Lesson 2673 — Finding Spanning Trees: Depth-First Search Lesson 2674 — Finding Spanning Trees: Breadth-First Search
exponent: tells you how many times to multiply a number by itself.; Lesson 132 — What is an Exponent?Lesson 137 — Working with Fractional Bases Lesson 140 — Integer Exponents in Context Lesson 310 — What is a Polynomial?Lesson 618 — Definition of Logarithm as Inverse of Exponential Lesson 1728 — Comparing Geometric and p-Series
Exponential: Lesson 650 — Comparing Linear, Exponential, and Logarithmic Growth
Exponential decay: happens when something is repeatedly divided or reduced by a constant factor.; Lesson 140 — Integer Exponents in Context Lesson 600 — What is an Exponential Function?Lesson 603 — Growth vs. Decay: The Role of the Base
Exponential distribution: Measures the *time between consecutive events*; Lesson 2902 — Relationship Between Exponential and Poisson
Exponential forcing: If **f**(t) = [2e³ᵗ, -e³ᵗ]ᵀ, guess **x_p** = **a**e³ ᵗ.; Lesson 2386 — Non-Homogeneous Systems: Undetermined Coefficients
exponential function: is a function where the variable appears in the *exponent*, not the base.; Lesson 600 — What is an Exponential Function?Lesson 668 — Geometric Sequences vs Exponential Functions
Exponential growth: happens when something multiplies by the same factor repeatedly.; Lesson 140 — Integer Exponents in Context Lesson 600 — What is an Exponential Function?Lesson 603 — Growth vs. Decay: The Role of the Base Lesson 650 — Comparing Linear, Exponential, and Logarithmic Growth
Exponential resonance: If your trial is *Ae^(rx)* and *e^(rx)* appears in the complementary solution, use *Axe^(rx)*.; Lesson 2354 — The Modification Rule for Resonance
Exponential terms: where the base contains n (like n^n or 2^n); Lesson 1747 — Comparing Ratio and Root Tests
Exponential(λ): Memoryless waiting time:; Lesson 2962 — MGFs of Common Distributions
Exponentials: Functions like f(x) = e ˣ or f(x) = 2ˣ are continuous on their entire domain (all real numbers).; Lesson 1354 — Continuity of Elementary Functions
Exponentials telescope: Expressions like `2^(n+1)/2^n` reduce to the base: `2`; Lesson 1748 — Series with Factorials and Exponentials
Exponents: – Evaluate powers (if present); Lesson 47 — Mixed Operations with Negative Numbers Lesson 513 — Mixed Operations with Radicals
Exponents next: `4² = 16`; Lesson 183 — Order of Operations in Evaluation
Exponents with negatives: `(-3)²` equals `9`, not `-9`; Lesson 184 — Evaluating with Negative Numbers
Express as prime powers: Write the result as a product of primes with their exponents.; Lesson 2725 — Prime Factorization: Finding the Decomposition
Express as sum/difference: 15° = 45° − 30°; Lesson 1055 — Evaluating Non-Standard Angles Using Sum Formulas
Express each variable: in terms of the free parameters; Lesson 2064 — Applications: Expressing Solutions as Linear Combinations
Express forces in components: Break each tension force into horizontal (i) and vertical (j) components using the angle each cable makes.; Lesson 1269 — Tension in Cables and Ropes
Express in one variable: using constraint equations (like a curve equation); Lesson 1512 — Distance Optimization Problems
Express pivot variables: in terms of free variables; Lesson 2130 — The Null Space
Expression inside a root: √(x² + 4) → u = x² + 4; Lesson 1580 — Identifying the Inner Function u
Extend that shape: parallel to the missing variable's axis (parallel to the z-axis); Lesson 1799 — Cylindrical Surfaces and Traces
Extend the pattern: using periodicity.; Lesson 1040 — Graphing with All Transformations Combined
Extend the perpendicular: the same distance on the other side; Lesson 927 — Reflections Across Arbitrary Lines
Extended Euclidean Algorithm: goes further: it finds integers *x* and *y* such that:; Lesson 2715 — Extended Euclidean Algorithm Lesson 2765 — Computing the Private Exponent d
Extending sides: Sometimes extending a line segment beyond its endpoint reveals exterior angles or creates useful intersections.; Lesson 769 — Auxiliary Lines and Constructions
Extending to a basis: means systematically adding more vectors (carefully chosen to remain independent) until you have enough vectors to span the whole space.; Lesson 2090 — Extending to a Basis
Extension: For any constants a and b:; Lesson 2945 — Properties of Independent Random Variables
exterior angle: .; Lesson 733 — Exterior Angle Theorem Lesson 810 — Exterior Angles of Polygons Lesson 811 — Exterior Angle Sum Theorem
Exterior Angle Sum: Always `360°` for any polygon; Lesson 813 — Using Angle Sums to Find Unknown Angles
Exterior Angle Theorem: states that an exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.; Lesson 733 — Exterior Angle Theorem
Extract: 5 · x² · y² = 5x²y²; Lesson 504 — Combining Simplification Techniques
Extract complete groups: from under the radical; Lesson 157 — Simplifying Higher-Order Radicals
Extract components: The columns of **V** are your principal components; Lesson 2289 — Principal Component Analysis (PCA) and SVD
Extract matrix properties: The polynomial immediately tells you:; Lesson 2188 — Using the Characteristic Polynomial in Practice
Extract the nth root: Take \(\sqrt[n]{|a_n|}\); Lesson 1746 — Applying the Root Test
extraneous: it appeared during solving but isn't a real solution.; Lesson 525 — Checking for Extraneous Solutions Lesson 637 — Checking for Extraneous Solutions
extraneous solutions: .; Lesson 481 — Checking for Extraneous Solutions Lesson 522 — Introduction to Radical Equations Lesson 524 — Solving Simple Radical Equations Lesson 637 — Checking for Extraneous Solutions Lesson 1091 — Extraneous Solutions and Verification
Extrapolation: means using your regression line to predict values outside the range of your original data.; Lesson 3071 — Cautions and Limitations
Extreme Value Theorem: guarantees that a continuous function on a closed and bounded region *must* have both an absolute maximum and an absolute minimum.; Lesson 1870 — Optimization on Closed and Bounded Regions Lesson 1871 — Global Extrema: Finding Absolute Maximum and Minimum
Extreme Value Theorem (EVT): guarantees that somewhere along your journey, you'll reach both a highest point (maximum elevation) and a lowest point (minimum elevation).; Lesson 1358 — The Extreme Value Theorem

F

F · n: captures how much of the field penetrates the surface (versus flowing parallel to it).; Lesson 2019 — Vector Fields and Surface Integrals
F · u: .; Lesson 1268 — Component of Force Along a Direction
F = 2: faces (the interior region and the exterior unbounded region); Lesson 2691 — Euler's Formula for Planar Graphs
F = 3: (the diagonal splits the interior into two regions); Lesson 2691 — Euler's Formula for Planar Graphs
F = s₁²/s₂²: Lesson 3047 — The F Distribution
f = λ g: Lesson 1875 — Parallel Gradients at Optimal Points Lesson 1876 — The Lagrange Multiplier λ
F ≤ 2E/3: Lesson 2692 — Consequences of Euler's Formula
F is NOT conservative: !; Lesson 1990 — Simply Connected Domains and Why They Matter
F statistic: is the ratio:; Lesson 3049 — One-Way ANOVA F Test Introduction
F test for variances: "We fail to reject the null hypothesis (F = 1.; Lesson 3050 — Interpreting Chi-Squared and F Test Results
F tests: shine when comparing variances (two samples) or means across multiple groups (ANOVA).; Lesson 3051 — Applications and Limitations
f_x: or **f_x(x,y)** — This is compact and convenient, especially when dealing with multiple partial derivatives.; Lesson 1823 — Notation for Partial Derivatives Lesson 1837 — The Total Differential
F-distribution: is a continuous probability distribution that emerges naturally when we compare the variability of two independent samples.; Lesson 2925 — The F-Distribution: Definition and Derivation Lesson 3096 — The F-Statistic and F- Distribution
F-statistic: answers this by forming a ratio:; Lesson 3096 — The F-Statistic and F-Distribution
F-test: specifically for the interaction term.; Lesson 3100 — Two-Way ANOVA With Interaction
f''(x): tells us about concavity:; Lesson 1500 — Inflection Points Lesson 1522 — Inflection Points
f''(x) < 0: means the function is concave down; Lesson 1500 — Inflection Points Lesson 1521 — Second Derivative and Concavity
f''(x) > 0: means the function is concave up; Lesson 1500 — Inflection Points Lesson 1521 — Second Derivative and Concavity
f'(x) = 0: Lesson 1388 — The Derivative of a Constant Function Lesson 1515 — Optimization on Open Intervals Lesson 1520 — Critical Points and Local Extrema
f(a) exists: the function has a defined value at *a*; Lesson 576 — Continuity of Functions Lesson 1350 — Definition of Continuity at a Point
f(g(x)): , we're composing two functions:; Lesson 1408 — Composition of Functions Review Lesson 1569 — Integrals with Variable Bounds
f(n): , where *n* represents the position number (1st, 2nd, 3rd, .; Lesson 1703 — Definition of a Sequence and Notation Lesson 1711 — Computing Limits Using L'Hôpital's Rule Lesson 2633 — Non-Homogeneous Recurrence Relations
f(x - ct): represents a wave traveling to the **right** (in the positive x-direction).; Lesson 2427 — D'Alembert's Solution Lesson 2428 — Initial Conditions and Wave Propagation
f(x, y): means "f is a function of the independent variables x and y.; Lesson 1802 — Introduction to Functions of Several Variables Lesson 1813 — Computing Limits Using Direct Substitution Lesson 1818 — Continuity on a Region Lesson 1836 — Linear Approximation in Two Variables Lesson 2306 — Existence and Uniqueness Theorems
f(x): , read it as "f of x.; Lesson 539 — Understanding Function Notation Lesson 1320 — Limit Notation and Reading Limits Lesson 1350 — Definition of Continuity at a Point Lesson 1379 — The Difference Quotient Lesson 1470 — Estimating Function Values with Linear Approximation Lesson 1666 — Revolving Around the y-axis Lesson 1693 — Work Done by a Variable Force Lesson 1711 — Computing Limits Using L'Hôpital's Rule (+2 more)
f(x) = e^x: .; Lesson 611 — The Natural Exponential Function f(x) = e^x Lesson 1438 — Higher-Order Derivatives of Exponential and Logarithmic Functions
F(x) = kx: (force proportional to displacement).; Lesson 1693 — Work Done by a Variable Force
f(x) = xⁿ: where n is a positive integer.; Lesson 1383 — Computing Derivatives from the Definition: Polynomial Functions Lesson 1394 — The Power Rule for Negative Integer Exponents
F₁: , **F₂**, **F₃**, etc.; Lesson 1264 — Equilibrium: Balancing Multiple Forces
F₂: , **F₃**, etc.; Lesson 1264 — Equilibrium: Balancing Multiple Forces
F₃: , etc.; Lesson 1264 — Equilibrium: Balancing Multiple Forces
fact family: is a group of related addition and subtraction facts that use the same three numbers.; Lesson 13 — The Relationship Between Addition and Subtraction Lesson 24 — The Relationship Between Multiplication and Division
factor: is a whole number that divides evenly into another number with no remainder.; Lesson 48 — Understanding Factors Lesson 353 — Using Division to Test Factors Lesson 355 — What is the Greatest Common Factor (GCF)?Lesson 385 — Factoring Perfect Square Trinomials Lesson 452 — Simplifying Before Multiplying Lesson 455 — Dividing and Simplifying in One Step Lesson 458 — Applications of Multiplying and Dividing Rationals Lesson 461 — Finding the Least Common Denominator (LCD) (+7 more)
Factor A effect: (main effect); Lesson 3100 — Two-Way ANOVA With Interaction
Factor all trinomials: in the numerator and denominator completely; Lesson 448 — Simplifying with Trinomial Factors
Factor and cancel: common terms that caused the 0/0; Lesson 1342 — Limits Involving Conjugates Lesson 1680 — Simplifying Arc Length Integrals
Factor and simplify: Write the left side as perfect squares and simplify the right side to find `r²`.; Lesson 1140 — Converting Between Circle Forms Lesson 1532 — Indeterminate Differences: ∞ - ∞
Factor B effect: (main effect); Lesson 3100 — Two-Way ANOVA With Interaction
Factor by grouping: Lesson 363 — The AC Method for General Trinomials
Factor completely: both the top and bottom; Lesson 445 — Simplifying by Canceling Common Factors Lesson 504 — Combining Simplification Techniques
Factor everything you can: Lesson 456 — Mixed Operations: Multiplication and Division Together
Factor it out: , placing it in front of parentheses; Lesson 360 — GCF with Four or More Terms Lesson 367 — Factoring Out a GCF Before Trinomial Factoring
Factor or combine fractions: to simplify; Lesson 1051 — Verifying Trigonometric Identities
Factor out coefficients: Pull out the coefficients from the squared terms; Lesson 1175 — Converting General Form to Standard Form
Factor out each GCF: from its respective group; Lesson 376 — Factoring Out the GCF from Each Group
Factor out negatives early: If you see **(a − b)** in one place and **(b − a)** in another, remember they're opposites: **(a − b) = −(b − a)**.; Lesson 457 — Multiplying and Dividing with Negative Signs
Factor the denominator: x² - 4 = (x + 2)(x - 2) [difference of squares]; Lesson 448 — Simplifying with Trinomial Factors
Factor the joint PDF: The joint probability density function f(x,y) must equal the product of the marginal PDFs: f(x,y) = fₓ(x) · f_Y(y); Lesson 2941 — Checking Independence for Continuous Random Variables
Factor the number: inside the radical (prime factorization helps!; Lesson 157 — Simplifying Higher-Order Radicals Lesson 415 — Simplifying Radical Answers
Factor the numerator: x² + 5x + 6 = (x + 2)(x + 3); Lesson 448 — Simplifying with Trinomial Factors
Factor the simpler trinomial: This factors to (u + 2)(u + 3); Lesson 372 — Factoring Higher-Degree Trinomials Using Substitution
Factor the trinomial inside: `x² + 4x + 3 = (x + 1)(x + 3)`; Lesson 367 — Factoring Out a GCF Before Trinomial Factoring
Factor Theorem: is a powerful shortcut that connects two concepts you already know: the Remainder Theorem and factoring.; Lesson 352 — The Factor Theorem
Factor using the pattern: (x + 4)(x - 4) = 0; Lesson 398 — Solving Special Forms: Difference of Squares
Factorial: n!; Lesson 2627 — What is a Recurrence Relation?
Factorials: (like n!; Lesson 1747 — Comparing Ratio and Root Tests
Factorials grow explosively fast: When you divide (n+1)!; Lesson 1743 — Applying the Ratio Test to Common Series
Factorials or exponentials: (n!; Lesson 1750 — Strategy: Choosing the Right Convergence Test
Factorials simplify beautifully: When you compute `a_(n+1)/a_n`, factorials like `(n+1)!; Lesson 1748 — Series with Factorials and Exponentials
Factoring: the trinomial into two binomials; Lesson 396 — Solving by Factoring: Simple Trinomials Lesson 1341 — Algebraic Manipulation for 0/0 Lesson 1366 — ε-δ Proofs for Quadratic Functions Lesson 2315 — Separable Equations with Algebraic Manipulation
factoring by grouping: which means grouping pairs of terms, factoring out the GCF from each pair, and then factoring out the common binomial factor.; Lesson 364 — Factoring by Grouping After Splitting the Middle Term Lesson 374 — Introduction to Factoring by Grouping
Factoring out: the GCF from each group separately; Lesson 374 — Introduction to Factoring by Grouping
Factoring works best when: Lesson 419 — Choosing Between Factoring and the Formula
Factors of 12: 1, 2, 3, 4, 6, 12 (limited list); Lesson 49 — Understanding Multiples
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24; Lesson 48 — Understanding Factors
fail to reject: H₀—not because you proved it's fair, but because you lack compelling evidence against it.; Lesson 3024 — The Logic of Hypothesis Testing Lesson 3030 — Interpreting Hypothesis Test Results
fail to reject H₀: (insufficient evidence to abandon the null hypothesis).; Lesson 3020 — Introduction to Hypothesis Testing Lesson 3024 — The Logic of Hypothesis Testing
Fails closure under addition: (1, 0) and (0, 1) are both in S, but their sum (1, 1) is not.; Lesson 2077 — Non-Examples and Verifying Vector Spaces
Failure: (often coded as 0); Lesson 2872 — The Bernoulli Trial and Bernoulli Distribution
Fall short: never reach the opposite side (no triangle); Lesson 1121 — Introduction to the Ambiguous Case (SSA)
false: .; Lesson 212 — Equations with No Solution Lesson 300 — Graphing Linear Inequalities in Two Variables Lesson 525 — Checking for Extraneous Solutions Lesson 2468 — Propositions and Truth Values Lesson 2469 — Logical Negation Lesson 2470 — Conjunction (AND)Lesson 2471 — Disjunction (OR)Lesson 2473 — Biconditional Statements (If and Only If) (+1 more)
Fan shape (expanding): Residuals spread out as fitted values increase—the "megaphone" pattern; Lesson 3085 — Detecting Non-Constant Variance (Heteroscedasticity)
Fast coordinate computation: via inner products; Lesson 2211 — Orthogonal Bases
feasible region: where all solutions satisfy every inequality at once.; Lesson 301 — Systems of Linear Inequalities Lesson 302 — Vertices of the Feasible Region Lesson 303 — Bounded vs Unbounded Feasible Regions Lesson 304 — Introduction to Linear Programming
Fermat primes: have the form \( F_n = 2^{2^n} + 1 \).; Lesson 2729 — Special Primes: Mersenne, Fermat, and Twin Primes
Fermat's Little Theorem: works when your modulus is **prime** (p): `a^(p-1) ≡ 1 (mod p)`; Lesson 2751 — Problem Solving with Fermat's and Euler's Theorems
Fewer assumptions: Less worry about violating test conditions; Lesson 3103 — Introduction to Nonparametric Methods
Fibonacci sequence: Lesson 2627 — What is a Recurrence Relation?
Fifteen: = ten + five; Lesson 2 — Counting from 11 to 100
Fifth derivative: f ⁵ (x) = 0 (and all higher derivatives are 0); Lesson 1397 — Higher-Order Derivatives
Fifth root: ⁵√32 = 2 because 2 × 2 × 2 × 2 × 2 = 32; Lesson 153 — Cube Roots and Higher-Order Roots
Filling containers: "How many liters fill a cylindrical tank 2m tall with radius 0.; Lesson 884 — Volume Word Problems and Applications
Final answer: `3(x + 1)(x + 3)`; Lesson 367 — Factoring Out a GCF Before Trinomial Factoring Lesson 585 — Writing Transformed Functions from Descriptions Lesson 1202 — Simplifying Complex Expressions Lesson 1338 — Combining Multiple Limit Laws Lesson 1411 — The Outside-Inside Method
Final answer: 54: Lesson 183 — Order of Operations in Evaluation
Final calculation: (7 × 6 × 5) / 6 = 210 / 6 = **35**; Lesson 2590 — Computing Basic Combinations
Final Formula: Lesson 1259 — Vector Projection Formula
Final numerical answers: – "The ladder is approximately 8.; Lesson 521 — Applications and When Not to Rationalize
Final statement: (if needed): Use CPCTC for specific parts; Lesson 748 — Writing Two-Column Congruence Proofs
Finally add: 18 + 4 = 22; Lesson 33 — Exponents in Order of Operations
find: it.; Lesson 356 — Finding the GCF of Numbers Lesson 778 — Word Problems Involving Right Triangles Lesson 1007 — Negative Angles and Trigonometric Values Lesson 1210 — Argument of a Complex Number Lesson 1344 — The Squeeze Theorem Lesson 1399 — The Product Rule Formula Lesson 1489 — Finding the c Guaranteed by the MVT Lesson 2274 — Eigenspaces and Multiplicity for Symmetric Matrices (+2 more)
Find A ⁻¹: Use the formula for 2×2 matrices or row reduction for larger ones.; Lesson 1307 — Using Inverses to Solve Matrix Equations
Find a trig relationship: connecting θ to the changing quantities; Lesson 1483 — Angle Related Rates
Find A(y): , the area of a horizontal slice at height y; Lesson 1695 — Pumping Liquids from Tanks
Find all critical points: in the open interval (a, b); Lesson 1508 — The Closed Interval Method Lesson 1515 — Optimization on Open Intervals Lesson 1871 — Global Extrema: Finding Absolute Maximum and Minimum
Find all eigenvalues: of the **n×n** matrix; Lesson 2190 — The Diagonalization Theorem
Find all intersection points: between the curves in your interval `[a, b]`; Lesson 1647 — Area with Multiple Regions Lesson 1650 — Area Enclosed by Intersecting Curves
Find all n roots: Substitute each value of k to get all solutions; Lesson 1232 — Solving Equations of the Form z^n = w
Find all solutions: for that compressed period; Lesson 1084 — Equations with Multiple Angles
Find angle B: Lesson 1094 — Solving AAS Triangles
Find bounds: Solve for intersection points or use given limits; Lesson 1652 — Applied Problems: Area Between Curves
Find common denominators: when adding or subtracting fractions; Lesson 185 — Evaluating Expressions with Fractions
Find constraints: – What relationships or limits exist between variables?; Lesson 1872 — Applications: Optimization Problems in Two Variables
Find corner points: Calculate coordinates of each vertex; Lesson 308 — Applications: Maximization Problems Lesson 309 — Applications: Minimization Problems
Find corresponding y-values: (if needed): Plug each x back into either original function; Lesson 1646 — Finding Intersection Points
Find critical points: Take the derivative of the perimeter function and set it equal to zero; Lesson 1510 — Geometric Optimization: Minimizing Perimeter Lesson 1511 — Box and Container Optimization Lesson 1512 — Distance Optimization Problems Lesson 1513 — Cost and Revenue Optimization Lesson 1514 — Optimization with Implicit Constraints Lesson 1867 — Applying the Second Derivative Test Lesson 1872 — Applications: Optimization Problems in Two Variables
Find degrees of freedom: df = n - 1; Lesson 3034 — One-Sample t-Test: Testing a Population Mean Lesson 3043 — Chi-Squared Goodness-of- Fit Test
Find du: Differentiate your choice of u; Lesson 1585 — u-Substitution with Exponential and Logarithmic Functions Lesson 1586 — Definite Integrals and Changing Limits
Find dx: dx = a cos θ dθ; Lesson 1613 — Substitution for √(a²-x²): Using x = a sin θ
Find dy/dx: using implicit differentiation (you've already learned this); Lesson 1455 — Second Derivatives by Implicit Differentiation
Find each derivative separately: Lesson 1400 — Applying the Product Rule to Polynomials
Find eigenvalues and eigenvectors: of the symmetric matrix A; Lesson 2277 — Principal Axes and Quadratic Forms Lesson 2296 — Diagonalization of Quadratic Forms
Find eigenvectors: Once you have eigenvalues, substitute each back into (A - λI)v = 0.; Lesson 2188 — Using the Characteristic Polynomial in Practice
Find f''(x): Compute the second derivative of your function; Lesson 1499 — Finding Intervals of Concavity Lesson 1521 — Second Derivative and Concavity Lesson 1522 — Inflection Points
Find f(x + h): Lesson 1382 — Computing Derivatives from the Definition: Quadratic Functions
Find intersection points: by setting the equations equal and solving:; Lesson 1656 — Finding Limits of Integration for Disk Method
Find its total area: using formulas you already know; Lesson 869 — Finding Area Using Subtraction Method
Find M: Determine the maximum of the (n+1)-th derivative on your interval; Lesson 1787 — Determining Required Degree for Given Accuracy
Find missing measurements first: before calculating area or perimeter; Lesson 866 — Perimeter and Area Problem Solving
Find p: From `4p`, calculate `p`.; Lesson 1151 — Graphing Parabolas from Standard Form
Find R(y): This is the horizontal distance from the y-axis to your curve — simply the x-value, so R(y) = x = f(y); Lesson 1655 — Disk Method: Rotation About the y-axis
Find s: s = (7 + 8 + 9) / 2 = 24 / 2 = 12; Lesson 1116 — Applying Heron's Formula
Find slope: m = (13 - 5)/(6 - 2) = 8/4 = 2; Lesson 259 — Writing Equations from Two Points
Find the area: of each individual shape using the formulas you've learned; Lesson 868 — Finding Area of Composite Figures by Addition
Find the argument (θ): This is the angle the point makes with the positive real axis.; Lesson 1215 — Converting from Rectangular to Polar Form
Find the base area: Use the appropriate area formula for that shape; Lesson 878 — Volume of Prisms with Any Base Lesson 889 — Surface Area of Pyramids
Find the bounds: What are the starting and ending points of your curve?; Lesson 1691 — Applications: Real-World Surfaces
Find the center point: Halfway between asymptotes, cotangent crosses through the midline at y = D; Lesson 1027 — Transformations of Cotangent
Find the change: Subtract the original value from the new value (ignore if negative for now); Lesson 101 — Percent Increase and Decrease
Find the constant: using given information (`k = xy`); Lesson 130 — Real-World Inverse Variation Problems
Find the critical value(s): under the null hypothesis that define your rejection region; Lesson 3058 — Calculating Power for Simple Tests
Find the derivative: f'(x) to get the instantaneous rate at any point.; Lesson 1489 — Finding the c Guaranteed by the MVT Lesson 1507 — Critical Points in Optimization Lesson 1676 — Computing Arc Length: Polynomial Functions
Find the eigenvectors: for each eigenvalue (build the eigenspaces); Lesson 2190 — The Diagonalization Theorem
Find the first angle: using the Law of Sines; Lesson 1124 — Solving for Two Possible Triangles
Find the first variable: (often z) from a simplified equation like `2z = 6`, giving you `z = 3`.; Lesson 296 — Back-Substitution in Three Variables
Find the gateway element: Often you'll use one law to find just *one* missing piece—the piece that unlocks the ability to use the other law.; Lesson 1111 — Complex Problems Combining Triangle Laws
Find the GCF: of the numerator and denominator; Lesson 65 — Simplifying Fractions Lesson 358 — Factoring Out the GCF from Polynomials Lesson 367 — Factoring Out a GCF Before Trinomial Factoring
Find the general antiderivative: with the constant C; Lesson 1544 — Initial Value Problems
Find the gradient: ∇F = ⟨F_x, F_y, F_z ⟩ at your point (x₀, y₀, z₀); Lesson 1835 — Normal Lines to Surfaces
Find the height: This is the perpendicular distance between the two bases; Lesson 878 — Volume of Prisms with Any Base
Find the hole: Set the canceled factor to zero: `x - 3 = 0`, so `x = 3`; Lesson 487 — Holes in Rational Functions
Find the horizontal line: – Imagine a flat line at eye level; Lesson 967 — Identifying Angles in Real-World Scenarios
Find the innermost limits: (may depend on both outer variables); Lesson 1928 — Changing the Order of Integration in Triple Integrals
Find the interior angle: of the triangle using bearing differences; Lesson 1101 — Law of Sines with Bearing and Direction
Find the intersection point: where the two lines cross; Lesson 274 — Solving Systems by Graphing: Basic Technique Lesson 927 — Reflections Across Arbitrary Lines
Find the lateral area: Imagine "unrolling" the side into a rectangle.; Lesson 888 — Surface Area of Cylinders
Find the LCD: The LCD of 4 and 6 is 12; Lesson 73 — Subtracting Fractions with Unlike Denominators Lesson 209 — Equations with Fractions: Clearing Denominators Lesson 463 — Adding Rationals with Unlike Denominators Lesson 464 — Subtracting Rationals with Unlike Denominators Lesson 467 — Adding and Subtracting Three or More Rationals Lesson 470 — Simplifying Complex Fractions by Multiplying by the LCD Lesson 480 — Solving Equations with Multiple Terms
Find the limit: Compute \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|}\); Lesson 1746 — Applying the Root Test
Find the lowest power: of each variable that appears in *every* term; Lesson 360 — GCF with Four or More Terms
Find the magnitude: of your vector using the formula you learned: ||**v**|| = √(v₁² + v₂² + v₃²); Lesson 1249 — Unit Vectors and Normalization Lesson 1849 — Directional Derivatives in Non-Unit Directions
Find the marginal PMF: $p_X(x)$ by summing the joint PMF over all $y$ values; Lesson 2930 — Conditional Distributions: Discrete Case
Find the midpoint: of each subinterval.; Lesson 1549 — Midpoint Riemann Sums
Find the minor: Cover up the element's row and column.; Lesson 2150 — Determinant of a 3×3 Matrix: Cofactor Expansion
Find the modulus (r): This is the distance from the origin to your point.; Lesson 1215 — Converting from Rectangular to Polar Form
Find the new limits: for the middle variable (may depend on the outer); Lesson 1928 — Changing the Order of Integration in Triple Integrals
Find the normal vector: to your parametric surface (using the cross product of partial derivatives); Lesson 2021 — Computing Flux for Parametric Surfaces
Find the observer: – Who or what is doing the looking?; Lesson 967 — Identifying Angles in Real-World Scenarios
Find the partial derivatives: ∂f/∂x and ∂f/∂y; Lesson 1853 — Computing Derivatives along Parametric Curves
Find the period: π/|B|; Lesson 1027 — Transformations of Cotangent
Find the prime factorization: of the number under the radical; Lesson 155 — Simplifying Square Roots
Find the radial limits: For each fixed angle θ, determine where the region starts (inner radius r₁) and ends (outer radius r₂).; Lesson 1917 — Integrating over Sectors and Wedges
Find the radius: from the center to that point; Lesson 1143 — Tangent Lines to Circles
Find the reciprocal: of the bottom fraction; Lesson 80 — Complex Fractions Lesson 537 — Solving Equations with Fractional Exponents
Find the reference angle: the acute angle formed between the terminal side and the nearest x-axis; Lesson 995 — Second Quadrant Points Lesson 997 — Fourth Quadrant Points
Find the reference solution(s): in the first period; Lesson 1082 — Finding All Solutions in an Interval
Find the relevant dimensions: (length, width, height, radius, slant height); Lesson 895 — Word Problems Involving Surface Area
Find the rise: How much did the y-coordinate change?; Lesson 244 — Rise Over Run: Understanding Slope Geometrically
Find the run: How much did the x-coordinate change?; Lesson 244 — Rise Over Run: Understanding Slope Geometrically
Find the sector area: using the central angle θ (in degrees or radians):; Lesson 843 — Segment Area Lesson 865 — Area of Segments
Find the slope: of the radius; Lesson 1143 — Tangent Lines to Circles
Find the supplementary angle: If B₁ is your first answer, calculate B₂ = 180° - B₁; Lesson 1124 — Solving for Two Possible Triangles
Find the transversal: crossing them; Lesson 715 — Alternate Interior Angles
Find the tree's shadow: Measure how long the tree's shadow is at the same time of day; Lesson 761 — Applications and Proofs with Similar Triangles
Find the triangle area: formed by the two radii and the chord:; Lesson 843 — Segment Area Lesson 865 — Area of Segments
Find the width: of the surface at depth y: w(y); Lesson 1699 — Fluid Pressure and Force on Submerged Surfaces
Find the x-coordinate: Look straight down (or up) from the point to the x-axis.; Lesson 240 — Reading Coordinates from a Graph
Find the x-intercept: Lesson 262 — Graphing Lines in Standard Form
Find the x-value: that makes the canceled factor equal zero—that's where the hole is located; Lesson 487 — Holes in Rational Functions
Find the y-coordinate: Look straight left (or right) from the point to the y-axis.; Lesson 240 — Reading Coordinates from a Graph
Find the y-intercept: Lesson 262 — Graphing Lines in Standard Form
Find triangles: Show those parts belong to two triangles; Lesson 747 — CPCTC: Corresponding Parts of Congruent Triangles
Find two numbers that: Lesson 363 — The AC Method for General Trinomials
Find two relationships: – Read carefully to spot two separate facts connecting your variables; Lesson 282 — Application Problems Using Systems and Substitution
Find where you land: on the circle; Lesson 1010 — Using the Unit Circle to Evaluate Trig Functions
Find x = a: on the horizontal axis; Lesson 1322 — Estimating Limits from Graphs
Find y-coordinate by substitution: f(2) = 2(2)² - 8(2) + 3 = 8 - 16 + 3 = -5; Lesson 429 — Finding the Vertex Using x = -b/(2a)
Find your reference angle: This is the acute angle between the terminal side and the nearest x-axis; Lesson 996 — Third Quadrant Points
Finding ∂f/∂x: Lesson 1824 — Computing First-Order Partial Derivatives
Finding ∂f/∂y: Lesson 1824 — Computing First-Order Partial Derivatives
Finding a side (SAS): If your known angle is obtuse, expect the opposite side to be longest; Lesson 1107 — Law of Cosines with Obtuse Angles
Finding an angle (SSS): If your calculation yields a negative cosine value, that angle is obtuse; Lesson 1107 — Law of Cosines with Obtuse Angles
Finding extremal dimensions: is common in geometry.; Lesson 1882 — Applications: Economics, Geometry, and Physics
Finding missing values: If you know two of V, E, F, you can calculate the third; Lesson 2691 — Euler's Formula for Planar Graphs
Finding p: Divide the coefficient of y by 4; Lesson 1147 — Finding the Focus and Directrix from Equations
Finding the parts: If angle PQR = 120° and QS bisects it, then angle PQS = angle SQR = 60°.; Lesson 710 — Angle Bisectors
Finding the whole: If ray BD bisects angle ABC and angle ABD = 35°, then angle ABC = 2 × 35° = 70°.; Lesson 710 — Angle Bisectors
Finding total height: Measure angles to the top and bottom of a building from the same spot.; Lesson 980 — Two Right Triangle Problems and Indirect Measurement
Finding variances: Var(g(X) + h(Y)) = Var(g(X)) + Var(h(Y)); Lesson 2946 — Independence and Functions of Random Variables
Finite number of extrema: f has finitely many peaks and valleys; Lesson 2460 — Convergence of Fourier Series
first: , you need to understand the *order* in which operations will be performed—just like reading a recipe before cooking.; Lesson 177 — Order of Operations with Variables Lesson 326 — Multiplying Two Binomials Using FOIL Lesson 335 — Square of a Binomial Lesson 558 — Function Composition Basics Lesson 929 — Composition of Translations and Reflections
First column: Use row operations to get a 1 in position (1,1) and zeros below it; Lesson 1305 — Row Reduction to Find Inverses of 3×3 Matrices
First derivative: f'(x) = 4x³ - 6x; Lesson 1397 — Higher-Order Derivatives Lesson 1504 — Applications: Real-World Monotonicity and Concavity
First Derivative Test: says: look at how f'(x) changes sign as you pass through a critical point.; Lesson 1496 — The First Derivative Test for Local Extrema Lesson 1501 — The Second Derivative Test for Local Extrema Lesson 1502 — Comparing First and Second Derivative Tests Lesson 1520 — Critical Points and Local Extrema Lesson 1523 — The Second Derivative Test for Extrema Lesson 1525 — Comprehensive Curve Sketching Algorithm
First draw: P(Red₁) = 5/8; Lesson 2797 — Conditional Probability in Sampling Without Replacement
first factor: is simple: **(a + b)** or **(a - b)** — it matches the sign between the cubes; Lesson 339 — Sum and Difference of Cubes Lesson 1338 — Combining Multiple Limit Laws
First group: `(6x + 9)`; Lesson 376 — Factoring Out the GCF from Each Group
First leading principal minor: (top-left 1×1): det([2]) = 2 > 0; Lesson 2294 — Principal Minors and Sylvester's Criterion
First moment (mean): Lesson 2961 — Finding Moments from MGFs
First moment about x-axis: $$M_x = \iint_R y \cdot \rho(x,y) \, dA$$; Lesson 1958 — Center of Mass in Two Dimensions
First moment about y-axis: $$M_y = \iint_R x \cdot \rho(x,y) \, dA$$; Lesson 1958 — Center of Mass in Two Dimensions
first moments: quantify how mass is distributed relative to coordinate axes using double or triple integrals.; Lesson 1957 — Moments and First Moments Lesson 1958 — Center of Mass in Two Dimensions Lesson 1959 — Center of Mass in Three Dimensions Lesson 1960 — Moments of Inertia
first number: (x) is called the **x-coordinate** – it shows horizontal position; Lesson 236 — Understanding Ordered Pairs and Coordinates Lesson 1221 — Multiplying Complex Numbers in Polar Form
first position: , you have **n choices** (any of the n books); Lesson 2579 — Counting Permutations of n Distinct Objects Lesson 2580 — Permutations of n Objects Taken r at a Time
First quartile: Q₁ (25th percentile): Solve F(x) = 0.; Lesson 2849 — Percentiles and Quantiles from the CDF
First relationship: "sum is 50" → x + y = 50; Lesson 282 — Application Problems Using Systems and Substitution
First rotation: (or reflection): The matrix **V^T** rotates the input space, aligning it with special directions; Lesson 2281 — Geometric Interpretation of SVD
first term: is always `x²` (the square of the first term); Lesson 401 — Perfect Square Trinomials Review Lesson 657 — Finding the Number of Terms in a Finite Sequence Lesson 663 — Writing Formulas for Geometric Sequences Lesson 693 — Stacking and Construction Problems Lesson 1542 — Antiderivatives of Polynomial Functions
First term times everything: Multiply *x* by each of the three terms: *x·x*, *x·1*, *x·4*; Lesson 329 — Multiplying Two Trinomials
First two, last two: (term₁ + term₂) + (term₃ + term₄); Lesson 375 — Grouping Terms in Pairs
first-order linear ODE: with integrating factor μ(t) = e^(rt/V).; Lesson 2326 — Applications: Mixing Problems Lesson 2449 — Solving the Temporal ODE
First-order PDE: Contains only first partial derivatives like ∂u/∂x or ∂u/∂t; Lesson 2409 — Classification: Order and Linearity
First-stage branches: show all possible outcomes and their probabilities; Lesson 2796 — Sequential Experiments and Tree Diagrams
five equations: Lesson 1884 — The Method with Two Constraints Lesson 1885 — Solving Systems with Multiple Lagrange Multipliers
Fixed (u = 0): Forces antisymmetric continuation → inverted reflection; Lesson 2429 — Wave Reflections at Boundaries
Fixing x: Set \(x = k\) to get a curve in the yz-plane; Lesson 1809 — Traces and Cross-Sections
Fixing y: Set \(y = k\) to get a curve in the xz-plane; Lesson 1809 — Traces and Cross-Sections
Fixing z: Set \(z = k\) (a constant) to get level curves in the xy-plane; Lesson 1809 — Traces and Cross-Sections
Flaws: in fabric per square meter; Lesson 2893 — Applications of the Poisson Distribution
flexibility: recognize what the expression needs.; Lesson 381 — Applications and Mixed Practice Lesson 3103 — Introduction to Nonparametric Methods
Flexible: (invariance lets you estimate transformed parameters freely); Lesson 3007 — Properties and Advantages of MLEs
Flip: the second fraction (the divisor); Lesson 455 — Dividing and Simplifying in One Step
Flip it: Write the reciprocal → 1/5²; Lesson 136 — Evaluating Negative Exponents
Flipping a coin: Lesson 2772 — Sample Spaces: The Set of All Possible Outcomes
floor function: ) takes any real number and returns the largest integer that is less than or equal to that number.; Lesson 595 — The Greatest Integer (Floor) Function Lesson 596 — The Ceiling and Rounding Functions
Floor Plans: An L-shaped apartment might break into two rectangles; Lesson 873 — Composite Figures in Real-World Contexts
Flow conservation: At every intermediate vertex v (neither source nor sink), the total flow entering v equals the total flow leaving v.; Lesson 2704 — Networks and Flows: Definitions and Notation
Flow f(u,v): The actual amount currently flowing along edge (u,v); Lesson 2704 — Networks and Flows: Definitions and Notation
Fluency: means answering quickly and accurately without needing to count or calculate slowly.; Lesson 25 — Division Facts and Basic Division Fluency
fluid dynamics: , the circulation form of Green's Theorem relates the microscopic rotation of fluid particles (curl) to the net circulation around a boundary.; Lesson 2003 — Applications and Extensions of Green's Theorem Lesson 2045 — Combining with Stokes' Theorem and Applications
Fluid flow: Field lines show particle trajectories in the velocity field; Lesson 1971 — Field Lines and Flow Lines Lesson 2408 — What is a Partial Differential Equation?Lesson 2436 — Introduction to Laplace's Equation
flux: of **F** through *S*.; Lesson 2019 — Vector Fields and Surface Integrals Lesson 2020 — Flux Through a Surface: Definition and Setup Lesson 2036 — Introduction to the Divergence Theorem
Focal Width = |4p|: Lesson 1153 — The Latus Rectum and Focal Width
foci: ) is constant.; Lesson 1130 — Geometric Definitions of Conics Lesson 1156 — Parts of an Ellipse: Vertices, Co-vertices, and Foci Lesson 1164 — Eccentricity of an Ellipse Lesson 1167 — Definition and Standard Form of a Hyperbola Lesson 1168 — Components of a Hyperbola: Center, Vertices, and Foci Lesson 1174 — Hyperbolas Centered at (h, k)
focus: ) and a fixed line (the **directrix**).; Lesson 1130 — Geometric Definitions of Conics Lesson 1146 — Standard Form: Vertex at Origin, Horizontal Axis Lesson 1147 — Finding the Focus and Directrix from Equations Lesson 1148 — Parabolas with Vertex at (h, k): Vertical Axis Lesson 1149 — Parabolas with Vertex at (h, k): Horizontal Axis Lesson 1153 — The Latus Rectum and Focal Width Lesson 1154 — Applications: Reflective Properties and Real-World Uses
Focus and directrix: You can find the vertex (halfway between them) and go from there; Lesson 1152 — Finding Equations from Geometric Information
FOIL: is a memory trick that helps you remember which terms to multiply:; Lesson 326 — Multiplying Two Binomials Using FOIL Lesson 369 — Checking Factorization by Multiplying
Folium of Descartes: , defined by:; Lesson 1457 — Applications and Complex Implicit Curves
Follow order: Each step must logically follow from previous statements; Lesson 748 — Writing Two-Column Congruence Proofs
Follow order of operations: just like with whole numbers—don't skip steps!; Lesson 185 — Evaluating Expressions with Fractions Lesson 540 — Evaluating Functions with Numeric Inputs
Follow the logical consequences: of that assumption; Lesson 2497 — Proof by Contradiction: Basic Idea
For (a - b)³: Lesson 338 — Cube of a Binomial
For (a + b)³: Lesson 338 — Cube of a Binomial
For ∫arcsin(x) dx: Lesson 1598 — Inverse Trigonometric Function Integrals
For ∫arctan(x) dx: Lesson 1598 — Inverse Trigonometric Function Integrals
For a: Multiply both sides by cos(mx) and integrate.; Lesson 2458 — Fourier Coefficients: The Formulas
For a binary tree: , you might prove:; Lesson 2516 — Structural Induction on Trees and Graphs
For a square: (where L = W = s): P = 4s and A = s²; Lesson 1478 — Related Rates with Rectangles and Squares
For a₀: Integrate both sides of the series from -π to π.; Lesson 2458 — Fourier Coefficients: The Formulas
For b: Similarly, multiply by sin(mx) and integrate:; Lesson 2458 — Fourier Coefficients: The Formulas
For Bijectivity: Lesson 2567 — Composition and Function Properties
For categorical data: Lesson 2997 — Choosing Appropriate Summaries and Displays
For continuous \(X\): Lesson 2935 — The Law of Total Probability for Random Variables
For continuous random variables: Identify the interval over which the variable can take values.; Lesson 2841 — Finding PMFs and PDFs from Problem Statements
For continuous X: Lesson 2854 — Expectation of Functions of Random Variables Lesson 2960 — Moment Generating Functions: Definition
For Convergence: If 0 ≤ a ≤ b for all n (beyond some point), and ∑b converges, then ∑a also converges.; Lesson 1734 — The Direct Comparison Test: Basic Principle
For cos(x): Lesson 1776 — Maclaurin Series for sin(x) and cos(x)
For cosine: cos(A - B) = cos A cos B + sin A sin B; Lesson 1059 — Sum and Difference Formulas in Equation Solving
For cosine functions: Lesson 1076 — Sum-to-Product Formulas
For cosine times cosine: Lesson 1058 — Simplifying Products Using Sum Formulas
For cosine times sine: Lesson 1058 — Simplifying Products Using Sum Formulas
For cube roots (³√): Multiply to create a perfect cube; Lesson 516 — Rationalizing with Higher-Index Roots
For curl: Lesson 2010 — Properties of Divergence and Curl
For decimals: Multiply each equation by a power of 10 (like 10, 100, or 1000) to shift all decimal points to the right.; Lesson 289 — Elimination with Fractions and Decimals
For directed graphs (digraphs): Lesson 2655 — Graph Representations: Adjacency Matrix
For discrete \(X\): Lesson 2935 — The Law of Total Probability for Random Variables
For discrete random variables: Read carefully to identify each possible outcome value, then determine the probability of each using counting techniques, basic probability rules, conditional probability, or symmetry arguments.; Lesson 2841 — Finding PMFs and PDFs from Problem Statements
For discrete X: Lesson 2854 — Expectation of Functions of Random Variables Lesson 2960 — Moment Generating Functions: Definition
For Divergence: If 0 ≤ b ≤ a for all n (beyond some point), and ∑b diverges, then ∑a also diverges.; Lesson 1734 — The Direct Comparison Test: Basic Principle Lesson 2010 — Properties of Divergence and Curl
For double angles: Lesson 1089 — Equations with Double or Half Angles
For each subsequent vector: , remove the parts that point in directions already covered:; Lesson 2220 — The Gram-Schmidt Algorithm: Overview
For Fibonacci: `r² - r - 1 = 0`, which solves to `r = (1±√5)/2`.; Lesson 2629 — Linear Homogeneous Recurrence Relations
For fourth roots (⁴√): Multiply to create a perfect fourth power; Lesson 516 — Rationalizing with Higher-Index Roots
For fractions: Multiply each equation by the least common denominator (LCD) of all fractions in that equation.; Lesson 289 — Elimination with Fractions and Decimals
For graphs: , you might build them by adding edges or vertices, proving that if a property holds before the addition, it holds after.; Lesson 2516 — Structural Induction on Trees and Graphs
For half angles: Lesson 1089 — Equations with Double or Half Angles
For horizontal distance: When two points share the same y-coordinate (same height), subtract their x-coordinates and take the absolute value.; Lesson 243 — Distance and the Coordinate Plane
For horizontal tangents: Lesson 1456 — Horizontal and Vertical Tangents Lesson 1467 — Horizontal and Vertical Tangent Lines
For Injectivity (one-to-one): Lesson 2567 — Composition and Function Properties
For large numbers: The exponent is positive.; Lesson 162 — What is Scientific Notation?
For mixture problems: If you mix 2 gallons of 20% juice with 3 gallons of 50% juice:; Lesson 194 — Money and Mixture Problems (Introduction)
For money problems: If you have 5 dimes and 3 quarters, the total value is:; Lesson 194 — Money and Mixture Problems (Introduction)
For products: Lesson 512 — Simplifying Products and Quotients of Radicals
For quantitative data: Lesson 2997 — Choosing Appropriate Summaries and Displays
For quotients: Lesson 512 — Simplifying Products and Quotients of Radicals
For sin(x): Lesson 1776 — Maclaurin Series for sin(x) and cos(x)
For sine: sin(A + B) = sin A cos B + cos A sin B; Lesson 1059 — Sum and Difference Formulas in Equation Solving
For sine functions: Lesson 1076 — Sum-to-Product Formulas
For sine times cosine: Lesson 1058 — Simplifying Products Using Sum Formulas
For sine times sine: Lesson 1058 — Simplifying Products Using Sum Formulas
For small numbers: The exponent is negative (remember negative exponents mean division!; Lesson 162 — What is Scientific Notation?
For Surjectivity (onto): Lesson 2567 — Composition and Function Properties
For tangent: tan(A + B) = (tan A + tan B)/(1 - tan A tan B); Lesson 1059 — Sum and Difference Formulas in Equation Solving
For the disk method: If your function is f(x) and you're rotating around y = k, the radius becomes the *vertical distance* from the line y = k to the curve: **R = |f(x) - k|**; Lesson 1661 — Rotation About Horizontal Lines y = k Lesson 1662 — Rotation About Vertical Lines x = h
For the GCD: Take the **minimum** exponent of each prime.; Lesson 2727 — Applications: GCD and LCM via Prime Factorization
For the LCM: Take the **maximum** exponent of each prime.; Lesson 2727 — Applications: GCD and LCM via Prime Factorization
For the washer method: You need both outer and inner radii measured from y = k:; Lesson 1661 — Rotation About Horizontal Lines y = k Lesson 1662 — Rotation About Vertical Lines x = h
For vertical distance: When two points share the same x-coordinate (same vertical line), subtract their y-coordinates and take the absolute value.; Lesson 243 — Distance and the Coordinate Plane
For vertical tangents: Lesson 1456 — Horizontal and Vertical Tangents Lesson 1467 — Horizontal and Vertical Tangent Lines
For θ: Determine rotational coverage.; Lesson 1941 — Setting Up Triple Integrals in Spherical Coordinates
For ρ: Ask "What's the inner and outer boundary of the region?; Lesson 1941 — Setting Up Triple Integrals in Spherical Coordinates
For φ: Identify the cone or angular constraint.; Lesson 1941 — Setting Up Triple Integrals in Spherical Coordinates
Force: 10 Newtons pulling at 45° upward; Lesson 1235 — Introduction to Vectors: Magnitude and Direction Lesson 1263 — Force as a Vector: Magnitude and Direction
Forced harmonic motion: occurs when we add a periodic driving term to our differential equation:; Lesson 2371 — Forced Harmonic Motion and Resonance
Forgetting domain restrictions: A negative value *anywhere* disqualifies the function; Lesson 2842 — Verifying Valid PMFs and PDFs
Forgetting parentheses: around negative substitutions; Lesson 184 — Evaluating with Negative Numbers
Forgetting the Original Limit: Lesson 1534 — Common Mistakes and When Not to Use L'Hôpital's Rule
Forgetting the Parentheses: Lesson 547 — Common Function Notation Mistakes
Forgetting the Second Solution: Lesson 1128 — Common Errors and Verification in SSA
Form: ∂u/∂t = k ∂²u/∂x²; Lesson 2410 — The Three Classical PDEs
Form 1: is great when your problem *already involves both sine and cosine* of the same angle.; Lesson 1064 — Double Angle Formulas for Cosine
Form 1 (Standard): Lesson 1070 — Half Angle Formula for Tangent
Form 2: works best when you only have information about *cosine* or when simplifying expressions with cosine terms.; Lesson 1064 — Double Angle Formulas for Cosine
Form 2 (Non-radical): Lesson 1070 — Half Angle Formula for Tangent
Form 3: is perfect when you only know about *sine* or when your expression features sine terms.; Lesson 1064 — Double Angle Formulas for Cosine
Form A - λI: by subtracting λ from each diagonal entry of A; Lesson 2178 — Computing Eigenvalues for 3×3 Matrices Lesson 2182 — Computing Characteristic Polynomials for 3×3 Matrices
Form A = U Σ V ᵀ: , using only the first k columns of U and V; Lesson 2288 — Low-Rank Approximation and Data Compression
Form a straight line: The non-shared sides of the angles create a straight line (180°); Lesson 708 — Linear Pairs
Form an augmented matrix: Place the C-basis vectors as columns on the left, and the B-basis vectors as columns on the right: `[C | B]`; Lesson 2252 — Computing Transition Matrices
Form matrix A: Place each vector as a column; Lesson 2069 — The Matrix Test for Linear Independence
Form the augmented matrix: [A|b]; Lesson 1318 — Applications: Real-World Matrix Systems Lesson 2142 — The Gauss-Jordan Method for Finding Inverses
Form the contrapositive: ¬Q → ¬P; Lesson 2495 — Proof by Contrapositive: Method and Strategy
Form the difference quotient: Lesson 1381 — Computing Derivatives from the Definition: Linear Functions Lesson 1382 — Computing Derivatives from the Definition: Quadratic Functions
Form the matrix: (**A** - λ**I**) by subtracting λ from each diagonal entry of **A**; Lesson 2170 — Finding Eigenvectors from Eigenvalues Lesson 2264 — Matrix Representation of Linear Transformations Lesson 2265 — Finding the Matrix of a Transformation with Respect to Standard Bases
Form the orthogonal matrix: `P` whose columns are orthonormal eigenvectors; Lesson 2296 — Diagonalization of Quadratic Forms
Form the right triangle: Lesson 968 — Drawing Diagrams for Angle Problems
Formal definition: Lesson 597 — The Sign Function Lesson 2525 — Intersection of Sets Lesson 2526 — Set Difference and Complement Lesson 2938 — Definition of Independence for Random Variables
Formally: If f(x) ≤ g(x) for all x in [a,b], then:; Lesson 1563 — Comparison Properties of Definite Integrals
Forms a tree: (connected and acyclic); Lesson 2672 — Spanning Trees of a Graph
formula: is a special type of equation that shows a relationship between multiple variables.; Lesson 187 — Using Formulas and Literal Equations Lesson 193 — Perimeter and Area Word Problems Lesson 691 — Depreciation and Asset Value Lesson 1703 — Definition of a Sequence and Notation Lesson 1919 — Computing Areas Using Polar Integrals Lesson 3052 — Type I Error: False Positives Lesson 3113 — Discrete Parameter Spaces and Bayesian Updating Lesson 3120 — Predictive Distributions: Prior and Posterior Predictive
Formula approach: Lesson 887 — Surface Area of Prisms
Formula pattern: If `y` depends on `u`, and `u` depends on `t`, then:; Lesson 1418 — Chain Rule Applications and Problem-Solving
Forward (Euclidean Algorithm): Lesson 2715 — Extended Euclidean Algorithm
Forward elimination: is the systematic process of using elementary row operations to eliminate (create zeros in) all entries *below* the main diagonal of a matrix.; Lesson 2108 — Forward Elimination Process
Four Color Theorem: states that every planar graph can be properly colored with at most four colors.; Lesson 2697 — The Four Color Theorem
Fourier and Laplace transforms: Convert PDEs into algebraic equations in transform space; Lesson 2455 — Limitations and Extensions of the Method
Fourier cosine series: (only cosine terms, no sines).; Lesson 2462 — Half-Range Expansions
Fourier series: becomes essential.; Lesson 2466 — Fourier Series and Partial Differential Equations
Fourier sine series: representation of $f(x)$!; Lesson 2451 — Applying Initial Conditions with Fourier Series Lesson 2462 — Half-Range Expansions
Fourier transform: , which handles non-periodic functions by treating them as having "infinite period.; Lesson 2467 — Applications and Extensions
Fourier's law: states that the heat flux (energy per unit area per unit time) is proportional to the negative temperature gradient:; Lesson 2416 — Introduction to the Heat Equation
Fourth and beyond: f ⁴ (x), f ⁵ (x), .; Lesson 1397 — Higher-Order Derivatives
Fourth derivative: f ⁴ (x) = 24; Lesson 1397 — Higher-Order Derivatives
Fourth root: ⁴√81 = 3 because 3 × 3 × 3 × 3 = 81; Lesson 153 — Cube Roots and Higher-Order Roots
Fraction form: Lesson 106 — Writing Ratios in Different Forms
Fraction form (3/4): looks like the fractions you already know, but here it represents a *comparison*, not a part of a whole.; Lesson 106 — Writing Ratios in Different Forms
Fractional approximation: π ≈ 22/7 (close, but not exact); Lesson 830 — Circles and Pi (π)
Fractional exponents: – These represent roots (like square roots), which aren't allowed in polynomials; Lesson 311 — Identifying Polynomials vs Non-Polynomials Lesson 536 — Negative Fractional Exponents Lesson 601 — Evaluating Exponential Expressions
Free (u_x = 0): Forces symmetric continuation → upright reflection; Lesson 2429 — Wave Reflections at Boundaries
Free variable: x₃; Lesson 2123 — Pivot and Free Variables from RREF Lesson 2487 — Binding Variables and Scope
free variables: variables that can take any value.; Lesson 1315 — Systems with Infinitely Many Solutions Lesson 2112 — Free Variables and Parametric Solutions Lesson 2117 — Leading Entries and Pivot Positions Lesson 2123 — Pivot and Free Variables from RREF
Frequency: is the reciprocal of period—it tells you how many complete cycles occur in a 2π interval.; Lesson 1016 — Period Changes: y = sin(Bx) and Frequency Lesson 2369 — Free Undamped Harmonic Motion Lesson 2992 — Histograms and Frequency Distributions
Frequency assignment: Assign radio frequencies to transmitters without interference; Lesson 2695 — Graph Coloring and the Chromatic Number
Friedman test: comes to the rescue.; Lesson 3108 — Friedman Test
Frobenius method: handles this by assuming a solution of the form:; Lesson 2401 — Regular Singular Points and the Method of Frobenius
From a graph: Look for peaks (maxima) and valleys (minima).; Lesson 573 — Extrema: Local and Absolute Maxima and Minima
From a table: Compare function values.; Lesson 573 — Extrema: Local and Absolute Maxima and Minima
From Acceleration to Velocity: Lesson 1545 — Applications: Position from Velocity, Velocity from Acceleration
From Cylindrical to Spherical: Lesson 1938 — Spherical Coordinates: Definition and Conversion
From Exponential → Poisson: Lesson 2902 — Relationship Between Exponential and Poisson
From Poisson → Exponential: Lesson 2902 — Relationship Between Exponential and Poisson
From Rectangular to Spherical: Lesson 1938 — Spherical Coordinates: Definition and Conversion
From set notation: If you list the outcomes and find *A* and *B* share no common elements, they're mutually exclusive.; Lesson 2779 — Mutually Exclusive Events
From Spherical to Cylindrical: Lesson 1938 — Spherical Coordinates: Definition and Conversion
From Spherical to Rectangular: Lesson 1938 — Spherical Coordinates: Definition and Conversion
From the left: (smaller x-values getting bigger); Lesson 1322 — Estimating Limits from Graphs Lesson 1326 — Limits Approaching from Left and Right Lesson 1330 — Notation for One-Sided Limits
From the right: (larger x-values getting smaller); Lesson 1322 — Estimating Limits from Graphs Lesson 1326 — Limits Approaching from Left and Right Lesson 1330 — Notation for One-Sided Limits
From Velocity to Position: Lesson 1545 — Applications: Position from Velocity, Velocity from Acceleration
From Venn diagrams: Two circles (events) that don't overlap represent mutually exclusive events.; Lesson 2779 — Mutually Exclusive Events
FTC Part 1: tells us that if we define a function by integrating another function:; Lesson 1577 — Connecting Both Parts of the FTC Lesson 1578 — Applications and Problem Solving with FTC
FTC Part 2: tells us that if $F$ is *any* antiderivative of $f$, then:; Lesson 1577 — Connecting Both Parts of the FTC Lesson 1578 — Applications and Problem Solving with FTC
Fubini's Theorem: .; Lesson 1892 — Introduction to Double Integrals Lesson 1894 — Fubini's Theorem for Rectangles
Full column rank: When rank equals the number of columns.; Lesson 2136 — Full Rank Matrices Lesson 2137 — Applications to System Solvability
Full model: Your regression with all predictors (X₁, X₂, .; Lesson 3078 — F-Test for Overall Model Significance
full rank: when its rank equals the maximum value it could possibly have.; Lesson 2136 — Full Rank Matrices Lesson 2143 — Testing Invertibility Lesson 2165 — Determinant and Matrix Rank
Full rank (square matrices): When a square matrix has rank equal to both its number of rows and columns.; Lesson 2136 — Full Rank Matrices
Full row rank: When rank equals the number of rows.; Lesson 2136 — Full Rank Matrices
function composition: (plugging one function into another) and **inverse functions** (functions that "undo" each other), you can now combine these tools!; Lesson 568 — Compositions Involving Inverses Lesson 1408 — Composition of Functions Review
Function Spaces: Lesson 2076 — Examples of Vector Spaces
Functions of several variables: accept multiple inputs and return a single output.; Lesson 1802 — Introduction to Functions of Several Variables
functions of y: rather than x.; Lesson 1648 — Area Between Curves: Horizontal Slicing Lesson 1660 — Washer Method: Rotation About the y-axis
Functions that simplify: Some functions become constants or simpler expressions when one variable is integrated first; Lesson 1900 — Choosing Integration Order Strategically
Fundamental Theorem of Arithmetic: guarantees this never happens—and we prove it using a technique called **proof by contradiction** combined with properties of divisibility.; Lesson 2726 — Uniqueness of Prime Factorization: The Proof
Funnel shape: (spreading out or narrowing): Indicates heteroscedasticity—the variability of errors changes as predictions change.; Lesson 3084 — Residual Plots: Fitted Values vs. Residuals
Funnel shape (contracting): Residuals tighten as fitted values increase; Lesson 3085 — Detecting Non-Constant Variance (Heteroscedasticity)
Further algebraic manipulation: – If you need to add, subtract, or simplify expressions further, rationalized forms like `3/(√2)` becoming `(3√2)/2` keep you working symbolically and avoid rounding errors.; Lesson 521 — Applications and When Not to Rationalize
Further operations: Adding fractions with rationalized denominators is cleaner; Lesson 514 — Why Rationalize Denominators?

G

g(x): the "inner" function; Lesson 558 — Function Composition Basics Lesson 1408 — Composition of Functions Review
Gamma function: , defined as:; Lesson 1642 — Applications and Special Improper Integrals Lesson 2916 — Introduction to the Gamma Function
Gamma(α, β): Generalizes exponential:; Lesson 2962 — MGFs of Common Distributions
Gas vs entertainment: a = e + 50; Lesson 299 — Applications of Three-Variable Systems
Gauss-Jordan method: gives us a mechanical, reliable process: we start with the matrix A we want to invert, attach the identity matrix I to its right side (forming an augmented matrix [A|I]), then use elementary row operations to transform the left side into I.; Lesson 2142 — The Gauss-Jordan Method for Finding Inverses
GCF = 12xy³: Lesson 357 — Finding the GCF of Monomials
GCF = 6: (the largest); Lesson 52 — Greatest Common Factor (GCF)Lesson 355 — What is the Greatest Common Factor (GCF)?Lesson 356 — Finding the GCF of Numbers
GCF = 6x²: Lesson 357 — Finding the GCF of Monomials
General Addition Rule: for two events:; Lesson 2789 — Probability Bounds and Boole's Inequality
General case: Whenever you're unsure if events can happen together; Lesson 2785 — The General Addition Rule
general form: ax² + bx + c = 0, using letters instead of numbers?; Lesson 408 — Deriving the Quadratic Formula Lesson 918 — Completing the Square for Circles Lesson 1083 — General Solutions Using Period Lesson 1139 — General Form of a Circle Lesson 1140 — Converting Between Circle Forms Lesson 1141 — Finding Circle Properties from General Form Lesson 1150 — Converting to Standard Form by Completing the Square Lesson 1795 — Planes in 3D: Normal Form and General Equation
General pattern: For ⁿ√a ᵐ where m < n, multiply by ⁿ√(aⁿ ᵐ) to complete the power.; Lesson 516 — Rationalizing with Higher-Index Roots
general solution: that contains one or more arbitrary constants.; Lesson 2303 — General vs Particular Solutions Lesson 2304 — Initial Value Problems Lesson 2311 — Initial Value Problems with Separable ODEs Lesson 2323 — Solving Non-Homogeneous Linear ODEs Lesson 2324 — Initial Value Problems for Linear ODEs Lesson 2339 — Real Distinct Roots: General Solution Lesson 2340 — Repeated Real Roots: Reduction of Order Lesson 2343 — The Principle of Superposition (+4 more)
General solutions: capture this infinite family using a parameter (usually *n*) that represents any integer (.; Lesson 1083 — General Solutions Using Period Lesson 2343 — The Principle of Superposition
Generalized Pigeonhole Principle: goes further—it tells us *exactly how many* items must be in some container.; Lesson 2572 — Generalized Pigeonhole Principle
Geometric (exponential): Use when the change depends on the current population.; Lesson 690 — Population Growth and Decay Models
Geometric intuition: Orthogonal vectors point in "completely independent" directions; Lesson 2206 — Angle Between Vectors and Orthogonality
Geometric Mean Relations: involve the *altitude to the hypotenuse* and the segments it creates.; Lesson 805 — Comparing Pythagorean Theorem and Geometric Mean Methods
geometric multiplicity: of the eigenvalue.; Lesson 2172 — Eigenspaces and Their Dimension Lesson 2173 — Algebraic vs Geometric Multiplicity Lesson 2195 — When Diagonalization Fails Lesson 2196 — Geometric and Algebraic Multiplicity Lesson 2274 — Eigenspaces and Multiplicity for Symmetric Matrices
Geometric progression: a(n) = r · a(n-1) (constant ratio); Lesson 2627 — What is a Recurrence Relation?
geometric sequence: is a list of numbers where each term after the first is found by *multiplying* the previous term by a fixed, non-zero constant.; Lesson 660 — Introduction to Geometric Sequences Lesson 666 — Growth and Decay Applications Lesson 668 — Geometric Sequences vs Exponential Functions Lesson 691 — Depreciation and Asset Value
Geometric sequences: model populations that grow or shrink by a *fixed percentage* each time period.; Lesson 690 — Population Growth and Decay Models
geometric series: Lesson 688 — Annuities and Regular Payments Lesson 689 — Loan Amortization Lesson 1722 — Geometric Series: Definition and Formula Lesson 1728 — Comparing Geometric and p-Series Lesson 1735 — Choosing Comparison Series Effectively Lesson 1750 — Strategy: Choosing the Right Convergence Test Lesson 2506 — Induction with Geometric Series Lesson 2639 — Ordinary Generating Functions for Simple Sequences (+1 more)
Geometric series clues: Lesson 1730 — Mixed Examples: Known Series
Geometric(p): Counting trials until first success:; Lesson 2962 — MGFs of Common Distributions
Geometry: area problems, dimensions of rectangles; Lesson 400 — Applications and Word Problems
Gibbs phenomenon: , discovered by physicist Josiah Willard Gibbs in 1899.; Lesson 2464 — Gibbs Phenomenon
Gibbs sampling: Update one parameter at a time, cycling through conditionals; Lesson 3121 — Computational Tools: Introduction to MCMC and Bayesian Software
Given: A slope and a y-intercept; Lesson 256 — Writing Equations in Slope-Intercept Form Lesson 429 — Finding the Vertex Using x = - b/(2a)Lesson 736 — Properties of Equilateral Triangles Lesson 748 — Writing Two-Column Congruence Proofs Lesson 762 — Introduction to Geometric Proof Lesson 768 — Direct Proof Techniques Lesson 771 — Proving Properties of Parallel Lines Lesson 778 — Word Problems Involving Right Triangles (+7 more)
given information: , a **prove statement**, and a logical sequence of **statements with reasons** that build from known facts to the conclusion.; Lesson 762 — Introduction to Geometric Proof Lesson 763 — Two-Column Proof Format Lesson 2490 — What Does It Mean to Prove Something?
Glide Reflections: – A combination of a reflection followed by a translation parallel to the mirror line; Lesson 939 — Introduction to Rigid Motions (Isometries)
Goal: Show 2^(k+1) > (k+1)².; Lesson 2505 — Proving Inequalities by Induction
Going backward: If *F(s) = e^(-as)·G(s)*, then the inverse transform is ℒ⁻¹{*F(s)*} = *g(t - a)·u(t - a)*.; Lesson 2394 — Second Shifting Theorem and Unit Step Function
Good (independence): Lesson 3090 — Checking for Independence: Residual Sequence Plots
Good news: The process is *identical* to factoring single-variable trinomials.; Lesson 371 — Factoring Trinomials in Two Variables
Governing PDE: The heat equation *∂u*/*∂t* = *k* *∂²u*/*∂x²*; Lesson 2423 — Heat Equation on a Finite Rod
gradient: $\nabla f$ at work.; Lesson 1840 — Higher-Dimensional Linear Approximation Lesson 1841 — The Gradient Vector: Definition and Notation Lesson 1851 — Applications: Temperature Gradients and Heat Flow
Gradient fields: Flow lines point "downhill," perpendicular to level curves; Lesson 1971 — Field Lines and Flow Lines
gradient vector: `∇f = ⟨fₓ, fᵧ⟩`, the tangent plane is perpendicular to the vector ` ⟨fₓ(x₀, y₀), fᵧ(x₀, y₀), -1⟩`, giving the same equation in normal form.; Lesson 1833 — Equation of a Tangent Plane Lesson 1835 — Normal Lines to Surfaces Lesson 1845 — The Gradient Points in the Direction of Steepest Ascent
Gram-Schmidt algorithm: is a step-by-step procedure that transforms these vectors into a new set where each vector is orthogonal (perpendicular) to all the others, while preserving the span—the subspace they generate remains exactly the same.; Lesson 2220 — The Gram-Schmidt Algorithm: Overview
Graph behavior: The ceiling function creates a step graph that jumps at every integer, with the jump happening *at* the integer point moving upward.; Lesson 596 — The Ceiling and Rounding Functions
Graph coloring: Colors partition vertices based on "connected to same neighbors"; Lesson 2558 — Applications and Quotient Sets
Graph each inequality separately: on the same coordinate plane (shade above or below the boundary line, use solid or dashed lines appropriately); Lesson 301 — Systems of Linear Inequalities
Graph each piece separately: – Plot points or sketch the shape for each formula, but *only* over its designated interval; Lesson 592 — Graphing Piecewise Functions
Graph it: Plot (4, 0) and (0, 3) on your coordinate plane, then draw a straight line through both points.; Lesson 262 — Graphing Lines in Standard Form
Graph the feasible region: This shows all possible production combinations; Lesson 308 — Applications: Maximization Problems Lesson 309 — Applications: Minimization Problems
Graph the first equation: using any method you know (slope-intercept form, point-plotting, etc.; Lesson 274 — Solving Systems by Graphing: Basic Technique
Graph the second equation: on the same coordinate plane; Lesson 274 — Solving Systems by Graphing: Basic Technique
Graphing and estimation: – When plotting points or checking reasonableness, `0.; Lesson 521 — Applications and When Not to Rationalize
Graphing tools needed: Requires graph paper or technology; Lesson 281 — Comparing Graphing and Substitution Methods
Graphs with holes: If there's a hole at x = 3, you'd exclude that value: something like (−∞, 3) ∪ (3, ∞).; Lesson 552 — Domain from Graphs
greater than 1: (like `3x² + 10x + 8`), use either the **AC method** or **trial and error**, whichever feels more comfortable to you.; Lesson 373 — Applications and Strategy Selection for Trinomial Factoring Lesson 581 — Horizontal Stretches and Compressions Lesson 603 — Growth vs. Decay: The Role of the Base Lesson 1181 — Calculating Eccentricity for Hyperbolas
greatest common divisor: (abbreviated GCD) of two integers *a* and *b*, denoted gcd(*a*, *b*), is the largest positive integer that divides both *a* and *b* without leaving a remainder.; Lesson 2711 — Greatest Common Divisor (GCD): Definition Lesson 2712 — Finding GCD by Listing Divisors
Greatest Common Factor: (GCF) is the biggest factor that two or more numbers share.; Lesson 52 — Greatest Common Factor (GCF)
Greatest Common Factor (GCF): is the largest factor that two or more numbers share.; Lesson 355 — What is the Greatest Common Factor (GCF)?
Green's functions: Build solutions using integral representations; Lesson 2455 — Limitations and Extensions of the Method
Green's Theorem states: Lesson 1996 — Verifying Green's Theorem for Simple Regions
Groceries vs entertainment: g = 2e; Lesson 299 — Applications of Three-Variable Systems
Group factors: into sets matching the root's index; Lesson 157 — Simplifying Higher-Order Radicals
Group like variables: Place all x-terms together and all y-terms together; Lesson 1175 — Converting General Form to Standard Form
Group pairs: Put parentheses around the first two terms and the last two:; Lesson 364 — Factoring by Grouping After Splitting the Middle Term
Group Properties: When you multiply any two numbers in this set and reduce modulo *n*, you get another number in the set.; Lesson 2749 — Proof of Euler's Theorem
Group terms: Collect all `x` terms together, all `y` terms together, and move the constant to the right side.; Lesson 1140 — Converting Between Circle Forms
Group the first two: (2 + 3) + 4 = 5 + 4 = **9**; Lesson 37 — The Associative Property
Group the imaginary parts: (the coefficients of `i`); Lesson 1197 — Adding and Subtracting Complex Numbers
Group the last two: 2 + (3 + 4) = 2 + 7 = **9**; Lesson 37 — The Associative Property
Group the real parts: (the numbers without `i`); Lesson 1197 — Adding and Subtracting Complex Numbers
Group the X-values: For each y-value, find which x-values map to it (i.; Lesson 2868 — Functions of Random Variables: Discrete Case
Grouping: asks: "How many bags can I make?; Lesson 23 — Understanding Division as Sharing and Grouping
Grow: Look at all edges connecting vertices *inside* the MST to vertices *outside* the MST.; Lesson 2676 — Minimum Spanning Trees: Prim's Algorithm
growth: (things getting bigger).; Lesson 666 — Growth and Decay Applications Lesson 690 — Population Growth and Decay Models Lesson 1437 — Applications: Exponential Growth and Decay Rates Lesson 2316 — Applications: Exponential Growth and Decay
Growth rate: `N'(t) = N₀ · k · e^(kt) = k · N(t)`; Lesson 1437 — Applications: Exponential Growth and Decay Rates
Guarantees limited options: (keeps the number of holes manageable); Lesson 2576 — Constructing Pigeonholes Strategically
Guess a form: for the particular solution $y_p$ based on $g(t)$; Lesson 2357 — Higher-Order ODEs and Undetermined Coefficients

H

H ₐ: βᵢ ≠ 0: (the alternative: this predictor matters); Lesson 3077 — Hypothesis Testing for Individual Coefficients
H₀: β ᵢ = 0: (the null hypothesis: this predictor has no real effect); Lesson 3077 — Hypothesis Testing for Individual Coefficients
H₁: μ ≠ μ₀: .; Lesson 3026 — Two-Tailed z-Tests
half: of its intercepted arc.; Lesson 838 — Inscribed Angles Lesson 839 — Angles Inscribed in a Semicircle Lesson 1036 — Calculating Period from the B Coefficient
Half-integer value: Γ(1/2) = √π; Lesson 2916 — Introduction to the Gamma Function
Half-life: and **doubling time** are special time intervals in exponential models:; Lesson 643 — Half-Life and Doubling Time Lesson 646 — Carbon Dating and Radioactive Decay Applications
Half-planes perpendicular to z-axis: The equation **φ = π/2** gives the xy-plane (everything at 90° from the z-axis).; Lesson 1939 — Graphing Surfaces in Spherical Coordinates
Hall's Condition: For every subset S ⊆ X, we need |N(S)| ≥ |S|.; Lesson 2700 — Hall's Marriage Theorem
Hall's Marriage Theorem: answers: *When can everyone be matched to an acceptable partner?; Lesson 2700 — Hall's Marriage Theorem
Hamiltonian: Focus on *vertices* (visit every vertex once); Lesson 2683 — Hamiltonian Paths and Cycles: Definition and Examples Lesson 2686 — Ore's Theorem
Hamiltonian circuit: ) is a Hamiltonian path that returns to its starting vertex, forming a closed loop.; Lesson 2683 — Hamiltonian Paths and Cycles: Definition and Examples
Hamiltonian cycle: (or **Hamiltonian circuit**) is a Hamiltonian path that returns to its starting vertex, forming a closed loop.; Lesson 2683 — Hamiltonian Paths and Cycles: Definition and Examples
Hamiltonian Monte Carlo (HMC): Uses gradient information for efficient exploration; Lesson 3121 — Computational Tools: Introduction to MCMC and Bayesian Software
Hamiltonian path: in a graph is a path that visits each vertex in the graph exactly one time.; Lesson 2683 — Hamiltonian Paths and Cycles: Definition and Examples
Hamiltonian paths: focus on visiting every *vertex* exactly once.; Lesson 2683 — Hamiltonian Paths and Cycles: Definition and Examples
Handle constant terms: with ∫k dx = kx + C; Lesson 1542 — Antiderivatives of Polynomial Functions
Handle endpoints carefully: – This is crucial:; Lesson 592 — Graphing Piecewise Functions
Handle exponents first: 3² = 9; Lesson 33 — Exponents in Order of Operations
Handle negative bases carefully: use parentheses to avoid confusion; Lesson 141 — Mixed Practice with Integer Exponents
Handle parentheses first: (multiply or divide inside them if needed); Lesson 1202 — Simplifying Complex Expressions
Handle piecewise functions: naturally by integrating each piece separately; Lesson 1560 — Properties of Definite Integrals: Intervals
Handle ties: by assigning average ranks; Lesson 3106 — Mann-Whitney U Test (Wilcoxon Rank-Sum)
Handles ties: more gracefully with adjusted formulas (Tau-b, Tau-c); Lesson 3110 — Kendall's Tau
Handshaking Lemma: states:; Lesson 2650 — Degree of a Vertex and the Handshaking Lemma
harmonic function: .; Lesson 2436 — Introduction to Laplace's Equation Lesson 2444 — Mean Value Property and Applications
harmonic series: is one of the most famous examples in all of mathematics:; Lesson 1721 — The Harmonic Series Lesson 1727 — The Harmonic Series Lesson 1744 — When the Ratio Test is Inconclusive
harmonics: or **overtones**.; Lesson 2430 — Standing Waves and Normal Modes Lesson 2435 — Applications: Vibrating Strings, Membranes, and Sound
Has its derivative present: (or nearly present) elsewhere in the integrand; Lesson 1580 — Identifying the Inner Function u
Heading: (or **heading vector**): The direction and speed the vehicle is pointed, based on its own power; Lesson 1266 — Navigation Problems: Heading and Course
Heartbeat: 72 beats per minute (beats ÷ minutes); Lesson 109 — Introduction to Rates
Heat diffusion: through materials; Lesson 2408 — What is a Partial Differential Equation?
Heavier tails: More probability mass in the extremes, meaning more chance of observing values far from the mean; Lesson 2923 — The Student's t-Distribution: Definition Lesson 3032 — Why t-Tests? When the Population Variance is Unknown Lesson 3033 — The t-Distribution: Properties and Degrees of Freedom
height: (or altitude) of a trapezoid is the perpendicular distance between the two bases.; Lesson 821 — Trapezoids: Definitions and Basic Properties Lesson 860 — Area of Trapezoids Lesson 876 — Volume of Rectangular Prisms Lesson 886 — Volume of Oblique Solids Lesson 899 — Cavalieri's Principle with Cylinders Lesson 1112 — Area Formula Using Base and Height Lesson 1552 — Sigma Notation for Riemann Sums Lesson 1664 — Introduction to the Shell Method (+9 more)
Height (h): The perpendicular distance between that base and its opposite side; Lesson 858 — Area of Parallelograms Lesson 859 — Area of Triangles
Height patterns: where the function has high or low values; Lesson 1806 — Graphing Surfaces in 3D
Here's the key insight: Just as we found arc length by adding up infinitesimally small straight-line segments along a curve, we'll find surface area by adding up infinitesimally small *bands* (like thin ribbons) that wrap around the surface.; Lesson 1683 — Introduction to Surface Area of Revolution
Hermitian matrices: (A = A*) are normal over the complexes; Lesson 2276 — The Spectral Theorem for Normal Matrices
Hermitian operator: (the complex analogue of symmetric matrices).; Lesson 2279 — Spectral Theorem in Quantum Mechanics and Statistics
Heron's Formula: , one of the most beautiful results in geometry.; Lesson 1115 — Heron's Formula: Introduction and Derivation Lesson 1116 — Applying Heron's Formula
Hessian matrix: simply packages these four derivatives into a 2×2 matrix in a standard way.; Lesson 1864 — The Hessian Matrix Lesson 2298 — Applications to Optimization and Hessian Matrices
Heteroscedasticity: occurs when this variance changes systematically.; Lesson 3085 — Detecting Non-Constant Variance (Heteroscedasticity)
High variance: Values are widely dispersed (more variability); Lesson 2856 — Variance: Definition and Interpretation
High-leverage points: have extreme values of the predictor variable(s).; Lesson 3089 — Outliers in Regression
Higher degree in denominator: Limit = 0 (horizontal asymptote at y = 0); Lesson 1343 — Limits at Infinity and Horizontal Asymptotes
Higher degree in numerator: Limit = ±∞ (no horizontal asymptote); Lesson 1343 — Limits at Infinity and Horizontal Asymptotes
Higher desired power: → need more participants; Lesson 3059 — Sample Size Determination
Higher pH: = more basic/alkaline (bleach ≈ 13); Lesson 647 — pH and the Richter Scale
Higher α: (e.; Lesson 3055 — Significance Level and Error Control
Higher-frequency modes decay faster: If you think of $\lambda_n$ as measuring how "wiggly" a spatial pattern is, the wigglier patterns smooth out more quickly.; Lesson 2425 — Physical Interpretation and Decay Rates
Highest powers: 2¹, 3², x³; Lesson 461 — Finding the Least Common Denominator (LCD)
Histograms: reveal distribution shape, skewness, and modality; Lesson 2997 — Choosing Appropriate Summaries and Displays
HL Congruence Theorem: states:; Lesson 745 — HL (Hypotenuse-Leg) Congruence Theorem
hole: (or removable discontinuity) occurs in a rational function when the same factor appears in both the numerator and denominator, causing them to cancel out.; Lesson 487 — Holes in Rational Functions Lesson 1322 — Estimating Limits from Graphs
Holes: also create single missing x-values; Lesson 493 — Domain and Range from Graphs Lesson 1325 — Limits at Points of Discontinuity
Homogeneity: `T(c·u) = c·T(u)`; Lesson 2259 — Definition of Linear Transformations
Homogeneous: When $Q(x) = 0$, the equation becomes $\frac{dy}{dx} + P(x)y = 0$; Lesson 2319 — Homogeneous vs Non-Homogeneous Linear ODEs Lesson 2337 — Form and Classification of Second-Order Linear ODEs Lesson 2343 — The Principle of Superposition Lesson 2446 — When Separation of Variables Works Lesson 2633 — Non-Homogeneous Recurrence Relations
Homogeneous boundary conditions: X(0) = 0 and X(L) = 0; Lesson 2420 — Eigenvalue Problems from Boundary Conditions Lesson 2455 — Limitations and Extensions of the Method
homogeneous first-order linear ODE: has the standard form:; Lesson 2320 — Solving Homogeneous First-Order Linear ODEs Lesson 2327 — Applications: RC Circuits and Decay
Homogeneous solution: (a_n^(h)): The solution to the associated homogeneous recurrence (when you set the non- homogeneous term to zero); Lesson 2635 — General Solution of Non-Homogeneous Recurrences
Hooke's Law: Lesson 1694 — Hooke's Law and Spring Problems Lesson 2368 — Spring-Mass Systems: Setting Up the Differential Equation
Horizontal: Quick for simple problems; keeps everything visible in one line; Lesson 323 — Subtracting Polynomials Horizontally and Vertically Lesson 1171 — Graphing Hyperbolas Centered at the Origin Lesson 1172 — Hyperbolas with Horizontal vs Vertical Transverse Axes Lesson 1456 — Horizontal and Vertical Tangents Lesson 1468 — Normal Lines to Curves Lesson 1524 — Slant (Oblique) Asymptotes
horizontal asymptote: showing the limiting behavior!; Lesson 494 — Applications of Rational Functions Lesson 612 — Properties of e^x: Growth and Behavior Lesson 1328 — Limits at Infinity and Horizontal Asymptotes Lesson 1343 — Limits at Infinity and Horizontal Asymptotes
horizontal asymptotes: invisible horizontal lines the function approaches but may never quite reach.; Lesson 488 — Horizontal Asymptotes: Degree Rules Lesson 492 — End Behavior of Rational Functions Lesson 493 — Domain and Range from Graphs Lesson 494 — Applications of Rational Functions Lesson 1518 — Vertical and Horizontal Asymptotes Lesson 1525 — Comprehensive Curve Sketching Algorithm
horizontal axis: (x-axis) represents the **real part**; Lesson 1207 — Complex Numbers as Points in the Plane Lesson 1208 — The Argand Diagram Lesson 3084 — Residual Plots: Fitted Values vs. Residuals
Horizontal axis (real axis): represents the real part of the complex number; Lesson 1192 — Plotting Complex Numbers on the Complex Plane
Horizontal base: Height rises vertically upward (easiest to spot); Lesson 1112 — Area Formula Using Base and Height
Horizontal change: (difference in input values): `b - a`; Lesson 574 — Average Rate of Change
Horizontal component: F_x = ||**F**|| cos(θ); Lesson 1263 — Force as a Vector: Magnitude and Direction
Horizontal component (x-component): `v_x = |v| cos(θ)`; Lesson 1252 — Resolving Vectors into Components
Horizontal hyperbola: (opens left/right): vertices at (h ± a, k); Lesson 1168 — Components of a Hyperbola: Center, Vertices, and Foci Lesson 1169 — The Transverse and Conjugate Axes Lesson 1170 — Asymptotes of a Hyperbola
horizontal leg: (running parallel to the x-axis); Lesson 779 — Right Triangles on the Coordinate Plane Lesson 903 — Distance Formula in the Coordinate Plane Lesson 968 — Drawing Diagrams for Angle Problems Lesson 975 — Angles of Elevation: Definition and Setup Lesson 978 — Finding Distances Using Angles of Depression
horizontal line test: draw horizontal lines across the graph.; Lesson 562 — One-to-One Functions Lesson 2561 — Testing for Injectivity
Horizontal lines: run left-to-right, perfectly flat across the plane.; Lesson 263 — Horizontal and Vertical Lines
Horizontal major axis: `x²/a² + y²/b² = 1` (where a > b); Lesson 1158 — Graphing Ellipses Centered at the Origin Lesson 1162 — Finding the Equation from Given Information
Horizontal parabolas: Lesson 1151 — Graphing Parabolas from Standard Form
horizontal shift: (left or right) changes the x-coordinate; Lesson 922 — Translations in the Coordinate Plane Lesson 1031 — Transformations of Cosecant Lesson 1037 — Phase Shift: Horizontal Translation
Horizontal shifts: (except zero) always destroy both even and odd symmetry.; Lesson 587 — Even and Odd Functions Under Transformation Lesson 589 — Inverse Relationships in Transformations
Horizontal shifts and stretches/compressions: (these happen "inside" the function); Lesson 584 — Combining Multiple Transformations
Horizontal stretches/compressions: preserve the symmetry type but change the shape.; Lesson 587 — Even and Odd Functions Under Transformation Lesson 589 — Inverse Relationships in Transformations
Horizontal tangent lines: These occur where the curve momentarily "flattens out" — like reaching the top of a hill or the bottom of a valley.; Lesson 1467 — Horizontal and Vertical Tangent Lines
Horizontal tangents: occur when the slope is **zero**: dy/dx = 0; Lesson 1456 — Horizontal and Vertical Tangents Lesson 1467 — Horizontal and Vertical Tangent Lines
Horizontal traces: (fix z): \(k = x^2 + y^2\) gives circles of radius \(\sqrt{k}\); Lesson 1809 — Traces and Cross-Sections
Horizontal transverse axis: Lesson 1173 — Writing Equations from Given Information Lesson 1174 — Hyperbolas Centered at (h, k)
Horizontal/Slant: Apply degree rules (you've learned these!; Lesson 491 — Graphing Rational Functions with Asymptotes
horizontally: .; Lesson 579 — Horizontal Shifts of Functions Lesson 1172 — Hyperbolas with Horizontal vs Vertical Transverse Axes Lesson 1649 — Choosing Vertical vs Horizontal Slicing Lesson 1655 — Disk Method: Rotation About the y-axis
How it works: Lesson 1596 — Tabular Integration
How many ways: can we arrange k successes among n trials?; Lesson 2875 — Binomial Probability Mass Function
How to find it: Row reduce A to echelon form.; Lesson 2092 — Dimension of Subspaces
How to find them: Lesson 1518 — Vertical and Horizontal Asymptotes
How to solve: Lesson 2719 — Applications of the Euclidean Algorithm
How to use it: Look at your integrand.; Lesson 1591 — Choosing u and dv: The LIATE Rule Lesson 2908 — Using the Standard Normal Table
Hundredths place: (2nd position) = 1/100; Lesson 83 — Understanding Decimal Place Value
hyperbola: is the set of all points where the **absolute difference of distances** to two foci is constant.; Lesson 1130 — Geometric Definitions of Conics Lesson 1131 — The General Second-Degree Equation Lesson 1132 — Using the Discriminant to Classify Conics Lesson 1133 — Standard Position and Orientation Lesson 1167 — Definition and Standard Form of a Hyperbola Lesson 1179 — Eccentricity: Measuring the Shape of Conics Lesson 1182 — Focus-Directrix Definition of Conics
Hyperbolas: are essential in navigation systems like LORAN and GPS, which use the time difference of signals from two stations to locate positions along hyperbolic curves.; Lesson 1135 — Overview of Conic Applications
hyperbolic paraboloid: (saddle surface) has the form:; Lesson 1797 — Quadric Surfaces: Ellipsoids and Paraboloids Lesson 1807 — Common Surfaces: Planes, Paraboloids, and Saddles
hypotenuse: (the side opposite the right angle—always the longest side); Lesson 773 — What is the Pythagorean Theorem?Lesson 775 — Finding the Hypotenuse Lesson 779 — Right Triangles on the Coordinate Plane Lesson 794 — The 30-60-90 Triangle: Side Ratios Lesson 903 — Distance Formula in the Coordinate Plane Lesson 948 — Identifying Opposite, Adjacent, and Hypotenuse Lesson 950 — The Cosine Ratio (CAH)Lesson 953 — Finding Side Lengths Using Cosine (+6 more)
Hypotenuse ÷ 2: = shortest side; Lesson 795 — Finding Missing Sides in 30-60-90 Triangles
Hypotenuse-Leg (HL): theorem, △ OAP ≅ △ OBP; Lesson 848 — Properties of Tangents from an External Point
Hypotheses: Lesson 3028 — z-Test for Proportions
hypothesis: and q is the **consequence**.; Lesson 2472 — Conditional Statements (If-Then)Lesson 2504 — Proving Divisibility Properties Lesson 2514 — Proving Divisibility Results with Strong Induction
Hypothesis (P): The condition or assumption you start with; Lesson 2489 — Structure of Mathematical Statements
Hypothesis testing: Is there really a linear relationship, or could the observed slope just be due to random chance?; Lesson 3069 — Inference for the Slope
Hypothesize a median value: (call it M₀).; Lesson 3104 — Sign Test for Median

I

i = √(-1): Lesson 1188 — What Are Complex Numbers?Lesson 1189 — The Imaginary Unit i
I = mk²: .; Lesson 1962 — Radius of Gyration and Applications
I_A: or **1_A**, by:; Lesson 2830 — Indicator Random Variables
i-th row: from matrix **A**; Lesson 1282 — Computing the (i,j)-Entry of a Product Lesson 1284 — Multiplying Larger Matrices
i, −1, −i, 1: .; Lesson 1194 — Powers of i Lesson 1201 — Powers of i
I₀: = reference intensity = 10 ¹² watts/m² (the quietest sound a human can typically hear); Lesson 648 — Sound Intensity and Decibels
i² = -1: or equivalently, **i = √(-1)**; Lesson 1188 — What Are Complex Numbers?Lesson 1189 — The Imaginary Unit i
i² = −1: .; Lesson 1194 — Powers of i Lesson 1201 — Powers of i
IA = A: Lesson 1288 — Multiplying by the Identity Matrix Lesson 2103 — Special Matrices: Identity and Zero
IB = C: , which means **B = C**; Lesson 2139 — Properties of Invertible Matrices
Ideal pattern: A random scatter of points around the horizontal line at zero, with no discernible pattern.; Lesson 3084 — Residual Plots: Fitted Values vs. Residuals
Idempotence: P² = P (projecting twice is the same as projecting once); Lesson 2232 — Properties of Projection Matrices
Identify: what you know and what you're finding; Lesson 113 — Solving Rate Problems Lesson 130 — Real-World Inverse Variation Problems Lesson 192 — Age Problems and Consecutive Integer Problems Lesson 452 — Simplifying Before Multiplying Lesson 475 — Applications and Word Problems with Complex Fractions Lesson 598 — Writing Functions as Piecewise Definitions Lesson 659 — Applications of Arithmetic Sequences Lesson 712 — Solving Multi-Step Angle Problems (+11 more)
Identify all boundary pieces: (top, bottom, sides, etc.; Lesson 2042 — The Divergence Theorem for General Regions
Identify all denominators: in the equation; Lesson 209 — Equations with Fractions: Clearing Denominators Lesson 467 — Adding and Subtracting Three or More Rationals Lesson 470 — Simplifying Complex Fractions by Multiplying by the LCD Lesson 480 — Solving Equations with Multiple Terms
Identify all forces: Typically, you have two or more tension vectors pulling upward/outward, plus gravity pulling straight down.; Lesson 1269 — Tension in Cables and Ropes
Identify all four parameters: Lesson 1040 — Graphing with All Transformations Combined
Identify an "inner function": some expression that's making the integral complicated; Lesson 1579 — What is u-Substitution?
Identify b and c: Here b = 7 and c = 12; Lesson 362 — Factoring Trinomials with Leading Coefficient 1
Identify coefficients: a = 2, b = -7, c = 3; Lesson 414 — Solving Quadratics When a ≠ 1
Identify common factors: (polynomials that appear in both); Lesson 445 — Simplifying by Canceling Common Factors Lesson 448 — Simplifying with Trinomial Factors
Identify corresponding points: on both figures; Lesson 941 — Congruence Through Rigid Motions
Identify corresponding sides: these are sides that match up in the similar figures; Lesson 727 — Scale Factor Lesson 757 — Finding Missing Sides Using Similarity
Identify each basic solid: in the composite figure; Lesson 892 — Surface Area of Composite Solids
Identify each face: in the net; Lesson 893 — Nets and Surface Area
Identify each term: separately; Lesson 1393 — Combining Basic Rules for Polynomials
Identify free variables: from your reduced matrix; Lesson 1315 — Systems with Infinitely Many Solutions Lesson 2130 — The Null Space Lesson 2170 — Finding Eigenvectors from Eigenvalues
Identify leading coefficients: Top = 3, Bottom = 5; Lesson 489 — Finding Horizontal Asymptotes by Leading Coefficients
Identify like radicals: (same index and same radicand); Lesson 507 — Simplifying Before Adding or Subtracting
Identify like terms: Look for terms with the exact same variable raised to the same exponent; Lesson 179 — Combining Like Terms
Identify non-pivot columns: Column 3 has no leading 1.; Lesson 2123 — Pivot and Free Variables from RREF
Identify patterns: in how solutions behave; Lesson 2302 — Solutions and Solution Curves
Identify perfect square factors: (4, 9, 16, 25, 36, 49, etc.; Lesson 415 — Simplifying Radical Answers
Identify pivot columns: Columns 1, 2, and 4 have leading 1s.; Lesson 2123 — Pivot and Free Variables from RREF
Identify R(x): = distance from axis to outer curve; Lesson 1657 — Washer Method: Introduction and Setup
Identify restricted values: from the original equation (values that make any denominator zero); Lesson 481 — Checking for Extraneous Solutions Lesson 482 — Rational Equations Leading to Quadratics Lesson 530 — Mixed Radical and Rational Equations
Identify the angle: you're working with; Lesson 954 — Finding Side Lengths Using Tangent Lesson 1011 — Applications: Angles of Any Measure in Problems
Identify the angular range: Determine α and β by looking at which directions the wedge spans.; Lesson 1917 — Integrating over Sectors and Wedges
Identify the base: The base is the flat polygon at the top or bottom of the prism (triangle, pentagon, hexagon, etc.; Lesson 878 — Volume of Prisms with Any Base
Identify the basic shapes: The L could be split into two rectangles; Lesson 868 — Finding Area of Composite Figures by Addition
Identify the boundary curves: (where do x and y start and stop?; Lesson 1905 — Changing the Order of Integration
Identify the bounds: lower limit (start) and upper limit (stop); Lesson 670 — Evaluating Finite Sums with Sigma Notation
Identify the coefficient: of x in the denominator (that's your "a"); Lesson 1627 — Integrating Partial Fractions with Linear Terms
Identify the conjugate: of the numerator or denominator (whichever contains the radical); Lesson 1342 — Limits Involving Conjugates
Identify the curves: that bound your region (often given by equations like y = x² and y = 2x); Lesson 1656 — Finding Limits of Integration for Disk Method
Identify the dependency chain: `z` depends on `x` and `y`, which depend on `s` and `t`; Lesson 1854 — Chain Rule for Independent Variables
Identify the depth: at position y: usually h = y; Lesson 1699 — Fluid Pressure and Force on Submerged Surfaces
Identify the divisor: (the decimal you're dividing by); Lesson 91 — Dividing by Decimals
Identify the domain endpoints: (if they exist); Lesson 1507 — Critical Points in Optimization
Identify the feasible region: the area shaded by *every* inequality—this is where the solutions live; Lesson 301 — Systems of Linear Inequalities
Identify the GCF: of all coefficients (numbers); Lesson 360 — GCF with Four or More Terms
Identify the generating curve: What function, when rotated, produces your object?; Lesson 1691 — Applications: Real-World Surfaces
Identify the geometric shape: cylinder, cone, sphere, etc.; Lesson 1484 — Water Tank and Fluid Flow Problems
Identify the highest y-value: the curve reaches (if there is one); Lesson 554 — Range from Graphs
Identify the inner function: Look for composite functions (functions within functions); Lesson 1585 — u-Substitution with Exponential and Logarithmic Functions
Identify the inside function: – What's nested or "fed into" the outside?; Lesson 1411 — The Outside-Inside Method
Identify the largest number: this must be `c` (the hypotenuse candidate); Lesson 784 — Verifying Pythagorean Triples
Identify the lowest y-value: the curve reaches (if there is one); Lesson 554 — Range from Graphs
Identify the missing variable: (say, z is absent); Lesson 1799 — Cylindrical Surfaces and Traces
Identify the objective: What are you maximizing or minimizing?; Lesson 1511 — Box and Container Optimization
Identify the objective function: – What are you maximizing or minimizing?; Lesson 1872 — Applications: Optimization Problems in Two Variables
Identify the outer structure: – Is the whole expression a quotient?; Lesson 1406 — Combining Product and Quotient Rules
Identify the outermost operation: (addition, multiplication, division, etc.; Lesson 1338 — Combining Multiple Limit Laws
Identify the outside function: – What's the "last operation" applied?; Lesson 1411 — The Outside-Inside Method
Identify the parallel lines: first (often marked with arrows); Lesson 720 — Parallel Lines Cut by Multiple Transversals
Identify the parts: Separate rectangles, triangles, full circles, semicircles, or quarter-circles; Lesson 871 — Composite Figures with Circles and Semicircles
Identify the pattern: Look for an expression that appears both by itself and squared (or in another power relationship); Lesson 638 — Exponential and Logarithmic Equations with Quadratic Form
Identify the pigeons: These are usually points, line segments, or other geometric objects; Lesson 2575 — Pigeonhole Principle in Geometry
Identify the place value: you're rounding to (tenths, hundredths, etc.; Lesson 86 — Rounding Decimals
Identify the problem: The region cannot be expressed as a single Type I or Type II region; Lesson 1909 — Regions Requiring Multiple Integrals
Identify the radicand: the expression inside the square root; Lesson 550 — Domain Restrictions from Square Roots
Identify the region: 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, 0 ≤ z ≤ 2; Lesson 1932 — Applications: Density and Total Mass
Identify the region E: Determine the 3D solid's boundaries; Lesson 1956 — Computing Total Mass with Triple Integrals
Identify the region R: where the lamina exists (using the techniques for Type I or Type II regions you've learned); Lesson 1955 — Computing Total Mass with Double Integrals
Identify the right endpoint: of each subinterval: x₁, x₂, x₃, .; Lesson 1548 — Right Riemann Sums
Identify the segments: Look for corners or points where the parametrization changes formula; Lesson 1979 — Line Integrals Over Piecewise Smooth Curves
Identify the set A: the relation is defined on; Lesson 2542 — Reflexive Relations
Identify the shape: and what you're asked to find; Lesson 343 — Special Products in Word Problems Lesson 895 — Word Problems Involving Surface Area
Identify the situation: – What's being measured?; Lesson 873 — Composite Figures in Real-World Contexts
Identify the statement structure: P → Q; Lesson 2495 — Proof by Contrapositive: Method and Strategy
Identify the target value: (ideal measurement); Lesson 235 — Applications and Mixed Absolute Value Problems
Identify the three sides: of the first triangle; Lesson 741 — SSS (Side-Side-Side) Congruence Postulate
Identify the three unknowns: – assign variables like x, y, and z; Lesson 299 — Applications of Three-Variable Systems
Identify the Triangle(s): Lesson 982 — Mixed Application Problems and Problem-Solving Strategies
Identify the two factors: Lesson 1400 — Applying the Product Rule to Polynomials
Identify the two quantities: that change together; Lesson 129 — Real-World Direct Variation Problems
Identify the unknown: what varies?; Lesson 172 — Writing Expressions from Word Problems Lesson 189 — Writing Equations from Word Problems
Identify the y-interval: [c, d] where the region exists; Lesson 1655 — Disk Method: Rotation About the y-axis
Identify variables: (unknowns you need to find); Lesson 1318 — Applications: Real-World Matrix Systems
Identify what technique: the resulting integral needs; Lesson 1619 — Combining Trig Substitution with Other Techniques
Identify what you're minimizing: – cost, time, materials, waste, etc.; Lesson 309 — Applications: Minimization Problems
Identify what you're optimizing: (minimize surface area?; Lesson 896 — Optimization Problems with Surface Area
Identify what's been removed: (the "hole"); Lesson 869 — Finding Area Using Subtraction Method
Identify what's changing: and what you need to find (which rate?; Lesson 1478 — Related Rates with Rectangles and Squares
Identify your angle measurement: (radians or degrees); Lesson 842 — Sector Area Formula
Identify your function: and its period (usually 2π or 2L); Lesson 2459 — Computing Fourier Series for Simple Functions
Identify your known point: (a, f(a)) where you have exact data; Lesson 1474 — Tangent Line Approximation in Applied Problems
Identify your objective function: (what to maximize/minimize) and your constraint equation; Lesson 1514 — Optimization with Implicit Constraints
Identify your variables: – What are you looking for?; Lesson 282 — Application Problems Using Systems and Substitution
identity: .; Lesson 206 — Recognizing Identity and Contradiction Equations Lesson 213 — Equations with Infinitely Many Solutions Lesson 1275 — Properties of Matrix Addition Lesson 2098 — Matrix Addition and Subtraction Lesson 2524 — Union of Sets
Identity (infinitely many solutions): When simplifying both sides leads to a statement that's *always true*, like `5 = 5` or `0 = 0`, you have an **identity**.; Lesson 206 — Recognizing Identity and Contradiction Equations
identity matrix: is a special square matrix with 1s along its main diagonal (top-left to bottom-right) and 0s everywhere else.; Lesson 1288 — Multiplying by the Identity Matrix Lesson 1295 — Determinants of Special Matrices Lesson 1300 — Definition of an Inverse Matrix Lesson 2103 — Special Matrices: Identity and Zero Lesson 2138 — Definition of Matrix Inverse Lesson 2157 — Determinant of Special Matrices
Identity with 0: gcd(*a*, 0) = |*a*| for any nonzero *a*; Lesson 2711 — Greatest Common Divisor (GCD): Definition
If: `f` is continuous on `[a, b]` and we define a new function:; Lesson 1567 — The Fundamental Theorem of Calculus: Part 1 Lesson 1737 — Direct Comparison Test: Proving Divergence Lesson 2394 — Second Shifting Theorem and Unit Step Function Lesson 2679 — Euler's Theorem for Eulerian Circuits
If |r| < 1: , the series converges to: $$S = \frac{a}{1-r}$$; Lesson 1722 — Geometric Series: Definition and Formula Lesson 1723 — Convergence of Geometric Series
If |r| ≥ 1: , the series diverges (no finite sum exists); Lesson 1722 — Geometric Series: Definition and Formula Lesson 1723 — Convergence of Geometric Series
If a < h: The side is too short to reach—**zero triangles** exist.; Lesson 1123 — Using the Height to Determine Triangle Existence
If a = h: The side just barely touches at a right angle—**exactly one triangle** (a right triangle).; Lesson 1123 — Using the Height to Determine Triangle Existence
If a ≥ b: The side is long enough that it can only connect one way—**one triangle**.; Lesson 1123 — Using the Height to Determine Triangle Existence
if and only if: the remainder is zero when dividing.; Lesson 353 — Using Division to Test Factors Lesson 1714 — Sequence of Partial Sums Lesson 2196 — Geometric and Algebraic Multiplicity Lesson 2268 — Invertible Linear Transformations Lesson 2700 — Hall's Marriage Theorem Lesson 2759 — When Moduli Are Not Coprime
If B > 1: The graph is *compressed* horizontally.; Lesson 1016 — Period Changes: y = sin(Bx) and Frequency
If c is negative: factors have *opposite* signs: **(x + __)(x - __)**; Lesson 366 — Factoring Trinomials with Negative Coefficients
If c is positive: both factors have the *same* sign (both + or both -); Lesson 366 — Factoring Trinomials with Negative Coefficients
If curl: F** = 0 everywhere on a simply connected domain, then **F** is conservative.; Lesson 1987 — Testing for Conservative Fields: The Curl Test in ℝ³
If D < 0: The critical point is a **saddle point**; Lesson 1866 — The Second Derivative Test: Statement and Conditions
If D = 0: The test is **inconclusive**; Lesson 1866 — The Second Derivative Test: Statement and Conditions
If denominators are different: Find equivalent fractions with a common denominator (often the least common multiple), then compare numerators.; Lesson 68 — Ordering Multiple Fractions
If e = 1: You get a **parabola** (equal distances—just like you learned before!; Lesson 1182 — Focus-Directrix Definition of Conics
If e > 1: You get a **hyperbola** (closer to focus than directrix); Lesson 1182 — Focus-Directrix Definition of Conics
If f''(c) < 0: (negative): The function is **concave down** at c, so you have a **local maximum** (the curve is like a frown ☹); Lesson 1501 — The Second Derivative Test for Local Extrema Lesson 1523 — The Second Derivative Test for Extrema
If f''(c) = 0: The test is **inconclusive**—you must fall back to the First Derivative Test or other methods; Lesson 1501 — The Second Derivative Test for Local Extrema Lesson 1523 — The Second Derivative Test for Extrema
If f''(c) > 0: (positive): The function is **concave up** at c, so you have a **local minimum** (the curve is like a smile ☺ ); Lesson 1501 — The Second Derivative Test for Local Extrema Lesson 1523 — The Second Derivative Test for Extrema
If found: "Augment" the matching by flipping edge statuses along the path (add unmatched edges, remove matched ones).; Lesson 2702 — The Hungarian Algorithm for Maximum Bipartite Matching
If ker(T) = {0}: (only the zero vector maps to zero), then T is injective.; Lesson 2261 — The Kernel (Null Space) of a Linear Transformation
If not: , find a vector in the space that is *not* in the span of your current set; Lesson 2090 — Extending to a Basis
If not exact: Try finding an integrating factor to *make* it exact.; Lesson 2336 — Strategy Guide: Choosing the Right Solution Method
If p > 1: The series **converges**; Lesson 1726 — Convergence of p-Series
If p ≤ 1: The series **diverges**; Lesson 1726 — Convergence of p-Series
If they're dependent: , at least one vector is a linear combination of others; Lesson 2089 — Finding a Basis from a Spanning Set
Ignore: edges leading to already-visited vertices (these would create cycles); Lesson 2674 — Finding Spanning Trees: Breadth-First Search
Ignore interior edges: Where two shapes meet inside the composite figure, those shared edges are NOT part of the perimeter.; Lesson 870 — Perimeter of Composite Figures
Ignoring the Triangle Inequality: Lesson 1128 — Common Errors and Verification in SSA
image: .; Lesson 921 — Introduction to Geometric Transformations Lesson 2262 — The Range (Image) of a Linear Transformation
Image compression: Store large images with fewer numbers; Lesson 2288 — Low-Rank Approximation and Data Compression Lesson 2290 — Applications of SVD: Image Compression and Noise Reduction
Image point: (kx, ky); Lesson 935 — Dilations Centered at the Origin
imaginary part: (the coefficient of i); Lesson 1190 — Standard Form of Complex Numbers Lesson 1191 — Identifying Real and Imaginary Parts Lesson 1192 — Plotting Complex Numbers on the Complex Plane Lesson 1207 — Complex Numbers as Points in the Plane Lesson 1208 — The Argand Diagram
imaginary unit: , denoted as *i*, defined by:; Lesson 1188 — What Are Complex Numbers?Lesson 1189 — The Imaginary Unit i
Imperial units: Force in pounds (lb), distance in feet (ft) → Work in foot-pounds (ft·lb); Lesson 1692 — Work Done by a Constant Force
Implication (P → Q): The logical connection between them; Lesson 2489 — Structure of Mathematical Statements
implicit differentiation: .; Lesson 1431 — The Derivative of ln(x)Lesson 1448 — What is Implicit Differentiation?Lesson 1449 — The Basic Technique of Implicit Differentiation Lesson 1514 — Optimization with Implicit Constraints
Important caveat: These properties only apply when the original series *already converge*.; Lesson 1718 — Properties of Convergent Series Lesson 2349 — The Method of Undetermined Coefficients Overview Lesson 3064 — Interpreting Slope and Intercept
Important fact: π is an *irrational* number, meaning its decimal digits go on forever without repeating; Lesson 830 — Circles and Pi (π)
Important note: The zero vector has no well-defined direction—it's the only vector with magnitude zero.; Lesson 2047 — Vector Equality and Zero Vector
Important sign note: If the equation shows (x - 3), then h = 3 (positive).; Lesson 1137 — Graphing Circles from Standard Form
impossible event: (which never happens) and the **certain event** (which always happens).; Lesson 2776 — The Empty Event and Certain Event Lesson 2782 — Probability of the Empty Set and Impossible Events
Imprecise: Hard to read exact coordinates like (2.; Lesson 281 — Comparing Graphing and Substitution Methods
improper fraction: has a numerator that is *equal to or larger than* its denominator.; Lesson 62 — Proper, Improper, and Mixed Numbers Lesson 76 — Multiplying Mixed Numbers
improper integral: breaks one or both of these conditions:; Lesson 1633 — Introduction to Improper Integrals Lesson 1634 — Type 1 Improper Integrals: Infinite Upper Limit
Improper rational function: The degree of the numerator is *greater than or equal to* the degree of the denominator.; Lesson 1622 — Proper vs Improper Rational Functions for an n×n matrix), is the matrix equivalent of the number 1 in multiplication. Lesson 2103 — Special Matrices: Identity and Zero
In plain English: When you differentiate a definite integral with respect to its upper limit, you simply get the integrand evaluated at that upper limit.; Lesson 1568 — Evaluating d/dx of Definite Integrals Lesson 2619 — The General Inclusion-Exclusion Principle Lesson 2930 — Conditional Distributions: Discrete Case
In row echelon form: Lesson 2116 — Row Echelon Form: Definition and Properties
In short: If **A** is m×n and invertible, then m = n.; Lesson 2139 — Properties of Invertible Matrices
In-degree: the number of edges *entering* (pointing toward) a vertex; Lesson 2649 — Directed Graphs (Digraphs)Lesson 2650 — Degree of a Vertex and the Handshaking Lemma
incenter: is where the three **angle bisectors** meet.; Lesson 739 — Centroid, Orthocenter, Incenter, and Circumcenter Lesson 832 — Inscribed and Circumscribed Circles
Incident: edge: An edge is incident to the vertices it connects; Lesson 2647 — What is a Graph? Vertices and Edges Lesson 2651 — Adjacency and Incidence
included angle: (the angle formed between those two sides) of one triangle are congruent to two sides and the included angle of another triangle, then the triangles themselves are congruent.; Lesson 742 — SAS (Side-Angle-Side) Congruence Postulate Lesson 1104 — SAS Case: Finding the Third Side
incompressible: no net flow in or out; Lesson 2004 — The Divergence of a Vector Field Lesson 2006 — Interpreting Divergence Physically Lesson 2012 — Incompressible Fields and the Continuity Equation
inconclusive: you must fall back to the First Derivative Test or other methods; Lesson 1501 — The Second Derivative Test for Local Extrema Lesson 1502 — Comparing First and Second Derivative Tests Lesson 1742 — The Ratio Test: Statement and Logic Lesson 1744 — When the Ratio Test is Inconclusive Lesson 1750 — Strategy: Choosing the Right Convergence Test Lesson 1866 — The Second Derivative Test: Statement and Conditions
inconsistent: and has **no solution**.; Lesson 297 — Inconsistent Three-Variable Systems Lesson 1314 — Identifying Inconsistent Systems Lesson 2137 — Applications to System Solvability Lesson 2234 — The Least Squares Problem
Inconsistent (No Solution): If you see a row that looks like `[0 0 0 | b]` where `b ≠ 0`, the system is inconsistent.; Lesson 2111 — Consistent and Inconsistent Systems
Inconsistent System (No Solution): Look for a row that looks like `[0 0 0 .; Lesson 2125 — Recognizing Inconsistent and Dependent Systems
Incorporate initial conditions: The transforms of derivatives bring in initial values automatically, just like with single ODEs.; Lesson 2397 — Solving Systems of ODEs with Laplace Transforms
increasing: on an interval if `x₁ < x₂` means `f(x₁) ≤ f(x₂)`.; Lesson 571 — Monotonicity: Increasing and Decreasing Functions Lesson 1494 — Using the First Derivative to Determine Monotonicity Lesson 1495 — Finding Intervals of Increase and Decrease Lesson 1519 — First Derivative and Intervals of Increase/Decrease Lesson 1525 — Comprehensive Curve Sketching Algorithm Lesson 2869 — Functions of Random Variables: Continuous Case
Increasing (or non-decreasing): Each term is greater than or equal to the previous one: a₁ ≤ a₂ ≤ a₃ ≤ .; Lesson 1710 — Monotonic Sequences and Boundedness
Increasing intervals: Where f'(x) > 0; Lesson 1503 — Analyzing Function Behavior with Derivatives
Increasing or decreasing trends: Lesson 1704 — Visualizing Sequences: Terms and Graphs
indefinite: if **x**ᵀA**x** takes *both* positive and negative values for different choices of **x**.; Lesson 2292 — Positive Definite, Negative Definite, and Indefinite Matrices Lesson 2293 — Eigenvalue Criterion for Definiteness
independence: matters!; Lesson 2860 — Variance of Sums: Independent Case Lesson 2864 — Moment Generating Functions (MGFs)Lesson 3040 — Assumptions and Diagnostics for t-Tests Lesson 3068 — Assumptions of Simple Linear Regression Lesson 3094 — One-Way ANOVA: The Model and Assumptions Lesson 3101 — Checking ANOVA Assumptions Lesson 3111 — Choosing and Interpreting Nonparametric Tests
Independence → Zero Covariance: (always true); Lesson 2950 — Covariance and Independence
Independence violations: Repeated measurements or clustered data (students within schools) break both tests.; Lesson 3051 — Applications and Limitations
independent: .; Lesson 2067 — Testing Linear Independence in R² Lesson 2068 — Testing Linear Independence in R³ Lesson 2069 — The Matrix Test for Linear Independence Lesson 2802 — Definition of Independent Events Lesson 2803 — Independence vs. Disjoint Events Lesson 2810 — Applications: Reliability and Success Probabilities Lesson 2887 — Hypergeometric vs. Binomial: When to Use Each Lesson 2893 — Applications of the Poisson Distribution (+2 more)
Independent events: satisfy P(A ∩ B) = P(A) · P(B).; Lesson 2803 — Independence vs. Disjoint Events
indeterminate forms: .; Lesson 1339 — When Limit Laws Don't Apply Lesson 1340 — Indeterminate Forms: 0/0 and ∞/∞Lesson 1347 — Exponential and Logarithmic Limits Lesson 1527 — Indeterminate Forms 0/0 and ∞/∞Lesson 1814 — Limits Involving Indeterminate Forms
index: , and it tells you how many times to multiply the answer by itself:; Lesson 153 — Cube Roots and Higher-Order Roots Lesson 495 — Understanding Radicals and the Square Root Symbol Lesson 531 — Converting Between Radical and Exponential Form
index variable: (like *i*, *k*, or *n*) is just a placeholder that counts through values.; Lesson 671 — Index Shifts and Changing Bounds Lesson 678 — Finite Series and Sigma Notation
indicial equation: is an algebraic equation that determines the possible values of *r*.; Lesson 2402 — Frobenius Method: Finding Indicial Equations Lesson 2403 — Frobenius Solutions with Distinct Roots
Indirect measurement: uses right triangle trigonometry twice, working through two connected triangles to find your answer.; Lesson 980 — Two Right Triangle Problems and Indirect Measurement Lesson 1100 — Applications of the Law of Sines
Induction → Well-Ordering: Suppose some non-empty set S of natural numbers has no smallest element.; Lesson 2519 — Equivalence of Induction and Well-Ordering
Inductive: Assume 2^k > k² for some k ≥ 5.; Lesson 2505 — Proving Inequalities by Induction
inductive hypothesis: ).; Lesson 2501 — The Principle of Mathematical Induction Lesson 2504 — Proving Divisibility Properties Lesson 2508 — Induction Starting at Arbitrary Indices Lesson 2509 — Common Mistakes in Induction Proofs Lesson 2510 — Induction on Recursively Defined Sequences Lesson 2513 — The Strong Induction Principle Lesson 2514 — Proving Divisibility Results with Strong Induction Lesson 2515 — Strong Induction for Sequences and Recurrences
Inductive Step: Lesson 1225 — De Moivre's Theorem: Statement and Proof Lesson 2501 — The Principle of Mathematical Induction Lesson 2503 — Proving Summation Formulas Lesson 2504 — Proving Divisibility Properties Lesson 2505 — Proving Inequalities by Induction Lesson 2506 — Induction with Geometric Series Lesson 2508 — Induction Starting at Arbitrary Indices Lesson 2510 — Induction on Recursively Defined Sequences (+5 more)
Inefficient estimates: While coefficient estimates remain unbiased, they're no longer the most precise possible; Lesson 3085 — Detecting Non-Constant Variance (Heteroscedasticity)
Inequality notation: `-2 < x < 4`; Lesson 231 — Solving 'And' Compound Inequalities
Infinite limits of integration: One or both limits extend to infinity (∞ or -∞); Lesson 1633 — Introduction to Improper Integrals
infinite series: is what you get when you try to add up infinitely many terms.; Lesson 679 — Infinite Series: Convergence vs Divergence Lesson 1713 — Definition of an Infinite Series
Infinite Solutions (Dependent System): You get a true statement like `0 = 0` or `4 = 4`.; Lesson 290 — Special Cases: No Solution and Infinite Solutions
infinitely: in both directions; Lesson 696 — Lines: Infinite Straight Paths Lesson 699 — Planes: Flat Surfaces Extending Infinitely
Infinitely many real eigenvalues: λ₁ < λ₂ < λ₃ < .; Lesson 2405 — Eigenvalue Problems and Sturm-Liouville Theory
infinitely many solutions: .; Lesson 213 — Equations with Infinitely Many Solutions Lesson 229 — Absolute Value Equations with No Solution or All Solutions Lesson 276 — Graphing Systems with Parallel and Coincident Lines Lesson 280 — Identifying No Solution or Infinite Solutions via Substitution Lesson 290 — Special Cases: No Solution and Infinite Solutions Lesson 292 — Introduction to Three-Variable Systems Lesson 298 — Dependent Three- Variable Systems Lesson 1889 — Dealing with Redundant Constraints (+5 more)
Inflated standard errors: Coefficient estimates become less precise; Lesson 3080 — Multicollinearity: Detection and Consequences
inflection point: is a point on a curve where the **concavity changes**—that is, where the graph switches from curving upward (concave up) to curving downward (concave down), or vice versa.; Lesson 1500 — Inflection Points Lesson 1522 — Inflection Points
Inflection points: Where concavity changes (f''(x) = 0 or undefined, with sign change); Lesson 1503 — Analyzing Function Behavior with Derivatives Lesson 1525 — Comprehensive Curve Sketching Algorithm
influential point: is a data point that dramatically changes the regression line if removed.; Lesson 3071 — Cautions and Limitations Lesson 3087 — Influential Points and Leverage
Influential points: actually *do* pull the regression line significantly.; Lesson 3089 — Outliers in Regression
initial condition: .; Lesson 1544 — Initial Value Problems Lesson 1545 — Applications: Position from Velocity, Velocity from Acceleration Lesson 2304 — Initial Value Problems Lesson 2324 — Initial Value Problems for Linear ODEs Lesson 2422 — Determining Coefficients from Initial Conditions Lesson 2423 — Heat Equation on a Finite Rod
initial conditions: (or boundary conditions) that fix the values of the arbitrary constants.; Lesson 2303 — General vs Particular Solutions Lesson 2411 — Initial and Boundary Conditions Lesson 2417 — Initial and Boundary Conditions Lesson 2451 — Applying Initial Conditions with Fourier Series Lesson 2627 — What is a Recurrence Relation?Lesson 2628 — Initial Conditions and Solutions Lesson 2635 — General Solution of Non-Homogeneous Recurrences
Initial displacement: u(x,0) = f(x): the shape of the medium at time t = 0; Lesson 2428 — Initial Conditions and Wave Propagation
initial value problem: (IVP) gives you extra information: the value of the function at a specific point, called an **initial condition**.; Lesson 1544 — Initial Value Problems Lesson 2304 — Initial Value Problems Lesson 2311 — Initial Value Problems with Separable ODEs
Initial Value Problem (IVP): combines:; Lesson 2304 — Initial Value Problems Lesson 2324 — Initial Value Problems for Linear ODEs Lesson 2344 — Initial Value Problems for Second-Order ODEs
initial value problems (IVPs): where all conditions are given at a single point—typically telling you both the function value and its derivative at the start.; Lesson 2345 — Boundary Value Problems: Introduction Lesson 2404 — Boundary Value Problems: Introduction and Types
Initial velocity: u_t(x,0) = g(x): how fast each point is moving initially; Lesson 2428 — Initial Conditions and Wave Propagation
Initialize: Pick any starting vertex and add it to the MST.; Lesson 2676 — Minimum Spanning Trees: Prim's Algorithm Lesson 2706 — The Ford-Fulkerson Algorithm
injective: (one-to-one):; Lesson 2261 — The Kernel (Null Space) of a Linear Transformation Lesson 2559 — Introduction to Function Types Lesson 2560 — Injective Functions (One-to-One)Lesson 2564 — Bijective Functions (One-to-One Correspondence)Lesson 2565 — Inverse Functions and Bijections Lesson 2568 — Examples with Finite Sets Lesson 2569 — Applications to Infinite Sets
Injective (one-to-one): Different inputs produce different outputs; Lesson 2269 — Isomorphisms Between Vector Spaces
Injectivity (one-to-one): Each output comes from at most one input.; Lesson 2565 — Inverse Functions and Bijections
Inner: Multiply the inner terms: `3 · x = 3x`; Lesson 326 — Multiplying Two Binomials Using FOIL Lesson 335 — Square of a Binomial
Inner boundaries: (holes): traverse clockwise (negative orientation); Lesson 1999 — Green's Theorem for Regions with Holes
Inner curve: r = g₁(θ) (closer to the origin); Lesson 1918 — Regions Bounded by Polar Curves
Inner function: g: takes x as its input and feeds into f; Lesson 1409 — The Chain Rule Statement Lesson 1433 — Chain Rule with Exponential Functions
Inner integral: Hold x constant and integrate f(x,y) with respect to y from c to d.; Lesson 1894 — Fubini's Theorem for Rectangles Lesson 1899 — Double Integrals of Non-Separable Functions Lesson 1902 — Type I Regions: Vertically Simple Regions
Inner operation: Adding 5 to x → so `g(x) = x + 5`; Lesson 561 — Decomposing Functions
inner product: generalizes this concept to *any* vector space by requiring four essential properties that capture what makes a "product" between vectors meaningful.; Lesson 2200 — Definition of Inner Product Spaces Lesson 2201 — Standard Inner Products on ℝⁿ and ℂⁿ Lesson 2457 — Orthogonality of Trigonometric Functions
inner radius: r(x) is the distance from the axis to the inner curve; Lesson 1657 — Washer Method: Introduction and Setup Lesson 1658 — Washer Method: Identifying Outer and Inner Radii Lesson 1659 — Washer Method: Rotation About the x-axis
Inner radius r(x): The function that is *lower down* (smaller y-value) from the x-axis; Lesson 1658 — Washer Method: Identifying Outer and Inner Radii
Inner radius r(y): The function that is *closer left* (smaller x-value) from the y-axis; Lesson 1658 — Washer Method: Identifying Outer and Inner Radii
Inner variable (usually r): What are the smallest and largest radial distances?; Lesson 1936 — Setting Up Triple Integrals in Cylindrical Coordinates
Innermost integral: Integrate with respect to x (treating y and z as constants); Lesson 1923 — Setting Up Triple Integrals over Rectangular Boxes
Input: A connected graph with all vertices of even degree.; Lesson 2681 — Finding Eulerian Paths and Circuits: Algorithms
inscribed circle: (incircle)—the largest circle that fits inside the triangle.; Lesson 739 — Centroid, Orthocenter, Incenter, and Circumcenter Lesson 832 — Inscribed and Circumscribed Circles
Inscribed circles: sit *inside* a polygon and touch all its sides.; Lesson 832 — Inscribed and Circumscribed Circles
Inside: The expression inside the parentheses → g(x) = 3x² + 5; Lesson 1411 — The Outside-Inside Method Lesson 1993 — Introduction to Green's Theorem
Inside a circle: The region `x² + y² ≤ r²` can be described as `-√(r² - x²) ≤ y ≤ √(r² - x²)` for `-r ≤ x ≤ r`.; Lesson 1906 — Regions Bounded by Curves
Inside derivative: `d/dx[3x] = 3`; Lesson 1444 — Chain Rule with Inverse Trigonometric Functions
Inside the radius: If |x - c| < R, the series converges (absolutely); Lesson 1763 — Radius of Convergence
instantaneous rate of change: the rate at which a function is changing at a single point, not over an interval.; Lesson 1372 — The Concept of Instantaneous Rate of Change Lesson 1377 — Average Rate of Change Lesson 1378 — Instantaneous Rate of Change Lesson 1379 — The Difference Quotient Lesson 1380 — The Limit Definition of the Derivative Lesson 1382 — Computing Derivatives from the Definition: Quadratic Functions Lesson 1387 — Derivative Notation and Interpretation Lesson 1388 — The Derivative of a Constant Function (+2 more)
Instantaneous velocity: = limit as h→ 0 of `[s(t+h) - s(t)]/h`; Lesson 1376 — Applications: Velocity and Other Rates Lesson 1493 — Real-World Interpretation: Speed and Distance
Integral: V = ∫[0 to 2] 2πx · x² dx = 2π ∫[0 to 2] x³ dx = 2π[x⁴/4] from 0 to 2 = 8π; Lesson 1666 — Revolving Around the y-axis Lesson 1667 — Shell Method with Horizontal Shells Lesson 2855 — Definition of Expectation for Continuous Random Variables
Integral Test: creates a bridge between improper integrals and infinite series.; Lesson 1731 — The Integral Test: Statement and Conditions Lesson 1732 — Applying the Integral Test to p- Series and Beyond Lesson 1741 — Mixed Practice: Integral and Comparison Tests Lesson 1750 — Strategy: Choosing the Right Convergence Test
Integrate: the simpler expression in terms of u; Lesson 1579 — What is u-Substitution?Lesson 1600 — Powers of Sine and Cosine: Odd Powers Lesson 1601 — Powers of Sine and Cosine: Even Powers Lesson 1613 — Substitution for √(a²-x²): Using x = a sin θ Lesson 1655 — Disk Method: Rotation About the y-axis Lesson 1676 — Computing Arc Length: Polynomial Functions Lesson 1780 — Differentiation and Integration of Power Series Lesson 1956 — Computing Total Mass with Triple Integrals (+2 more)
Integrate and substitute back: Complete the integration and replace u with the original expression; Lesson 1585 — u-Substitution with Exponential and Logarithmic Functions
integrate both sides: with respect to x:; Lesson 2322 — Applying the Integrating Factor Lesson 2323 — Solving Non-Homogeneous Linear ODEs Lesson 2325 — Linear ODEs with Constant Coefficients
Integrate in order: , typically innermost to outermost; Lesson 1937 — Evaluating Triple Integrals Using Cylindrical Coordinates
Integrate ρ first: when the region is bounded by spheres; Lesson 1942 — Evaluating Triple Integrals Using Spherical Coordinates
Integrating: the function over the entire region (summing all values); Lesson 1931 — Average Value of a Function in Three Dimensions Lesson 2846 — CDF for Continuous Random Variables
Integration: `∫f(x)dx = C + Σ [c /(n+1)]·(x - a)ⁿ ¹` (same radius of convergence); Lesson 1780 — Differentiation and Integration of Power Series Lesson 2937 — Mixed Discrete-Continuous Joint Distributions
integration by parts: , which we apply by choosing the polynomial as `u` (since polynomials simplify when differentiated) and the trigonometric function as `dv` (since trig functions integrate nicely).; Lesson 1593 — Polynomial Times Trigonometric Functions Lesson 1598 — Inverse Trigonometric Function Integrals Lesson 2365 — Handling Logarithmic and Rational Forcing Functions
Intercept: Lesson 3063 — Calculating the Regression Coefficients
Intercept-only model: Just the mean of Y (no predictors at all); Lesson 3078 — F-Test for Overall Model Significance
intercepted arc: .; Lesson 836 — Central Angles and Intercepted Arcs Lesson 845 — Congruent Arcs and Central Angles
Intercepts: Lesson 612 — Properties of e^x: Growth and Behavior Lesson 1517 — Domain, Intercepts, and Symmetry Lesson 1525 — Comprehensive Curve Sketching Algorithm
interior angle: of a triangle is simply one of the three angles formed *inside* the triangle where two sides meet.; Lesson 807 — Interior Angles of a Triangle Lesson 811 — Exterior Angle Sum Theorem
Interior Angle Sum: `(n - 2) × 180°` where n is the number of sides; Lesson 813 — Using Angle Sums to Find Unknown Angles
Interior Angle Sum Formula: Lesson 814 — Determining the Number of Sides from Angle Information
Intermediate Value Theorem (IVT): formalizes this intuition mathematically.; Lesson 1356 — The Intermediate Value Theorem
interpret: any map or blueprint using proportions.; Lesson 120 — Maps and Scale Drawings Lesson 235 — Applications and Mixed Absolute Value Problems Lesson 659 — Applications of Arithmetic Sequences Lesson 1318 — Applications: Real-World Matrix Systems Lesson 1698 — Applications of Average Value Lesson 2335 — Applications of Exact and Bernoulli Equations
Interpret in context: Relate the mathematical result back to the real-world scenario (height, distance, velocity component, etc.; Lesson 1011 — Applications: Angles of Any Measure in Problems
Interpret the result: – Translate your answer back to the real situation (square feet of tile needed, linear feet of fencing required, etc.; Lesson 873 — Composite Figures in Real-World Contexts Lesson 1746 — Applying the Root Test
Interpret your results: in context (units, meaning, limitations); Lesson 1474 — Tangent Line Approximation in Applied Problems
Interpretability: Parameters have natural meanings.; Lesson 3114 — Conjugate Priors: Beta-Binomial Model
Interpretation: Lesson 3091 — Multicollinearity Diagnostics
Interpretation caution: A significant Kruskal-Wallis test tells you groups differ *somewhere*, but not *where*—you'll need follow-up comparisons.; Lesson 3111 — Choosing and Interpreting Nonparametric Tests
interquartile range (IQR): , which measures the spread of the middle 50% of your data.; Lesson 2990 — Percentiles and Quartiles Lesson 2991 — Box Plots and Outlier Detection
Intersecting Chords Theorem: states that if you multiply the lengths of the two segments of one chord, you'll get the same result as multiplying the lengths of the two segments of the other chord.; Lesson 853 — Intersecting Chords Theorem
Intersecting lines: They meet at exactly **one point**.; Lesson 700 — Intersection of Lines and Planes
Intersecting non-perpendicular lines: also meet at **one point** → **one solution**; Lesson 272 — Systems and Geometric Reasoning with Parallel/Perpendicular Lines
intersection: where both conditions are satisfied.; Lesson 231 — Solving 'And' Compound Inequalities Lesson 1883 — The Need for Multiple Constraints Lesson 1886 — Geometric Interpretation of Multiple Constraints Lesson 2083 — Intersection of Subspaces Lesson 2525 — Intersection of Sets Lesson 2529 — Laws of Set Operations Lesson 2530 — Problem Solving with Venn Diagrams
Intersection ( ∩): The event "A and B" occurs when *both* events happen simultaneously; Lesson 2777 — Operations on Events: Union and Intersection
Intersection over union: *A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)*; Lesson 2529 — Laws of Set Operations
interval: or a union of intervals on the real number line.; Lesson 2826 — The Range of a Continuous Random Variable Lesson 2834 — Computing Probabilities from PMFs
Interval notation: gives you a cleaner, more compact way to express the same solution set by showing only the *range* of values.; Lesson 222 — Interval Notation for Solutions Lesson 231 — Solving 'And' Compound Inequalities Lesson 232 — Solving 'Or' Compound Inequalities
Intervals of Increase/Decrease: The parabola increases on one side of the vertex and decreases on the other; Lesson 433 — Key Features Summary and Complete Graphing
Intuition: If you flip a fair coin 100 times, you expect 50 heads on average.; Lesson 2862 — Expectation and Variance in Common Distributions
Intuitive analogy: Imagine tapping a table with a hammer.; Lesson 2396 — The Dirac Delta Function
Invalid standard errors: Your confidence intervals and hypothesis tests for coefficients become unreliable; Lesson 3085 — Detecting Non-Constant Variance (Heteroscedasticity)
invariant under linear transformations: if you change X to aX + b (scale and shift), the correlation remains unchanged.; Lesson 2951 — Correlation Coefficient: Definition Lesson 2952 — Properties of Correlation
Inverse exists: If T is an isomorphism, then T ¹ is also an isomorphism; Lesson 2269 — Isomorphisms Between Vector Spaces
inverse functions: (functions that "undo" each other), you can now combine these tools!; Lesson 568 — Compositions Involving Inverses Lesson 1431 — The Derivative of ln(x)
Inverse scaling: Divide the moduli; Lesson 1224 — Geometric Interpretation of Polar Operations
inverse trig functions: come in.; Lesson 955 — Finding Angles Using Inverse Trig Functions Lesson 960 — Finding Missing Angles in Right Triangles
Inverse variation: works the opposite way: when one variable increases, the other *decreases* so that their *product* stays constant.; Lesson 125 — Inverse Variation Lesson 127 — Combined Variation Lesson 128 — Finding the Constant of Variation
Invert back: Use inverse Laplace transforms and partial fractions to recover the time-domain solutions x(t), y(t), etc.; Lesson 2397 — Solving Systems of ODEs with Laplace Transforms
Invert the transformation: Solve for x = g ¹(y); Lesson 2870 — The Jacobian Method for Transformations
Invertibility: If the determinant is zero, the matrix is singular (not invertible).; Lesson 2148 — What is a Determinant?Lesson 2188 — Using the Characteristic Polynomial in Practice
Invertibility testing: If det(A) ≠ 0 and det(B) ≠ 0, then det(AB) ≠ 0, so **AB** is invertible; Lesson 2159 — Determinant of a Product
invertible: (meaning it has an inverse matrix) if and only if its determinant is **nonzero**.; Lesson 1297 — Using Determinants to Test Invertibility Lesson 1300 — Definition of an Inverse Matrix Lesson 1301 — When Does an Inverse Exist?Lesson 1307 — Using Inverses to Solve Matrix Equations Lesson 1316 — Solving with Matrix Inverses Lesson 2136 — Full Rank Matrices Lesson 2138 — Definition of Matrix Inverse Lesson 2268 — Invertible Linear Transformations (+1 more)
Investment growth: Continuous compound interest follows `V(t) = V₀e^(rt)`, and `V'(t)` shows how fast your money is growing.; Lesson 1437 — Applications: Exponential Growth and Decay Rates
Inward orientation: Occasionally used; would reverse all signs; Lesson 2023 — Flux Through Closed Surfaces
Irregular singular point: otherwise; Lesson 2401 — Regular Singular Points and the Method of Frobenius
irrotational: (remember: conservative fields are irrotational, which we tested earlier with the curl test).; Lesson 2007 — The Curl of a Vector Field Lesson 2011 — Irrotational Fields and Conservative Vector Fields
Isolate: the radical term; Lesson 529 — Word Problems Involving Radical Equations
Isolate one radical: on one side of the equation; Lesson 527 — Equations Requiring Multiple Squaring Steps
Isolate the exponential term: (get e^(something) by itself); Lesson 615 — Natural Exponential Equations
Isolate the radical: on one side of the equation; Lesson 524 — Solving Simple Radical Equations Lesson 528 — Radical Equations with Higher Index Roots
Isolate the radical term: if it's not already alone.; Lesson 530 — Mixed Radical and Rational Equations
Isolate the remaining radical: if one exists; Lesson 527 — Equations Requiring Multiple Squaring Steps
Isolate the single logarithm: on one side of the equation; Lesson 636 — Solving Logarithmic Equations with Multiple Logs
Isolate the term: with the fractional exponent (get it alone on one side); Lesson 537 — Solving Equations with Fractional Exponents
Isolate the trig function: if needed; Lesson 1084 — Equations with Multiple Angles
isolate the variable: get the variable alone on one side of the equation.; Lesson 200 — Multiplication and Division Properties of Equality Lesson 211 — Multi-Step Equations Requiring Multiple Operations Lesson 216 — Mixed Practice: Complex Linear Equations Lesson 221 — Inequalities with Variables on Both Sides
Isolate x: using simple algebra:; Lesson 631 — Using Logarithms to Solve Exponential Equations
Isolated values: Each possible value is separated from the next—no "in-between" values exist; Lesson 2823 — Discrete Random Variables: Definition and Examples
isometries: (same measure).; Lesson 928 — Properties of Reflections Lesson 939 — Introduction to Rigid Motions (Isometries)Lesson 941 — Congruence Through Rigid Motions
isometry: ) is a movement that slides, flips, or turns a figure without changing its dimensions.; Lesson 724 — Congruence Transformations Lesson 924 — Properties of Translations Lesson 940 — Composition of Rigid Motions
Isomorphism preserves everything: Linear independence, spanning, bases, dimension—all structural properties transfer perfectly; Lesson 2269 — Isomorphisms Between Vector Spaces
Isosceles trapezoids: and **kites** are cousins in this extended family with their own unique characteristics.; Lesson 825 — The Quadrilateral Family Tree
Isosceles Triangle: Lesson 730 — Classifying Triangles by Sides Lesson 734 — Base Angles of Isosceles Triangles
Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent.; Lesson 734 — Base Angles of Isosceles Triangles
iterated integral: two successive single integrals:; Lesson 1892 — Introduction to Double Integrals Lesson 1893 — Setting Up Double Integrals over Rectangles Lesson 1894 — Fubini's Theorem for Rectangles
IX = X: , this simplifies to:; Lesson 1307 — Using Inverses to Solve Matrix Equations Lesson 1316 — Solving with Matrix Inverses

J

J = [∂x/∂u ∂x/∂v]: Lesson 1945 — Transformations and the Jacobian Matrix
j-th column: from matrix **B**; Lesson 1282 — Computing the (i,j)-Entry of a Product Lesson 1284 — Multiplying Larger Matrices
Jacobian: of the transformation from rectangular to cylindrical coordinates.; Lesson 1935 — Volume Elements in Cylindrical Coordinates Lesson 2870 — The Jacobian Method for Transformations
Jacobian determinant: quantifies exactly this scaling factor.; Lesson 1946 — The Jacobian Determinant Lesson 1947 — Change of Variables Formula for Double Integrals Lesson 1951 — Jacobians in Three Dimensions Lesson 1953 — Applications: Area and Volume Distortion
Jacobian matrix: captures how the transformation behaves locally—how small changes in (u, v) affect (x, y).; Lesson 1945 — Transformations and the Jacobian Matrix
JAGS/BUGS: Specify models in a simple language; software runs Gibbs sampling; Lesson 3121 — Computational Tools: Introduction to MCMC and Bayesian Software
Joint PMF-PDF: *p_X,Y(x, y) = P(X = x) · f_{Y|X}(y|x)*; Lesson 2937 — Mixed Discrete-Continuous Joint Distributions
Joint variation: combines these ideas: a variable varies *directly* with **two or more** other variables simultaneously.; Lesson 126 — Joint Variation Lesson 128 — Finding the Constant of Variation
Jordan canonical form: (beyond this lesson's scope), or work directly with the matrix without diagonalizing it.; Lesson 2195 — When Diagonalization Fails
jump discontinuities: where the discrete masses occur, and **smooth increases** where the continuous portions lie.; Lesson 2840 — Mixed Random Variables Lesson 2850 — Mixed Random Variables and Their CDFs
jump discontinuity: like a staircase with a sudden step up or down.; Lesson 593 — Continuity of Piecewise Functions Lesson 1333 — One-Sided Limits at Jump Discontinuities Lesson 2464 — Gibbs Phenomenon
Just touch: (exactly one triangle); Lesson 1096 — The SSA Configuration Introduction Lesson 1121 — Introduction to the Ambiguous Case (SSA)
Justify everything: Every statement needs a valid reason; Lesson 748 — Writing Two-Column Congruence Proofs

K

k = 1: is where we start; Lesson 678 — Finite Series and Sigma Notation Lesson 934 — Introduction to Dilations and Scale Factor Lesson 935 — Dilations Centered at the Origin Lesson 1229 — Finding All nth Roots Using Polar Form
k = 4: and; Lesson 1276 — Scalar Multiplication of Matrices Lesson 2602 — Finding Specific Terms in Binomial Expansions
k > 1: The figure enlarges (gets bigger).; Lesson 934 — Introduction to Dilations and Scale Factor Lesson 935 — Dilations Centered at the Origin
Keep: the first number; Lesson 44 — Subtracting Positive and Negative Numbers
Keep going: 4?; Lesson 48 — Understanding Factors
Keep the common exponent: Lesson 167 — Adding and Subtracting in Scientific Notation
Keep the denominator: – The bottom number stays the same; Lesson 69 — Adding Fractions with Like Denominators
Keep the first vector: as-is: **u**₁ = **v**₁; Lesson 2220 — The Gram-Schmidt Algorithm: Overview
Keep unlike terms separate: Constants stand alone, and different variables don't combine; Lesson 179 — Combining Like Terms
Kendall's Tau: takes a completely different approach: it examines every possible pair of observations and counts whether they're **concordant** (rankings agree) or **discordant** (rankings disagree).; Lesson 3110 — Kendall's Tau
kernel: ) of a matrix A is the collection of all vectors **x** that satisfy the homogeneous equation:; Lesson 2130 — The Null Space Lesson 2261 — The Kernel (Null Space) of a Linear Transformation
Key detail: The value `a` is always under the positive term and represents the distance from center to each vertex.; Lesson 1167 — Definition and Standard Form of a Hyperbola
Key distinction: This is *different* from (f ∘ g)⁻¹, which would be the inverse of the entire composition.; Lesson 568 — Compositions Involving Inverses
Key features: Lesson 1798 — Quadric Surfaces: Hyperboloids and Cones Lesson 2835 — Tabular and Graphical Representations of PMFs
Key insight: This equation has integer solutions if and only if `gcd(a, b)` divides `c`.; Lesson 2719 — Applications of the Euclidean Algorithm Lesson 2809 — Common Misconceptions About Independence Lesson 2871 — Order Statistics
Key pattern: The asymptote slopes use the coefficients from the denominators.; Lesson 1170 — Asymptotes of a Hyperbola
Key points: Lesson 773 — What is the Pythagorean Theorem?Lesson 1026 — Graphing the Cotangent Function
Key properties: Lesson 2522 — Universal Set and Empty Set
Key relationship: For any eigenvalue, geometric multiplicity ≤ algebraic multiplicity.; Lesson 2173 — Algebraic vs Geometric Multiplicity
Key relationships: Lesson 1969 — Vector Fields in Polar and Cylindrical Coordinates
Key series technique: Recall that the geometric series gives us:; Lesson 2881 — Expectation of the Geometric Distribution
Kinetic Energy: of a variable-mass system uses integration.; Lesson 1702 — Other Physical Applications: Energy and Probability Lesson 2432 — Energy Conservation in Wave Motion
Known: `l = 12`, `P = 40`; Lesson 193 — Perimeter and Area Word Problems Lesson 3016 — Confidence Interval for the Difference of Two Means
König's Theorem states: In any bipartite graph, the size of a maximum matching equals the size of a minimum vertex cover.; Lesson 2703 — König's Theorem and Vertex Covers
Kruskal-Wallis test: is the nonparametric cousin of one-way ANOVA.; Lesson 3107 — Kruskal-Wallis Test
Kruskal's algorithm: finds an MST using a simple greedy strategy:; Lesson 2675 — Minimum Spanning Trees: Kruskal's Algorithm
Kuratowski's Theorem: (beyond our current scope) states that any non-planar graph contains a subdivision of K₅ or K₃,₃ —making them the fundamental building blocks of non-planarity.; Lesson 2693 — K₅ and K₃,₃ are Non-Planar
Kurtosis: measures *tail behavior*: Does the distribution have heavy tails (prone to extreme values) or light tails?; Lesson 2861 — Higher Moments: Skewness and Kurtosis Lesson 2863 — Higher Moments: Skewness and Kurtosis
Kurtosis < 3: Light tails (platykurtic)—less probability in extremes; Lesson 2863 — Higher Moments: Skewness and Kurtosis
Kurtosis = 3: Normal distribution (reference point); Lesson 2863 — Higher Moments: Skewness and Kurtosis
Kurtosis > 3: Heavy tails (leptokurtic)—more probability in extreme values; Lesson 2863 — Higher Moments: Skewness and Kurtosis

L

L < 1: , the series **converges absolutely**; Lesson 1742 — The Ratio Test: Statement and Logic Lesson 1744 — When the Ratio Test is Inconclusive Lesson 1745 — The Root Test: Statement and Motivation Lesson 1746 — Applying the Root Test Lesson 1748 — Series with Factorials and Exponentials
L = 1: , the test is **inconclusive** (try another test); Lesson 1742 — The Ratio Test: Statement and Logic Lesson 1744 — When the Ratio Test is Inconclusive Lesson 1745 — The Root Test: Statement and Motivation Lesson 1746 — Applying the Root Test Lesson 1748 — Series with Factorials and Exponentials
L > 1: (or L = ∞), the series **diverges**; Lesson 1742 — The Ratio Test: Statement and Logic Lesson 1744 — When the Ratio Test is Inconclusive Lesson 1745 — The Root Test: Statement and Motivation Lesson 1746 — Applying the Root Test Lesson 1748 — Series with Factorials and Exponentials
L'Hôpital's Rule: Suppose functions f(x) and g(x) are differentiable near point a (except possibly at a itself), and:; Lesson 1528 — Statement of L'Hôpital's Rule Lesson 1711 — Computing Limits Using L'Hôpital's Rule
L(x, y): called the **linearization**.; Lesson 1836 — Linear Approximation in Two Variables
L(x): is your approximation for f(x); Lesson 1469 — Linear Approximation Formula
L[u] = 0: Lesson 2413 — Superposition Principle for Linear PDEs
Label carefully: Mark corresponding vertices clearly.; Lesson 758 — Similar Triangles in Nested Figures
Label everything: Lesson 968 — Drawing Diagrams for Angle Problems
Label the radius: (distance to axis) and **height** (length of rectangle); Lesson 1665 — Setting Up a Shell Integral
Label the sides: relative to the given angle (opposite, adjacent, hypotenuse); Lesson 974 — Setting Up Right Triangle Problems from Word Descriptions
Label your final answer: with correct units; Lesson 123 — Real-World Proportion Problems
Label your points: Lesson 245 — Calculating Slope from Two Points
Lagrange Remainder: gives you a formula for that error, and more importantly, lets you calculate the *worst-case scenario*—the maximum error you might encounter.; Lesson 1786 — Using the Lagrange Remainder to Bound Error Lesson 1791 — Error Analysis in Numerical Computation
Lagrange's Theorem: From group theory, Lagrange's theorem tells us that for any element *g* in a finite group of order *k*, we have *g^k = e* (where *e* is the identity).; Lesson 2749 — Proof of Euler's Theorem
lamina: ) with density that depends on position, you can find its mass and balance point using double integrals.; Lesson 1921 — Applications: Mass and Center of Mass in Polar Lesson 1958 — Center of Mass in Two Dimensions Lesson 1961 — Computing Moments of Inertia
Land Surveying: A triangular plot has sides of 85 meters and 120 meters meeting at a 65° angle.; Lesson 1120 — Real-World Area Applications
Landscape Design: Planning a triangular garden bed where you can measure the base along a fence (12 feet) and the perpendicular distance to the opposite corner (8 feet)?; Lesson 1120 — Real-World Area Applications
Landscaping: A flower bed could be a rectangle minus a semicircular cutout; Lesson 873 — Composite Figures in Real-World Contexts
Laplace transform: is an integral operator that takes a function of time, *f(t)*, and transforms it into a function of a new variable *s*, denoted *F(s)*.; Lesson 2388 — Definition of the Laplace Transform
Laplacian: in two dimensions, capturing how the membrane curves in both the x and y directions.; Lesson 2433 — The Two-Dimensional Wave Equation
Laplacian operator: ∇²:; Lesson 2436 — Introduction to Laplace's Equation
Large errors hurt more: A point far from the line gets heavily penalized (distance of 2 becomes 4, distance of 3 becomes 9); Lesson 3062 — The Least Squares Criterion
Large p-value: Insufficient evidence; you can't rule out that the coefficient is just noise; Lesson 3077 — Hypothesis Testing for Individual Coefficients
Large sample size: Typically n ≥ 30, so the Central Limit Theorem ensures your sampling distribution is approximately normal; Lesson 3031 — Choosing Between z-Tests and Other Tests
Large samples: (n ≥ 30): t and z give similar results, but t is still more appropriate; Lesson 3032 — Why t-Tests? When the Population Variance is Unknown
Larger: than your original series (term-by-term); Lesson 1736 — Direct Comparison Test: Proving Convergence Lesson 1737 — Direct Comparison Test: Proving Divergence
Larger effect sizes: → need fewer participants to detect; Lesson 3059 — Sample Size Determination
Last: Multiply the last terms: `3 · 5 = 15`; Lesson 326 — Multiplying Two Binomials Using FOIL Lesson 335 — Square of a Binomial
last term: is always `a²` (the square of the second term); Lesson 401 — Perfect Square Trinomials Review Lesson 657 — Finding the Number of Terms in a Finite Sequence Lesson 693 — Stacking and Construction Problems
Lateral surface area: πr*ℓ* (imagine the curved surface unwrapped); Lesson 890 — Surface Area of Cones
Law of Cosines: comes to the rescue.; Lesson 1102 — Introduction to the Law of Cosines Lesson 1103 — Deriving the Law of Cosines Lesson 1105 — SSS Case: Finding an Angle Lesson 1106 — Choosing Between Law of Sines and Law of Cosines Lesson 1111 — Complex Problems Combining Triangle Laws
Law of Large Numbers: (both weak and strong forms) guarantees that as we take more and more independent samples, the sample average converges to the true expected value.; Lesson 2976 — Monte Carlo Methods and Simulation
Law of Sines: is your key tool for solving these general triangles.; Lesson 1092 — The Law of Sines Statement Lesson 1106 — Choosing Between Law of Sines and Law of Cosines Lesson 1111 — Complex Problems Combining Triangle Laws
Law of Total Expectation: when dealing with expectations) says you can break this problem into simpler pieces by conditioning on another variable \(X\).; Lesson 2935 — The Law of Total Probability for Random Variables
Law of Total Probability: .; Lesson 2798 — Partition Problems and Conditional Probability Lesson 2811 — The Law of Total Probability Lesson 2813 — Motivation for Bayes' Theorem Lesson 2816 — Bayes' Theorem with Multiple Hypotheses Lesson 2935 — The Law of Total Probability for Random Variables
Layer multiple techniques: combine inclusion-exclusion with permutations, combinations, the pigeonhole principle, or derangements; Lesson 2626 — Complex Applications and Problem-Solving Strategies
LCD: 12; Lesson 82 — Word Problems Involving Fraction Operations Lesson 209 — Equations with Fractions: Clearing Denominators Lesson 461 — Finding the Least Common Denominator (LCD)Lesson 480 — Solving Equations with Multiple Terms
leading coefficient: is the number multiplied by the variable(s) in that leading term.; Lesson 317 — Leading Coefficient and Leading Term Lesson 422 — Parabola Basics: Shape and Orientation
leading entry: or **pivot**) is to the right of the leading entry in the row above it; Lesson 1311 — Row Reduction to Echelon Form Lesson 2109 — Pivot Positions and Leading Entries Lesson 2117 — Leading Entries and Pivot Positions
Least Common Denominator (LCD): to rewrite both fractions as equivalent fractions with the same denominator.; Lesson 73 — Subtracting Fractions with Unlike Denominators Lesson 205 — Equations with Fractions or Decimals Lesson 461 — Finding the Least Common Denominator (LCD)Lesson 479 — Multiplying by the LCD Method
Least Common Multiple (LCM): is the smallest positive number that is a multiple of two or more given numbers.; Lesson 54 — Least Common Multiple (LCM)Lesson 71 — Finding the Least Common Denominator (LCD)Lesson 287 — Multiplying Both Equations to Enable Elimination
Least squares solution x̂: produces **Ax̂**; Lesson 2236 — Least Squares via Orthogonal Projection
Leave non-perfect parts: under the radical; Lesson 415 — Simplifying Radical Answers
Leave unpaired factors: under the radical; Lesson 155 — Simplifying Square Roots
left: of zero; Lesson 40 — Representing Negative Numbers on the Number Line Lesson 41 — Comparing and Ordering Negative Numbers Lesson 163 — Converting from Standard Form to Scientific Notation Lesson 164 — Converting from Scientific Notation to Standard Form Lesson 219 — The Multiplication/Division Sign Rule Lesson 239 — Plotting Points on the Coordinate Plane Lesson 1020 — Comparing Sine and Cosine: The Phase Relationship Lesson 1025 — Transformations of Tangent: Phase Shift and Vertical Shift (+8 more)
Left column (Statements): Sequential facts, moving from given to conclusion; Lesson 763 — Two-Column Proof Format
Left distributive: `A(B + C) = AB + AC`; Lesson 1287 — The Distributive Property for Matrices
Left Null Space: (N(A ᵀ)): All vectors **y** where A ᵀ**y** = **0**; lives in ℝᵐ; Lesson 2096 — Bases for Important Subspaces Lesson 2131 — The Left Null Space Lesson 2134 — Orthogonality of Fundamental Subspaces Lesson 2283 — Left and Right Singular Vectors
Left Null Space, N(A^T): Lesson 2132 — The Four Fundamental Subspaces
Left Rectangles: Use the function value at the *left* endpoint of each subinterval as the rectangle's height; Lesson 1546 — Approximating Area Under a Curve with Rectangles
Left Riemann Sum: specifically uses the function value at the *left* edge of each subinterval to determine each rectangle's height.; Lesson 1547 — Left Riemann Sums Lesson 1552 — Sigma Notation for Riemann Sums
Left side: The double angle formula for sine directly states that sin(2x) = 2sin(x)cos(x); Lesson 1071 — Verifying Identities with Double and Half Angle Formulas Lesson 2001 — Connection to Curl in Two Dimensions Lesson 2026 — Introduction to Stokes' Theorem Lesson 2038 — Statement of the Divergence Theorem Lesson 2043 — Physical Interpretation: Conservation Laws
Left singular vectors (U): The first r columns of **U** span the **column space** of **A**.; Lesson 2283 — Left and Right Singular Vectors
Left sum: Uses the left edge of each interval to set rectangle height; Lesson 1550 — Comparing Left, Right, and Midpoint Approximations
left to right: when mixing addition and subtraction; Lesson 17 — Adding and Subtracting Three or More Numbers Lesson 456 — Mixed Operations: Multiplication and Division Together
Left-hand limit: lim(x→ a ) f(x) — the superscript minus means "from the left"; Lesson 1326 — Limits Approaching from Left and Right Lesson 1329 — Understanding One-Sided Limits Lesson 1331 — Evaluating One-Sided Limits Graphically Lesson 1333 — One-Sided Limits at Jump Discontinuities
Left-skewed: Mean < Median < Mode (mean pulled left by extreme lows); Lesson 2993 — Shape of Distributions: Symmetry and Skewness
Left-skewed (negatively skewed): A long tail stretches to the left.; Lesson 2993 — Shape of Distributions: Symmetry and Skewness
Leg formula: leg = √(hypotenuse segment × whole hypotenuse); Lesson 804 — Finding Missing Lengths with Geometric Means
Legendre's equation: for the polar angle θ; Lesson 2454 — Separation in Other Coordinate Systems
legs: (the sides that form the right angle); Lesson 773 — What is the Pythagorean Theorem?Lesson 775 — Finding the Hypotenuse Lesson 821 — Trapezoids: Definitions and Basic Properties
Leibniz Test: ) states:; Lesson 1752 — The Alternating Series Test (Leibniz Test)
Lemniscate of Bernoulli: Lesson 1457 — Applications and Complex Implicit Curves
Lemniscates: (like r² = cos(2θ)): Figure-eight loops.; Lesson 1919 — Computing Areas Using Polar Integrals
length: (or magnitude) of a vector and use that to compute the distance between any two points in space.; Lesson 2053 — Magnitude and Distance Lesson 2171 — Geometric Interpretation of Eigenvectors
Let x = -2: Lesson 1624 — Solving for Coefficients: Substitution Method
Let x = 3: Lesson 1624 — Solving for Coefficients: Substitution Method
Level 0: the starting vertex; Lesson 2674 — Finding Spanning Trees: Breadth-First Search
level curve: is the set of all points (x, y) where the function equals a constant value k.; Lesson 1800 — Level Curves and Contour Maps Lesson 1805 — Level Curves and Contour Maps Lesson 1966 — Gradient Vector Fields
Level k: all vertices at distance exactly *k* from the start; Lesson 2674 — Finding Spanning Trees: Breadth-First Search
leverage: in regression.; Lesson 3087 — Influential Points and Leverage Lesson 3088 — Cook's Distance and Influence Measures
Leverage values: identify high-leverage points (points with unusually extreme X values); Lesson 3089 — Outliers in Regression
LIATE: is a memory tool that ranks function types from highest to lowest priority for choosing *u*:; Lesson 1591 — Choosing u and dv: The LIATE Rule
Like radicals: are radicals with the *same index* (the little number outside) and the *same radicand* (the number inside).; Lesson 158 — Adding and Subtracting Radicals Lesson 506 — Identifying Like and Unlike Radicals Lesson 513 — Mixed Operations with Radicals Lesson 520 — Simplifying After Rationalizing
Like terms: are terms in an algebraic expression that have **identical variable parts**—meaning they contain the exact same variables raised to the exact same powers.; Lesson 173 — Like Terms Lesson 179 — Combining Like Terms Lesson 186 — Simplifying Before Evaluating Lesson 318 — What is a Polynomial? Terms and Like Terms Lesson 319 — Combining Like Terms in a Polynomial Lesson 320 — Adding Polynomials Horizontally Lesson 332 — Simplifying Products by Combining Like Terms
Likelihood: How probable the observed data is under different parameter values.; Lesson 3112 — Prior, Likelihood, and Posterior: Bayes' Theorem Revisited Lesson 3113 — Discrete Parameter Spaces and Bayesian Updating Lesson 3114 — Conjugate Priors: Beta-Binomial Model Lesson 3115 — Conjugate Priors: Normal-Normal Model
limit: answers the question: "What value does a function approach as the input gets arbitrarily close to some target number?; Lesson 1319 — What is a Limit? The Approaching Idea Lesson 1325 — Limits at Points of Discontinuity Lesson 1372 — The Concept of Instantaneous Rate of Change Lesson 1705 — Limits of Sequences: Intuitive Concept
Limit Comparison: = More robust when you need to "cancel out" complexity algebraically; Lesson 1740 — Strategic Selection Between Comparison Tests Lesson 1750 — Strategy: Choosing the Right Convergence Test
Limit Comparison Test: solves this by looking at how two functions *behave similarly* as they approach infinity (or a discontinuity).; Lesson 1641 — Limit Comparison Test for Improper Integrals Lesson 1738 — The Limit Comparison Test: Motivation and Statement Lesson 1740 — Strategic Selection Between Comparison Tests Lesson 1741 — Mixed Practice: Integral and Comparison Tests
Limit to zero: lim(n → ∞) b = 0; Lesson 1753 — Verifying Conditions for the Alternating Series Test
Limitations: Lesson 281 — Comparing Graphing and Substitution Methods
Limited curves: If a curve starts at x = 2 and ends at x = 7, the domain is [2, 7].; Lesson 552 — Domain from Graphs
Limits: The ε-δ definition of a limit uses nested quantifiers:; Lesson 2486 — Nested Quantifiers in Mathematics
limits at infinity: , we're asking: "What does f(x) approach as x becomes extremely large?; Lesson 1328 — Limits at Infinity and Horizontal Asymptotes Lesson 1347 — Exponential and Logarithmic Limits
Limits of Integration: Lesson 1665 — Setting Up a Shell Integral Lesson 2959 — Convolution Formula for Continuous RVs
line: is a perfectly straight path that extends forever in both directions.; Lesson 696 — Lines: Infinite Straight Paths Lesson 697 — Line Segments and Rays Lesson 708 — Linear Pairs Lesson 920 — Intersection of Lines and Circles Lesson 2059 — The Span of a Set of Vectors Lesson 2061 — Geometric Span in R²: Lines and Planes Lesson 2062 — Geometric Span in R³: Lines, Planes, and All of R³ Lesson 2584 — Circular Permutations
Line AB: (written in words); Lesson 696 — Lines: Infinite Straight Paths
Line segment AB: Starts at A, ends at B.; Lesson 697 — Line Segments and Rays
Line up the numbers: by place value (ones under ones, tens under tens); Lesson 14 — Adding and Subtracting with Larger Numbers
Linear: Lesson 650 — Comparing Linear, Exponential, and Logarithmic Growth Lesson 2269 — Isomorphisms Between Vector Spaces Lesson 2336 — Strategy Guide: Choosing the Right Solution Method Lesson 2355 — Superposition for Multiple Terms Lesson 2390 — Linearity and First Shifting Theorem Lesson 2440 — Superposition and Non-Homogeneous Boundaries Lesson 2446 — When Separation of Variables Works Lesson 2950 — Covariance and Independence (+1 more)
linear combination: of these unit vectors.; Lesson 1239 — Unit Vector Notation: i, j, and k Lesson 1247 — Linear Combinations of Vectors Lesson 2055 — What is a Linear Combination?Lesson 2056 — Computing Linear Combinations in R² Lesson 2709 — Divisibility: Definition and Basic Properties Lesson 2715 — Extended Euclidean Algorithm Lesson 2966 — Linear Combinations of Random Variables
Linear dependence: means there's redundancy: at least one vector doesn't add new directional information because it already lies in the span of the others.; Lesson 2066 — Linear Dependence and Redundant Vectors
Linear functions: like `f(x) = 2x + 3` can output any real number, so their range is all real numbers (unless restricted by context).; Lesson 553 — Finding Range from Equations Lesson 1646 — Finding Intersection Points
Linear Growth: increases by a *constant amount* each step.; Lesson 650 — Comparing Linear, Exponential, and Logarithmic Growth
Linear independence: ensures efficiency: no basis vector is redundant or expressible using the others.; Lesson 2086 — What is a Basis?Lesson 2087 — Verifying a Set is a Basis Lesson 2144 — The Invertible Matrix Theorem
Linear only: ρ = 0 means no *linear* relationship, but nonlinear dependence may exist; Lesson 2956 — Examples and Applications of Correlation
linear pair: is a special arrangement of two adjacent angles that share a common side and whose other sides form a straight line.; Lesson 708 — Linear Pairs Lesson 709 — Vertical Angles Lesson 718 — Using Angle Relationships to Find Measures Lesson 733 — Exterior Angle Theorem Lesson 810 — Exterior Angles of Polygons
Linear pairs: always sum to 180°; Lesson 718 — Using Angle Relationships to Find Measures
Linear PDE: The unknown function u and all its derivatives appear to the first power only, with no products between them.; Lesson 2409 — Classification: Order and Linearity
Linear programming: is a method for solving real-world optimization problems.; Lesson 304 — Introduction to Linear Programming
Linear regression: answers this by finding the line that minimizes the total squared vertical distance from each data point to the line.; Lesson 2238 — Applications: Linear Regression
linear transformation: is a special type of function that maps vectors from one vector space to another while preserving the fundamental operations: vector addition and scalar multiplication.; Lesson 2259 — Definition of Linear Transformations Lesson 2260 — Examples of Linear Transformations Lesson 2867 — Linear Transformations of Random Variables
Linear, homogeneous PDEs: with constant coefficients; Lesson 2455 — Limitations and Extensions of the Method
linearity: , and it mirrors what you already know from derivatives: the derivative of a sum equals the sum of derivatives.; Lesson 1540 — Sum and Difference Rules for Antiderivatives Lesson 1559 — Properties of Definite Integrals: Linearity Lesson 2202 — Inner Products on Function Spaces Lesson 2355 — Superposition for Multiple Terms Lesson 2635 — General Solution of Non-Homogeneous Recurrences Lesson 2853 — Properties of Expectation: Linearity Lesson 3068 — Assumptions of Simple Linear Regression Lesson 3084 — Residual Plots: Fitted Values vs. Residuals
linearly dependent: .; Lesson 2065 — Definition of Linear Independence Lesson 2066 — Linear Dependence and Redundant Vectors Lesson 2163 — Determinants and Linear Independence
linearly independent: if none of the vectors can be written as a linear combination of the others.; Lesson 2065 — Definition of Linear Independence Lesson 2066 — Linear Dependence and Redundant Vectors Lesson 2067 — Testing Linear Independence in R² Lesson 2068 — Testing Linear Independence in R³ Lesson 2072 — Linear Independence in General Vector Spaces Lesson 2086 — What is a Basis?Lesson 2088 — Standard Bases Lesson 2163 — Determinants and Linear Independence (+1 more)
Lines (not horizontal): usually (-∞, ∞); Lesson 554 — Range from Graphs
Lines through the origin: any line passing through (0,0), like the span of a single nonzero vector; Lesson 2080 — Subspaces of R^2 and R^3
List all elements: of A; Lesson 2542 — Reflexive Relations
List all integers: from 2 to *n*.; Lesson 2723 — The Sieve of Eratosthenes
List all possible outcomes: in the sample space; Lesson 2824 — The Range of a Discrete Random Variable
List-able: You can write out all possible values, even if it takes forever; Lesson 2823 — Discrete Random Variables: Definition and Examples
literal equation: is any equation with more than one variable.; Lesson 187 — Using Formulas and Literal Equations Lesson 214 — Literal Equations: Solving for a Specified Variable
Load Distribution: Lesson 981 — Applications in Construction and Engineering
Local maxima and minima: Using the First Derivative Test; Lesson 1503 — Analyzing Function Behavior with Derivatives
local maximum: is a point where the function value is higher than all nearby points—like standing on a hill that might not be the tallest mountain around, but you're at the top of *that* hill.; Lesson 573 — Extrema: Local and Absolute Maxima and Minima Lesson 1496 — The First Derivative Test for Local Extrema Lesson 1501 — The Second Derivative Test for Local Extrema Lesson 1520 — Critical Points and Local Extrema Lesson 1523 — The Second Derivative Test for Extrema Lesson 1863 — Classifying Critical Points: Local Extrema and Saddle Points Lesson 1866 — The Second Derivative Test: Statement and Conditions
local minimum: is a point where the function value is lower than all nearby points—like standing in a valley that might not be the deepest point overall, but it's the lowest spot in *that* area.; Lesson 573 — Extrema: Local and Absolute Maxima and Minima Lesson 1496 — The First Derivative Test for Local Extrema Lesson 1501 — The Second Derivative Test for Local Extrema Lesson 1520 — Critical Points and Local Extrema Lesson 1523 — The Second Derivative Test for Extrema Lesson 1863 — Classifying Critical Points: Local Extrema and Saddle Points Lesson 1866 — The Second Derivative Test: Statement and Conditions Lesson 1867 — Applying the Second Derivative Test (+1 more)
Locate: the angle vertex that both sides touch; Lesson 1114 — Choosing the Correct Angle in SAS Area Formula
Locate asymptotes: Solve B(x - C) = 0 and B(x - C) = π to find the first two asymptotes; Lesson 1027 — Transformations of Cotangent
Locate critical values: Find where *f''(x) = 0* or where *f''(x)* is undefined (these are *possible* inflection points); Lesson 1499 — Finding Intervals of Concavity
Locate the decimal point: in the number you're dividing (inside the division bracket); Lesson 90 — Dividing Decimals by Whole Numbers
Locate the focus: Move `p` units from the vertex along the axis of symmetry (in the opening direction).; Lesson 1151 — Graphing Parabolas from Standard Form
log₁₀(100) = 2: Lesson 618 — Definition of Logarithm as Inverse of Exponential Lesson 619 — Evaluating Simple Logarithms
log₂(8) = 3: Lesson 618 — Definition of Logarithm as Inverse of Exponential Lesson 619 — Evaluating Simple Logarithms
Logarithmic: Lesson 650 — Comparing Linear, Exponential, and Logarithmic Growth
Logarithmic Growth: increases rapidly at first, then slows down significantly.; Lesson 650 — Comparing Linear, Exponential, and Logarithmic Growth
Logarithms: Functions like f(x) = ln(x) are continuous wherever they're *defined*.; Lesson 1354 — Continuity of Elementary Functions Lesson 1803 — Domain and Range of Multivariable Functions
Logic: ∀x P(x); Lesson 2483 — Translating Between English and Quantifiers
logically equivalent: if they produce identical truth values in every possible scenario.; Lesson 2475 — Logical Equivalence Lesson 2494 — Contrapositive: Logical Equivalence Lesson 2519 — Equivalence of Induction and Well-Ordering
Look: for special product patterns; Lesson 342 — Using Special Products to Simplify Expressions
Look again: for the u-substitution pattern; Lesson 1587 — u-Substitution Requiring Algebraic Manipulation
Look at the denominator(s): of your rational expression; Lesson 477 — Finding Restricted Values and Domain
Look at what's given: Count sides and angles.; Lesson 1106 — Choosing Between Law of Sines and Law of Cosines
Look for common factors: that appear in both; Lesson 487 — Holes in Rational Functions
Look for connections: between angle pairs; Lesson 712 — Solving Multi-Step Angle Problems
Look for eigenvectors first: If your transformation has enough linearly independent eigenvectors, use them as your basis.; Lesson 2257 — Finding Convenient Bases
Look for pairs: of identical factors (each pair is a perfect square!; Lesson 155 — Simplifying Square Roots
Look for parallel lines: When a line inside a triangle runs parallel to one of its sides, it often creates a smaller similar triangle.; Lesson 758 — Similar Triangles in Nested Figures
Look for patterns: that allow grouping; Lesson 519 — Complex Denominators with Multiple Radicals
Look for separable integrands: that factor into functions of ρ, φ, and θ separately; Lesson 1942 — Evaluating Triple Integrals Using Spherical Coordinates
Look for shared angles: If two triangles share a common angle (at a vertex they both touch), that's one pair of congruent angles right away.; Lesson 758 — Similar Triangles in Nested Figures
Look for symmetry first: If a figure has symmetrical parts, you can calculate one section and multiply rather than computing each piece separately.; Lesson 874 — Optimizing Composite Figure Problems
Look for the overlap: where all shaded regions intersect; Lesson 301 — Systems of Linear Inequalities
Look for zeros: Lesson 2152 — Choosing the Best Row or Column for Expansion
Look up: probabilities using the standard normal table; Lesson 2909 — Finding Probabilities with Normal Distributions
Look vertically: Scan up and down the y-axis; Lesson 554 — Range from Graphs
Loops: (edges from a vertex to itself); Lesson 2648 — Types of Graphs: Simple, Multi-, and Pseudographs
LORAN navigation systems: use the hyperbola's definition: the difference in distances to two transmitters is constant along a hyperbolic curve.; Lesson 1187 — Real-World Conic Applications Review
Lotka-Volterra equations: capture this dance mathematically using two coupled first-order ODEs:; Lesson 2376 — Predator-Prey Models: The Lotka-Volterra Equations
Low variance: Values cluster tightly around the mean (predictable); Lesson 2856 — Variance: Definition and Interpretation
Lower fence: Q1 − 1.; Lesson 2991 — Box Plots and Outlier Detection
Lower peak: The center is slightly flatter than the normal curve; Lesson 2923 — The Student's t-Distribution: Definition
Lower pH: = more acidic (lemon juice ≈ 2); Lesson 647 — pH and the Richter Scale
Lower Sum: L = Σ m_i · Δx, where m_i is the minimum value of *f* on the *i*-th subinterval; Lesson 1554 — Upper and Lower Sums
Lower sums: use the **minimum** function value on each subinterval as the rectangle height, guaranteeing an underestimate.; Lesson 1554 — Upper and Lower Sums
lower triangular: matrices have zeros above it.; Lesson 1295 — Determinants of Special Matrices Lesson 2154 — Determinant of Triangular Matrices
Lower α: (e.; Lesson 3055 — Significance Level and Error Control
LU decomposition: or other factorizations; Lesson 2147 — Computational Considerations and Singular Matrices
Lucas's Theorem: for prime p = 2.; Lesson 2606 — Binomial Coefficients Modulo 2 and Sierpiński's Triangle

M

m × n: (m rows, n columns); Lesson 1281 — Matrix Multiplication Definition and Compatibility Lesson 2097 — Matrix Notation and Dimensions
m = 2: Lesson 249 — Comparing Steepness Using Slope Lesson 267 — Slopes of Perpendicular Lines
m = 3: (because |-5| = 5, and 5 > 3); Lesson 249 — Comparing Steepness Using Slope Lesson 257 — Point-Slope Form: y - y₁ = m(x - x₁)Lesson 266 — Writing Equations of Parallel Lines
M = E ⁻¹C: .; Lesson 1308 — Applications of Inverse Matrices
M = log₁₀(A/A₀): Lesson 647 — pH and the Richter Scale
M_ij: .; Lesson 2151 — Minors and Cofactors
M_X(t) = E[e^(tX)]: Lesson 2864 — Moment Generating Functions (MGFs)
M-augmenting path: (relative to matching M) is a path that:; Lesson 2701 — Augmenting Paths and Berge's Theorem
M''(0) = E[X²]: (the second moment); Lesson 2865 — Using MGFs to Find Moments
M'(0) = E[X]: (the first moment, or mean); Lesson 2865 — Using MGFs to Find Moments
M(t) = E[e^(tX)]: .; Lesson 2865 — Using MGFs to Find Moments
M^(k)(0) = E[X^k]: (the k-th moment); Lesson 2865 — Using MGFs to Find Moments
Machine learning: Reduce dimensionality while preserving information; Lesson 2288 — Low-Rank Approximation and Data Compression Lesson 2299 — Covariance Matrices and Applications
Maclaurin series: (just a Taylor series centered at the origin).; Lesson 1773 — The Taylor Series Formula Lesson 1774 — Maclaurin Series: Taylor Series at Zero
magnitude: (size/length) and **direction**.; Lesson 1235 — Introduction to Vectors: Magnitude and Direction Lesson 1237 — Component Form of Vectors in 2D Lesson 1240 — Finding Magnitude from Components Lesson 1246 — Scalar Multiplication of Vectors Lesson 1248 — Magnitude (Norm) of a Vector Lesson 1252 — Resolving Vectors into Components Lesson 1965 — Evaluating Vector Fields at Points Lesson 2007 — The Curl of a Vector Field (+4 more)
Magnitude (Length): Using the 3D distance formula:; Lesson 1794 — Vectors in 3D Space
Magnitude of λ: Tells you how sensitive your optimal value is to small changes in the constraint; Lesson 1876 — The Lagrange Multiplier λ
major axis: is the longer dimension (length 2a), and the **minor axis** is the shorter one (length 2b).; Lesson 1155 — Definition and Standard Form of an Ellipse Lesson 1159 — Horizontal vs. Vertical Ellipses Lesson 1163 — The Major and Minor Axes
Make a decision: If your p-value is less than your significance level α, or if z falls in the critical region, you reject H₀; Lesson 3025 — One-Sample z-Test for Means
Make an initial guess: for the firing angle (the unknown initial condition); Lesson 2406 — Solving Boundary Value Problems: Shooting Method Concept
Make integrands easier: (functions often become simpler in the right coordinates); Lesson 1944 — Motivation for Change of Variables in Multiple Integrals
Makes overlap meaningful: (so multiple objects in one hole gives you useful information); Lesson 2576 — Constructing Pigeonholes Strategically
Making Tables: Organize information systematically.; Lesson 195 — Checking Solutions and Problem-Solving Strategies
Manufacturing: Evaluating five machine settings on product quality; Lesson 3102 — ANOVA Applications and Experimental Design
Manufacturing defects: in a production run (e.; Lesson 2893 — Applications of the Poisson Distribution
Manufacturing Tolerances: A machine cuts rods to 50 cm with an error uniformly distributed between -0.; Lesson 2897 — Applications of the Uniform Distribution
Marginal cost: is the derivative C'(x)—it tells you how much one additional unit costs to produce.; Lesson 1513 — Cost and Revenue Optimization Lesson 1830 — Applications: Marginal Analysis in Economics
Marginal PDF of Y: *f_Y(y) = Σ_x p_X,Y(x, y)* (sum over *X*); Lesson 2937 — Mixed Discrete-Continuous Joint Distributions
Marginal PMF of X: *p_X(x) = ∫ p_X,Y(x, y) dy* (integrate out *Y*); Lesson 2937 — Mixed Discrete-Continuous Joint Distributions
Marginal probabilities: (for just X or just Y): Sum across the other variable; Lesson 2927 — Joint Probability Distributions: Discrete Case
Marginal revenue: is R'(x)—how much revenue one more sale generates.; Lesson 1513 — Cost and Revenue Optimization
Mark: the right angle (90°) where they meet; Lesson 849 — Tangent-Radius Perpendicularity Theorem
Mark edges: used to reach new vertices—these become tree edges; Lesson 2674 — Finding Spanning Trees: Breadth-First Search
Mark the bearings: as angles from north; Lesson 1101 — Law of Sines with Bearing and Direction
Mark the observer's position: Lesson 968 — Drawing Diagrams for Angle Problems
Mark the radius: by counting r units up, down, left, and right from the center; Lesson 1137 — Graphing Circles from Standard Form
Mark the reflected point: P' so that M is the midpoint of segment PP'; Lesson 927 — Reflections Across Arbitrary Lines
Mark the vertices: a units away on the transverse axis; Lesson 1171 — Graphing Hyperbolas Centered at the Origin
Market research: Checking if product preference is independent of age group (chi-squared test for independence); Lesson 3051 — Applications and Limitations
Mass: is found by integrating the density function over the region:; Lesson 1921 — Applications: Mass and Center of Mass in Polar Lesson 1922 — Introduction to Triple Integrals Lesson 1958 — Center of Mass in Two Dimensions Lesson 2018 — Surface Integrals of Scalar Functions
Master Theorem: gives us a shortcut to determine asymptotic behavior by comparing the "splitting work" (controlled by *a* and *b*) with the "combining work" (*f(n)*).; Lesson 2636 — Divide-and-Conquer Recurrences
Match exactly: The binomials must be identical, including signs; Lesson 377 — Identifying the Common Binomial Factor
Match the conclusion's definition: Since *2k²* is an integer, *n²* has the form *2m* where *m = 2k²*, so *n²* is even by definition.; Lesson 2492 — Direct Proof: Working with Definitions
Match them: to the three sides of the second triangle; Lesson 741 — SSS (Side-Side-Side) Congruence Postulate
Matching forms: When comparing two series, aligning their indices makes patterns clearer; Lesson 1720 — Reindexing and Shifting Series
Material quantities: "How much soil is needed for a rectangular garden bed 6ft × 4ft × 1.; Lesson 884 — Volume Word Problems and Applications
Materials engineering: Maximize strength-to-weight ratio given cost and manufacturing constraints; Lesson 1890 — Application: Optimization in Physics and Engineering
Mathematical tractability: Squared terms have nice calculus properties that let us find the minimum algebraically; Lesson 3062 — The Least Squares Criterion
Mathematically: , this function works for *any* real number.; Lesson 557 — Real-World Context and Restricted Domains
matrix: is a rectangular arrangement of numbers (or expressions) organized into rows and columns.; Lesson 1271 — What is a Matrix? Dimensions and Notation Lesson 2097 — Matrix Notation and Dimensions
Matrix addition: add all the scaled matrices together; Lesson 1278 — Linear Combinations of Matrices
Matrix D: This is a diagonal matrix where the diagonal entries are the eigenvalues, placed in the *same order* as their corresponding eigenvectors appear in **P**.; Lesson 2191 — Constructing P and D from Eigenvectors
Matrix P: Form this by placing the eigenvectors of **A** as *columns*.; Lesson 2191 — Constructing P and D from Eigenvectors
Maximally acyclic: – It has no cycles, but adding any edge creates one; Lesson 2668 — Definition and Properties of Trees
Maximize: profit, revenue, nutrition, or production output; Lesson 305 — The Objective Function
maximum: (greatest) or **minimum** (smallest) value of something you care about.; Lesson 304 — Introduction to Linear Programming Lesson 436 — Revenue and Profit Maximization Lesson 1501 — The Second Derivative Test for Local Extrema Lesson 1554 — Upper and Lower Sums Lesson 1786 — Using the Lagrange Remainder to Bound Error Lesson 2727 — Applications: GCD and LCM via Prime Factorization Lesson 2990 — Percentiles and Quartiles
Maximum Likelihood Estimation: finds parameter values that make the observed data "most likely" by maximizing the likelihood function.; Lesson 3008 — Comparing MLE and Method of Moments
Maximum Likelihood Estimation (MLE): and the **Method of Moments (MoM)** produce point estimates for population parameters, but they take fundamentally different approaches:; Lesson 3008 — Comparing MLE and Method of Moments
Maximum Principle Connection: Since the center value equals the boundary average, a harmonic function cannot have a local maximum or minimum in the interior (unless it's constant everywhere).; Lesson 2444 — Mean Value Property and Applications
Maximum rate of change: = ||∇f|| (the length of the gradient vector); Lesson 1846 — Maximum and Minimum Rates of Change
Maximum value: Changes from 1 to 1 + D; Lesson 1018 — Vertical Shift: y = sin(x) + D and the Midline Lesson 1034 — Finding Amplitude from Equations and Graphs
mean: of a discrete random variable tells you the "average" value you'd expect if you could repeat the random experiment infinitely many times.; Lesson 2852 — Definition of Expectation (Mean) for Discrete Random Variables Lesson 2855 — Definition of Expectation for Continuous Random Variables Lesson 2897 — Applications of the Uniform Distribution Lesson 2900 — Mean and Variance of Exponential Distributions Lesson 2905 — Parameters: Mean and Standard Deviation Lesson 2924 — Properties and Uses of the t-Distribution Lesson 2926 — Properties and Applications of the F-Distribution Lesson 2984 — CLT for Sample Sums (+3 more)
Mean (Expectation): E[X] = 1/λ; Lesson 2900 — Mean and Variance of Exponential Distributions
mean (expected value): is:; Lesson 2886 — Mean and Variance of the Hypergeometric Distribution Lesson 2891 — Mean and Variance of the Poisson Distribution Lesson 2920 — Properties and Applications of the Beta Distribution
Mean = 0: (when df > 1); Lesson 3033 — The t-Distribution: Properties and Degrees of Freedom
Mean and variance: E[χ²(k)] = *k* and Var(χ²(k)) = 2*k*; Lesson 2922 — Properties and Applications of Chi-Squared
Mean of S: E[S ] = nμ (not just μ); Lesson 2984 — CLT for Sample Sums
Mean Squares: = SS divided by df (a variance estimate); Lesson 3097 — The ANOVA Table and Hypothesis Testing
Mean Value Theorem: states:; Lesson 1486 — Statement and Geometric Interpretation of the MVT
Mean Value Theorem (MVT): isn't just abstract math—it has a powerful real-world interpretation when applied to motion.; Lesson 1493 — Real-World Interpretation: Speed and Distance
Means add linearly: The new mean is just *a* times the first mean plus *b* times the second mean; Lesson 2912 — Linear Combinations of Normal Variables
Measure: or note the dimensions needed for each piece; Lesson 883 — Volume of Composite Solids
Measure dimensions: Rectangle A is 5 ft × 3 ft; Rectangle B is 4 ft × 2 ft; Lesson 868 — Finding Area of Composite Figures by Addition
Measurement: r = 10 cm, dr = ±0.; Lesson 1473 — Applications of Differentials in Measurement
Mechanical Design: Analyzing rotating machinery—flywheels, gears, turbines; Lesson 1962 — Radius of Gyration and Applications
Mechanical systems: combine both.; Lesson 1270 — Applications in Physics and Engineering
Mechanics: Optimize trajectories under force equilibrium and momentum conservation; Lesson 1890 — Application: Optimization in Physics and Engineering
median: is a segment that connects a vertex to the **midpoint of the opposite side**.; Lesson 738 — Medians, Altitudes, and Angle Bisectors Lesson 2849 — Percentiles and Quantiles from the CDF Lesson 2988 — Measures of Central Tendency: Mean, Median, Mode Lesson 2990 — Percentiles and Quartiles Lesson 2997 — Choosing Appropriate Summaries and Displays
Median (Q2): middle value; Lesson 2990 — Percentiles and Quartiles
medians: meet (remember: medians connect vertices to midpoints of opposite sides).; Lesson 739 — Centroid, Orthocenter, Incenter, and Circumcenter Lesson 907 — Distance and Midpoint with Special Triangles
Medical lithotripsy: Breaking up kidney stones by focusing shock waves from one focus; the waves concentrate at the other focus where the stone is located; Lesson 1165 — Reflective Property of Ellipses
Medical research: Testing whether treatment outcomes differ across multiple patient groups (ANOVA F test); Lesson 3051 — Applications and Limitations
Medicine: Comparing four drug dosages on patient recovery times; Lesson 3102 — ANOVA Applications and Experimental Design
Medium side ÷ √3: = shortest side; Lesson 795 — Finding Missing Sides in 30-60-90 Triangles
memoryless property: states that for a geometric random variable (counting trials until first success), the probability of waiting an additional *k* trials is the same regardless of how many trials have already failed.; Lesson 2883 — Memoryless Property and Applications Lesson 2902 — Relationship Between Exponential and Poisson
Memoryless within intervals: Knowing you're past a certain point doesn't change probabilities for the remaining segment; Lesson 2897 — Applications of the Uniform Distribution
Menger's Theorem: states: *The maximum number of edge-disjoint paths from s to t equals the minimum number of edges whose removal disconnects s from t* (called a minimum *s-t* edge cut).; Lesson 2708 — Edge-Disjoint Paths and Menger's Theorem
Mersenne primes: have the form \( M_p = 2^p - 1 \), where \( p \) itself is prime.; Lesson 2729 — Special Primes: Mersenne, Fermat, and Twin Primes
Messy quotients with radicals: When numerators and denominators both have complicated expressions; Lesson 1465 — Applications and Complex Examples
Meteor strikes: in a region per year; Lesson 2893 — Applications of the Poisson Distribution
Method: Divide the part by the percent (in decimal form).; Lesson 100 — Finding the Whole Given a Part and Percent
Method 1 (Division): Use synthetic or long division and read off the remainder at the end.; Lesson 351 — The Remainder Theorem
Method 1 (Parentheses first): Lesson 38 — The Distributive Property
Method 1 (Ratios): Check if `5 : 5√3 : 10` simplifies to `x : x√3 : 2x`.; Lesson 799 — Comparing Special Triangles with Pythagorean Theorem
Method 2 (Pythagorean): Verify it's a right triangle: `5² + (5√3)² = 25 + 75 = 100 = 10²`; Lesson 799 — Comparing Special Triangles with Pythagorean Theorem
Method 2 (Remainder Theorem): Simply calculate **P(2)**:; Lesson 351 — The Remainder Theorem
Method 2: Geometric Construction: Lesson 1142 — Circles from Three Points
method of moments: is one of the oldest and most intuitive techniques for estimating parameters.; Lesson 3001 — Introduction to Method of Moments Lesson 3002 — Method of Moments for One Parameter Lesson 3003 — Method of Moments for Multiple Parameters Lesson 3008 — Comparing MLE and Method of Moments
Method of Moments (MoM): produce point estimates for population parameters, but they take fundamentally different approaches:; Lesson 3008 — Comparing MLE and Method of Moments
Method of Undetermined Coefficients: with a smart guess: assume the particular solution is also a polynomial.; Lesson 2350 — Particular Solutions for Polynomial Right-Hand Sides Lesson 2360 — The Variation of Parameters Formula for Second-Order ODEs
Metropolis-Hastings: Propose moves, accept or reject based on posterior density ratios; Lesson 3121 — Computational Tools: Introduction to MCMC and Bayesian Software
Middle integral: Integrate the result with respect to y (treating z as constant); Lesson 1923 — Setting Up Triple Integrals over Rectangular Boxes
Middle steps: Use definitions, postulates, and theorems you've learned; Lesson 748 — Writing Two-Column Congruence Proofs
Middle variable (usually θ): What angular range does the region sweep?; Lesson 1936 — Setting Up Triple Integrals in Cylindrical Coordinates
midline: (the horizontal center of the wave) to the highest or lowest point.; Lesson 1015 — Amplitude: Vertical Stretching with y = A·sin(x)Lesson 1018 — Vertical Shift: y = sin(x) + D and the Midline Lesson 1039 — Vertical Shift: D Parameter Lesson 1041 — Writing Equations from Transformed Graphs
Midline (D): Average depth = (20 + 8)/2 = 14 feet; Lesson 1042 — Applications: Modeling Periodic Phenomena
midpoint formula: calculates this center by finding the average of the x-coordinates and the average of the y- coordinates separately.; Lesson 905 — Midpoint Formula: Finding the Center Point Lesson 907 — Distance and Midpoint with Special Triangles Lesson 919 — Writing Equations of Circles from Given Information
Midpoint Rectangles: Use the *midpoint* of each subinterval; Lesson 1546 — Approximating Area Under a Curve with Rectangles
Midpoint Riemann sums: offer a third approach: use the **middle point** of each subinterval to determine the rectangle's height.; Lesson 1549 — Midpoint Riemann Sums
Midpoint sums: typically give better approximations than left or right sums.; Lesson 1550 — Comparing Left, Right, and Midpoint Approximations
Min-cut: A different optimization problem (minimize cut capacity); Lesson 2705 — The Max-Flow Min-Cut Theorem
Minimally connected: – It's connected, but removing any single edge disconnects it; Lesson 2668 — Definition and Properties of Trees
Minimize: cost, waste, time, or pollution; Lesson 305 — The Objective Function Lesson 1890 — Application: Optimization in Physics and Engineering
minimum: (smallest) value of something you care about.; Lesson 304 — Introduction to Linear Programming Lesson 1367 — Common Techniques and Restrictions on δ Lesson 1501 — The Second Derivative Test for Local Extrema Lesson 1554 — Upper and Lower Sums Lesson 2727 — Applications: GCD and LCM via Prime Factorization Lesson 2990 — Percentiles and Quartiles
Minimum rate of change: = 0, occurring in directions **perpendicular** to ∇f; Lesson 1846 — Maximum and Minimum Rates of Change
Minimum value: 0, occurring at x = 0; Lesson 594 — The Absolute Value Function Lesson 1018 — Vertical Shift: y = sin(x) + D and the Midline Lesson 1034 — Finding Amplitude from Equations and Graphs
minor axis: is the shorter one (length 2b).; Lesson 1155 — Definition and Standard Form of an Ellipse Lesson 1163 — The Major and Minor Axes
Minor₁: (for `a`): Cross out row 1, column 1 → `| e f |`; Lesson 1292 — Determinant of a 3×3 Matrix: Expansion by Minors
Minor₂: (for `b`): Cross out row 1, column 2 → `| d f |`; Lesson 1292 — Determinant of a 3×3 Matrix: Expansion by Minors
Minor₃: (for `c`): Cross out row 1, column 3 → `| d e |`; Lesson 1292 — Determinant of a 3×3 Matrix: Expansion by Minors
Misapplying the Quotient Rule: Lesson 150 — Common Mistakes with Exponent Rules
Misleading p-values: Individual predictors may appear non-significant even when the overall model is significant; Lesson 3080 — Multicollinearity: Detection and Consequences Lesson 3085 — Detecting Non-Constant Variance (Heteroscedasticity)
Miss entirely: (no triangle possible); Lesson 1096 — The SSA Configuration Introduction
Missing probability mass/density: If the sum/integral ≠ 1, it's invalid; Lesson 2842 — Verifying Valid PMFs and PDFs
MISSISSIPPI: has:; Lesson 2613 — Multinomial Coefficients and Word Arrangements
Mix of strategies: Sometimes you can compare pairs quickly (like when one fraction is obviously smaller) before doing all the conversions.; Lesson 68 — Ordering Multiple Fractions
Mixed boundaries: A region might be bounded by a parabola below and a circle above—sketch first, then identify which description (Type I or II) is simpler.; Lesson 1906 — Regions Bounded by Curves
Mixed complexity: Products involving exponentials, trig functions, and polynomials all at once; Lesson 1465 — Applications and Complex Examples
Mixed operations: means you'll encounter problems where you need to multiply, divide, simplify, *and then* add or subtract radicals, all in one expression.; Lesson 513 — Mixed Operations with Radicals
Mixed or Complex Forms: Lesson 1610 — Strategy Selection for Trigonometric Integrals
mixed partial derivatives: .; Lesson 1826 — Higher-Order Partial Derivatives Lesson 2328 — Exact Differential Equations: Definition and Test
mixed random variable: has a probability distribution that assigns positive probability to specific isolated points (like discrete variables) *and* spreads probability continuously over intervals (like continuous variables).; Lesson 2831 — Mixed and Other Types of Random Variables Lesson 2850 — Mixed Random Variables and Their CDFs
Mixing problems: Liquid enters a tank at some concentration, mixes instantly, and drains out.; Lesson 2317 — Applications: Mixing and Velocity Problems
Mixing up radius functions: Both methods require correct identification of the distance from the axis of rotation.; Lesson 1690 — Comparing Surface Area and Volume Calculations
Mixture problems: involve combining substances (like chemicals or liquids) with different concentrations or properties.; Lesson 439 — Work and Mixture Rate Problems
Mixtures: Creating paint colors, concrete, or juice concentrates in correct ratios.; Lesson 123 — Real-World Proportion Problems
MLE: Also gives the biased version, but theoretical properties are better understood; Lesson 3008 — Comparing MLE and Method of Moments
Möbius strip: , are *non-orientable*.; Lesson 2017 — Orientation of Surfaces
Mode: Useful for categorical data or understanding the most typical occurrence.; Lesson 2988 — Measures of Central Tendency: Mean, Median, Mode
Model validation: If sample mean and variance differ dramatically, the Poisson may not be appropriate; Lesson 2891 — Mean and Variance of the Poisson Distribution
Modeling real phenomena: Knowing which effects matter most long-term; Lesson 1535 — Applications to Growth Rates and Asymptotic Behavior
Modification needed: guess $y_p = Ate^{2t}$ instead of $Ae^{2t}$; Lesson 2357 — Higher-Order ODEs and Undetermined Coefficients
Modified p-series: ∑ 1/n² (converges); Lesson 1744 — When the Ratio Test is Inconclusive
modular inverse: of *a* modulo *n* is a number *x* such that *ax* ≡ 1 (mod *n*).; Lesson 2736 — Finding Modular Inverses Lesson 2765 — Computing the Private Exponent d
modulus: ) of a complex number measures how far that number is from the origin (0, 0) on the complex plane.; Lesson 1203 — Absolute Value (Modulus) of Complex Numbers Lesson 1209 — Modulus of a Complex Number Lesson 1215 — Converting from Rectangular to Polar Form Lesson 1216 — Finding the Modulus (Absolute Value)Lesson 2766 — RSA Encryption
Modulus of each root: ³√8 = 2; Lesson 1229 — Finding All nth Roots Using Polar Form
MoM: Uses the biased sample variance; Lesson 3008 — Comparing MLE and Method of Moments
Moment: ∫₀⁴ x(2 + x) dx = ∫₀⁴ (2x + x²) dx = [x² + x³/3]₀⁴ = 16 + 64/3 ≈ 37.; Lesson 1700 — Center of Mass for One-Dimensional Systems Lesson 1957 — Moments and First Moments
Moment about the x-axis: $M_y = \iint_R x \cdot \rho(r, \theta) \, r \, dr \, d\theta$; Lesson 1921 — Applications: Mass and Center of Mass in Polar
Moment about the y-axis: $M_x = \iint_R y \cdot \rho(r, \theta) \, r \, dr \, d\theta$; Lesson 1921 — Applications: Mass and Center of Mass in Polar
Moment Generating Function (MGF): is a special function that encodes information about all moments (mean, variance, skewness, kurtosis, and beyond) of a random variable into one compact formula.; Lesson 2864 — Moment Generating Functions (MGFs)Lesson 2960 — Moment Generating Functions: Definition
moments: , which measure how mass is distributed relative to an axis.; Lesson 1701 — Moments and Centroids of Planar Regions Lesson 1921 — Applications: Mass and Center of Mass in Polar Lesson 1958 — Center of Mass in Two Dimensions
Moments of inertia: are **second moments**—instead of multiplying density by distance, we multiply by *distance squared*.; Lesson 1960 — Moments of Inertia
Money: problems may require non-negative values; Lesson 557 — Real-World Context and Restricted Domains
monomial: (a single term like `3x` or `-5y²`) by a **polynomial** (multiple terms like `2x + 7` or `x² - 4x + 1`), you must distribute the monomial to *every* term inside the polynomial.; Lesson 325 — Multiplying a Monomial by a Polynomial Lesson 345 — Long Division with Polynomials Lesson 446 — Simplifying with Monomial Factors
monotonic: (always increasing or always decreasing) function:; Lesson 1550 — Comparing Left, Right, and Midpoint Approximations Lesson 1710 — Monotonic Sequences and Boundedness Lesson 3109 — Spearman's Rank Correlation
Monotonicity: describes how a function behaves as you move from left to right along its graph.; Lesson 571 — Monotonicity: Increasing and Decreasing Functions Lesson 1494 — Using the First Derivative to Determine Monotonicity
Monotonicity (increasing/decreasing): Shows whether quantities are growing or shrinking; Lesson 1504 — Applications: Real-World Monotonicity and Concavity
More Complex Examples: like `f(x) = |x - 3|` become:; Lesson 598 — Writing Functions as Piecewise Definitions
More efficient: than general LU decomposition (about half the computational work); Lesson 2295 — The Cholesky Decomposition
Mortgages: Calculate whether you can afford a house; Lesson 689 — Loan Amortization
Motion along paths: A particle moves along a curve.; Lesson 1418 — Chain Rule Applications and Problem-Solving
Move constant terms: to the other side; Lesson 211 — Multi-Step Equations Requiring Multiple Operations
Move down one row: and repeat the process for the remaining submatrix; Lesson 2119 — Converting to Row Echelon Form
Move the constant: Get the constant on the right side; Lesson 1175 — Converting General Form to Standard Form
Move the decimal point: in BOTH the divisor and dividend the same number of places to the right; Lesson 91 — Dividing by Decimals
Move up one row: and substitute the value(s) you just found; Lesson 1312 — Back Substitution from Echelon Form Lesson 2110 — Back Substitution
Move variable terms: to one side using addition/subtraction; Lesson 211 — Multi-Step Equations Requiring Multiple Operations
Move variables: `6x - x = 2 + 4`, so `5x = 6`; Lesson 216 — Mixed Practice: Complex Linear Equations
multinomial coefficient: and counts how many ways you can arrange n objects where r₁ are of type 1, r₂ are of type 2, and so on—exactly like permutations with repetition!; Lesson 2610 — The Multinomial Theorem Lesson 2612 — Counting Arrangements with Multiple Types Lesson 2613 — Multinomial Coefficients and Word Arrangements
multinomial coefficients: ways to arrange items when you have multiple categories.; Lesson 2608 — Multinomial Theorem Preview via Trinomial Expansion Lesson 2609 — From Binomial to Multinomial Coefficients Lesson 2611 — Calculating Multinomial Coefficients Lesson 2614 — Distribution Problems with Constraints Lesson 2615 — Relationship to Combinations and Stars-and-Bars
Multinomial Theorem: comes in.; Lesson 2610 — The Multinomial Theorem
Multiple edges: between the same two vertices; Lesson 2648 — Types of Graphs: Simple, Multi-, and Pseudographs
Multiple factors with powers: When you have three or more factors being multiplied, each raised to powers; Lesson 1465 — Applications and Complex Examples
Multiple linear regression: extends the simple model to handle multiple predictors simultaneously.; Lesson 3072 — Introduction to Multiple Linear Regression Lesson 3075 — Interpretation of Regression Coefficients
Multiple possible outcomes: – There must be at least two different results that could occur; Lesson 2771 — What is a Random Experiment?
Multiples of 12: 12, 24, 36, 48, 60.; Lesson 49 — Understanding Multiples
Multiples of 3: 3, 6, 9, **12**, 15, 18, 21, **24**, 27, 30, 33, **36**.; Lesson 53 — Common Multiples
Multiples of 4: 4, 8, **12**, 16, 20, **24**, 28, 32, **36**, 40.; Lesson 53 — Common Multiples
multiplication: Lesson 36 — The Commutative Property Lesson 37 — The Associative Property Lesson 82 — Word Problems Involving Fraction Operations Lesson 172 — Writing Expressions from Word Problems Lesson 183 — Order of Operations in Evaluation Lesson 534 — Multiplying and Dividing with Fractional Exponents Lesson 1224 — Geometric Interpretation of Polar Operations Lesson 2739 — Properties of Congruence Relations
Multiplication and Division: – Work left to right, applying sign rules; Lesson 47 — Mixed Operations with Negative Numbers Lesson 81 — Operations with Multiple Fractions Lesson 513 — Mixed Operations with Radicals
Multiplication equation: Lesson 201 — Solving One-Step Multiplication/Division Equations
Multiplication phrases: Lesson 188 — Translating Words into Mathematical Expressions
Multiplication Property example: Lesson 200 — Multiplication and Division Properties of Equality
Multiplication Property of Equality: says you can multiply both sides by the same nonzero number, and the equation stays balanced.; Lesson 200 — Multiplication and Division Properties of Equality Lesson 209 — Equations with Fractions: Clearing Denominators
Multiplication Rule: gives us exactly that:; Lesson 2795 — The Multiplication Rule Lesson 2796 — Sequential Experiments and Tree Diagrams Lesson 2813 — Motivation for Bayes' Theorem
multiplicative identity: because it preserves a number's identity in multiplication.; Lesson 39 — Identity and Inverse Properties Lesson 2103 — Special Matrices: Identity and Zero
multiplicative inverse: a number that multiplies with it to give one:; Lesson 39 — Identity and Inverse Properties Lesson 2735 — Division and Modular Inverses
Multiply: Multiply your answer by the divisor and write the result below.; Lesson 29 — Long Division Algorithm Lesson 76 — Multiplying Mixed Numbers Lesson 80 — Complex Fractions Lesson 178 — Reading and Interpreting Algebraic Notation Lesson 350 — Performing Synthetic Division Lesson 362 — Factoring Trinomials with Leading Coefficient 1 Lesson 452 — Simplifying Before Multiplying Lesson 455 — Dividing and Simplifying in One Step (+9 more)
Multiply all denominators together: to get the new denominator; Lesson 451 — Multiplying Rational Expressions: Basic Method
Multiply all numerators together: to get the new numerator; Lesson 451 — Multiplying Rational Expressions: Basic Method
Multiply along each diagonal: and sum everything up.; Lesson 1596 — Tabular Integration
Multiply both parts: (2/x) · (5/5) = (2·5)/(x·5) = 10/(5x); Lesson 462 — Building Equivalent Rationals
Multiply by 2: (6, 8, 10); Lesson 785 — Generating Multiples of Basic Triples Lesson 888 — Surface Area of Cylinders
Multiply by 3: (9, 12, 15); Lesson 785 — Generating Multiples of Basic Triples
Multiply by conjugates: to eliminate denominators; Lesson 1051 — Verifying Trigonometric Identities
Multiply by the conjugate: If your denominator is `c + di`, multiply both numerator and denominator by its conjugate `c - di`; Lesson 1206 — Rationalization and Standard Form Lesson 1346 — Other Trigonometric Limits Lesson 1385 — Computing Derivatives from the Definition: Root Functions
multiply by the reciprocal: .; Lesson 78 — Dividing Fractions Lesson 454 — Dividing Rational Expressions: Multiply by Reciprocal
Multiply by x: Lesson 1782 — Finding Taylor Series by Known Series Methods
Multiply every term: (including constants) by the LCD; Lesson 480 — Solving Equations with Multiple Terms
Multiply or divide: to make unwanted units cancel; Lesson 113 — Solving Rate Problems Lesson 458 — Applications of Multiplying and Dividing Rationals
Multiply out any products: using FOIL; Lesson 1202 — Simplifying Complex Expressions
Multiply the coefficients: together; Lesson 165 — Multiplying Numbers in Scientific Notation Lesson 508 — Multiplying Radicals with the Same Index
Multiply the denominators: (bottom numbers) together; Lesson 75 — Multiplying Fractions
Multiply the moduli: r₁ × r₂; Lesson 1221 — Multiplying Complex Numbers in Polar Form
Multiply the numerators: (top numbers) together; Lesson 75 — Multiplying Fractions
Multiply the radicands: (numbers under the radical symbols); Lesson 508 — Multiplying Radicals with the Same Index
Multiply the recurrence relation: by xⁿ and sum over all valid n; Lesson 2643 — Solving Recurrence Relations with Generating Functions
Multiply the results together: (by the multiplication principle); Lesson 2596 — Multiple Selection Problems
Multiply through: by e^(ax): the left side becomes d/dx[y·e^(ax)]; Lesson 2325 — Linear ODEs with Constant Coefficients
Multiply top and bottom: Multiply both the numerator and denominator by that same factor; Lesson 462 — Building Equivalent Rationals
Multiply what remains: Lesson 456 — Mixed Operations: Multiplication and Division Together
Multivariable functions: extend function notation to handle this.; Lesson 543 — Function Notation with Multiple Variables
Multivariate statistics: Describes probability distributions (multivariate Gaussian); Lesson 2299 — Covariance Matrices and Applications
Must: have 0/0 or ∞/∞ form first; Lesson 1528 — Statement of L'Hôpital's Rule
must diverge: .; Lesson 686 — The nth Term Test for Divergence Lesson 1716 — The nth Term Test for Divergence
Mutual Independence: (also called *full independence*): Not only is every pair independent, but *every subset* of the events satisfies the product rule.; Lesson 2807 — Pairwise vs. Mutual Independence Lesson 2944 — Pairwise vs Mutual Independence
mutually exclusive: (outcomes don't overlap—exactly one occurs per experiment).; Lesson 2772 — Sample Spaces: The Set of All Possible Outcomes Lesson 2779 — Mutually Exclusive Events Lesson 2780 — Partitions of the Sample Space Lesson 2785 — The General Addition Rule

N

N (A): to emphasize its orthogonal relationship.; Lesson 2131 — The Left Null Space
n → ∞: (n approaches infinity), something remarkable occurs: the width of each rectangle shrinks toward zero, and your approximation becomes increasingly accurate.; Lesson 1553 — Riemann Sums as n Approaches Infinity Lesson 2892 — Poisson as a Limit of Binomial
n = p: Lesson 1281 — Matrix Multiplication Definition and Compatibility Lesson 2100 — Matrix Multiplication: Definition
n total objects: where:; Lesson 2585 — Permutations of Objects with Repetition Lesson 2612 — Counting Arrangements with Multiple Types
N(A^T): or sometimes **N ⊥(A)** to emphasize its orthogonal relationship.; Lesson 2131 — The Left Null Space
N(v): , is the set of all vertices adjacent to `v`.; Lesson 2651 — Adjacency and Incidence
n×n: matrix **A**, the following are **equivalent**:; Lesson 2144 — The Invertible Matrix Theorem Lesson 2190 — The Diagonalization Theorem
Natural independence: occurs when the physical setup of the experiment guarantees that stages don't influence each other.; Lesson 2808 — Independence in Multi-Stage Experiments
Navigation: determining distances between ships, aircraft, or landmarks; Lesson 1100 — Applications of the Law of Sines
Navigation with bearings: A ship travels 50 km on a bearing of 040°, then changes course and travels 30 km on a bearing of 130°.; Lesson 1109 — Applications: Distance and Navigation Problems
Near-singular matrices: (those "almost" singular with very small determinants) amplify small errors into huge ones; Lesson 2147 — Computational Considerations and Singular Matrices
Necessity ( ): If a perfect matching exists, then for any S ⊆ X, those |S| vertices must be matched to |S| distinct vertices in Y.; Lesson 2700 — Hall's Marriage Theorem
Negate: the off-diagonal elements (change signs of `b` and `c`); Lesson 1302 — Finding the Inverse of a 2×2 Matrix
Negating a Universal Statement: Lesson 2484 — Negating Quantified Statements
Negating an Existential Statement: Lesson 2484 — Negating Quantified Statements
negative: .; Lesson 45 — Multiplying Positive and Negative Numbers Lesson 138 — Exponents with Negative Bases Lesson 218 — Solving One-Step Inequalities Lesson 246 — Positive, Negative, Zero, and Undefined Slopes Lesson 250 — Slope and Direction of Lines Lesson 359 — Factoring Out Negative GCFs Lesson 362 — Factoring Trinomials with Leading Coefficient 1 Lesson 366 — Factoring Trinomials with Negative Coefficients (+11 more)
Negative (clockwise) orientation: is the opposite direction.; Lesson 1995 — Orientation and Simple Closed Curves
Negative angle: rotate clockwise; Lesson 999 — Negative Angles and Clockwise Rotation
Negative angles rotate clockwise: from the positive x-axis.; Lesson 1007 — Negative Angles and Trigonometric Values
Negative case: x = -7; Lesson 227 — Solving Basic Absolute Value Equations
Negative covariance: When X increases, Y tends to decrease (and vice versa); Lesson 2947 — Covariance: Definition and Interpretation
Negative D: For `y = cos(x) - 2`, the midline shifts down to `y = -2`; Lesson 1039 — Vertical Shift: D Parameter
negative definite: if **x**ᵀA**x** < 0 for *all* nonzero vectors **x**.; Lesson 2292 — Positive Definite, Negative Definite, and Indefinite Matrices Lesson 2293 — Eigenvalue Criterion for Definiteness
Negative divergence (div: F** < 0):** The field is **converging** — more fluid is arriving than leaving.; Lesson 2006 — Interpreting Divergence Physically
Negative exponents: = create fractions (reciprocals); Lesson 135 — Negative Integer Exponents Lesson 141 — Mixed Practice with Integer Exponents Lesson 311 — Identifying Polynomials vs Non-Polynomials Lesson 536 — Negative Fractional Exponents Lesson 601 — Evaluating Exponential Expressions
Negative flux: field flows *in* (opposite **n**); Lesson 2019 — Vector Fields and Surface Integrals
Negative numbers: (-1, -2, -3, .; Lesson 40 — Representing Negative Numbers on the Number Line Lesson 43 — Adding Positive and Negative Numbers
Negative numbers outside: Lesson 174 — The Distributive Property with Variables
Negative orientation: the opposite choice, "inward" normal; Lesson 2017 — Orientation of Surfaces
Negative real parts: solutions decay → **stable equilibrium**; Lesson 2278 — Applications to Differential Equations and Dynamics
negative reciprocals: of each other.; Lesson 267 — Slopes of Perpendicular Lines Lesson 268 — Writing Equations of Perpendicular Lines Lesson 269 — Determining Parallel, Perpendicular, or Neither
Negative residual: Your model *over*-predicted (actual value was lower than predicted); Lesson 3083 — What Are Residuals?
Negative result: Lesson 138 — Exponents with Negative Bases
Negative Semidefinite: All eigenvalues are **non-positive** (λ ≤ 0), at least one is zero; Lesson 2293 — Eigenvalue Criterion for Definiteness
Negative slope: Line goes downhill from left to right (rise is negative); Lesson 244 — Rise Over Run: Understanding Slope Geometrically Lesson 246 — Positive, Negative, Zero, and Undefined Slopes
negative to positive: , the function was falling then started rising — you've found a **local minimum** (a valley); Lesson 1496 — The First Derivative Test for Local Extrema Lesson 1502 — Comparing First and Second Derivative Tests
Negative λ: Gradients point in opposite directions (often a local minimum); Lesson 1876 — The Lagrange Multiplier λ
Negative/Zero Exponents: – Convert and simplify at the end; Lesson 149 — Combining Multiple Exponent Rules
neighborhood: of a vertex `v`, denoted **N(v)**, is the set of all vertices adjacent to `v`.; Lesson 2651 — Adjacency and Incidence Lesson 2700 — Hall's Marriage Theorem
neither: Lesson 569 — Even and Odd Functions Lesson 1496 — The First Derivative Test for Local Extrema Lesson 1520 — Critical Points and Local Extrema Lesson 2006 — Interpreting Divergence Physically
Nested exponents: where taking the nth root simplifies dramatically; Lesson 1747 — Comparing Ratio and Root Tests
Network design: Connecting *n* computers in a tree topology (no redundant paths) requires exactly *n - 1* cables.; Lesson 2669 — Counting Vertices and Edges in Trees Lesson 2682 — Applications of Eulerian Paths: The Königsberg Bridge Problem
Neumann: Fixed heat flux/insulated boundary (e.; Lesson 2417 — Initial and Boundary Conditions
Neumann (Type II): Specifies the derivative (slope/flux); Lesson 2404 — Boundary Value Problems: Introduction and Types
Neumann boundary conditions: .; Lesson 2424 — Insulated Boundary Conditions
Neutrally stable: (neither attracting nor repelling); Lesson 2383 — Phase Portraits and Stability Analysis
Never mix modifications: – If one term needs `x`, the whole group does; Lesson 2358 — Common Mistakes and Problem-Solving Strategies
New argument: π/4 + π/6 = 3π/12 + 2π/12 = 5π/12; Lesson 1221 — Multiplying Complex Numbers in Polar Form
New info: You tested positive; Lesson 2801 — Updating Probabilities with New Information
New modulus: 2 × 3 = 6; Lesson 1221 — Multiplying Complex Numbers in Polar Form
New point: A'(−1, 9): Lesson 923 — Vector Notation for Translations
Newton's law of cooling: for temperature differences; Lesson 2327 — Applications: RC Circuits and Decay Lesson 2374 — Exponential Growth and Decay Models
Newton's Second Law: states that force equals mass times acceleration:; Lesson 2368 — Spring-Mass Systems: Setting Up the Differential Equation
No: Since 25 ≠ 36, these cannot form a right triangle.; Lesson 784 — Verifying Pythagorean Triples Lesson 1193 — Equality of Complex Numbers
No amplitude: (the function is unbounded); Lesson 1028 — Graphing the Secant Function
No change: to the determinant; Lesson 2155 — Computing Determinants via Row Reduction
No graphing required: Just algebra—works anywhere with pencil and paper; Lesson 281 — Comparing Graphing and Substitution Methods
No horizontal asymptote: Lesson 488 — Horizontal Asymptotes: Degree Rules
No horizontal/slant asymptote: If the numerator's degree is *two or more* higher than the denominator's, the function grows without bound (shoots up or down to infinity).; Lesson 492 — End Behavior of Rational Functions
No loops: (edges from a vertex to itself); Lesson 2648 — Types of Graphs: Simple, Multi-, and Pseudographs
No multiplication symbol: Write `xy`, not `x × y` or `x · y`; Lesson 178 — Reading and Interpreting Algebraic Notation
No overlap: The angles shouldn't cover any of the same area.; Lesson 705 — Adjacent Angles
No real solutions: The line misses the circle entirely; Lesson 920 — Intersection of Lines and Circles
No restrictions: Box can hold 0, 1, 2, 3,.; Lesson 2642 — Using Generating Functions to Count Distributions
no solution: (also written as the empty set ∅); Lesson 206 — Recognizing Identity and Contradiction Equations Lesson 212 — Equations with No Solution Lesson 213 — Equations with Infinitely Many Solutions Lesson 223 — Special Cases: All Reals and No Solution Lesson 229 — Absolute Value Equations with No Solution or All Solutions Lesson 272 — Systems and Geometric Reasoning with Parallel/Perpendicular Lines Lesson 280 — Identifying No Solution or Infinite Solutions via Substitution Lesson 290 — Special Cases: No Solution and Infinite Solutions (+9 more)
No Solution (Inconsistent System): You get a false statement like `0 = 5` or `3 = -2`.; Lesson 290 — Special Cases: No Solution and Infinite Solutions
No triangle: The opposite side is too short to reach; Lesson 1096 — The SSA Configuration Introduction Lesson 1122 — The Ambiguous Case with Acute Angles
No triangle exists when: `a < b·sin(A)`; Lesson 1097 — No Solution Case in SSA
No unique solution: to the augmented system; Lesson 1889 — Dealing with Redundant Constraints
No visual feedback: You don't see the geometric relationship between lines; Lesson 281 — Comparing Graphing and Substitution Methods
No x-intercepts: The parabola floats entirely above or below the x-axis (discriminant < 0); Lesson 431 — x-Intercepts and Zeros of Quadratic Functions
Noise Reduction: Real images often have noise—random pixel variations that don't represent true content.; Lesson 2290 — Applications of SVD: Image Compression and Noise Reduction
Non-example: Adding a constant vector (translation) is *not* linear because `T(u + v) = u + v + c ≠ T(u) + T(v) = (u + c) + (v + c)`.; Lesson 2259 — Definition of Linear Transformations Lesson 2562 — Surjective Functions (Onto)Lesson 2735 — Division and Modular Inverses
Non-exclusive choices: A student taking calculus OR physics (some take both); Lesson 2785 — The General Addition Rule
Non-homogeneous: When $Q(x) \neq 0$, there's a non-zero "driving" or "forcing" term on the right side; Lesson 2319 — Homogeneous vs Non-Homogeneous Linear ODEs Lesson 2337 — Form and Classification of Second-Order Linear ODEs Lesson 2348 — Non-Homogeneous ODEs: Structure of Solutions
non-homogeneous recurrence relation: has the form:; Lesson 2633 — Non-Homogeneous Recurrence Relations Lesson 2635 — General Solution of Non- Homogeneous Recurrences
non-negative: Lesson 557 — Real-World Context and Restricted Domains Lesson 2293 — Eigenvalue Criterion for Definiteness Lesson 3047 — The F Distribution
Non-negativity: *p*(*x*) ≥ 0 for all values *x*; Lesson 2832 — Introduction to Probability Mass Functions (PMFs)Lesson 2842 — Verifying Valid PMFs and PDFs Lesson 2922 — Properties and Applications of Chi-Squared Lesson 2927 — Joint Probability Distributions: Discrete Case Lesson 2928 — Joint Probability Distributions: Continuous Case
Non-Polynomials (not allowed): Lesson 311 — Identifying Polynomials vs Non-Polynomials
Non-proportional example: Lesson 122 — Proportional Relationships in Tables and Graphs
none: Lesson 488 — Horizontal Asymptotes: Degree Rules Lesson 2621 — Counting with None of the Properties Lesson 2622 — Derangements and the Derangement Formula Lesson 2624 — Euler's Totient Function via Inclusion-Exclusion Lesson 2625 — Counting Surjective Functions
Nonlinear PDE: The highest-order derivatives appear nonlinearly—raised to powers, multiplied together, or inside nonlinear functions.; Lesson 2409 — Classification: Order and Linearity
Nonlinear PDEs: (like Burger's equation); Lesson 2455 — Limitations and Extensions of the Method
Nonparametric methods: are statistical tests that don't assume a particular distribution shape.; Lesson 3103 — Introduction to Nonparametric Methods
nonsingular: ).; Lesson 1300 — Definition of an Inverse Matrix Lesson 2138 — Definition of Matrix Inverse
Nonsingular (invertible): det(A) ≠ 0, inverse exists; Lesson 1301 — When Does an Inverse Exist?
norm: ) of a vector is simply its length—the straight-line distance from its tail to its tip.; Lesson 1248 — Magnitude (Norm) of a Vector Lesson 2053 — Magnitude and Distance Lesson 2203 — Norm Induced by an Inner Product
Normal equation: Variable remains with a coefficient you can divide by → one solution; Lesson 206 — Recognizing Identity and Contradiction Equations
normal equations: .; Lesson 2235 — Normal Equations Lesson 2238 — Applications: Linear Regression Lesson 2247 — QR and Least Squares Problems Lesson 2248 — Computational Advantages of QR Decomposition Lesson 3074 — Least Squares Estimation in Multiple Regression
normal line: is perpendicular to the tangent line at that same point.; Lesson 1468 — Normal Lines to Curves Lesson 1835 — Normal Lines to Surfaces
Normal Probability Plot: (also called a **Quantile-Quantile plot** or **Q-Q plot**) gives you an immediate visual assessment.; Lesson 3086 — Normal Probability Plots (Q-Q Plots)
normal vector: a vector that's perpendicular to every direction within the plane itself.; Lesson 1795 — Planes in 3D: Normal Form and General Equation Lesson 1847 — The Gradient is Perpendicular to Level Curves and Surfaces Lesson 2015 — The Tangent Plane and Normal Vectors to Surfaces Lesson 2017 — Orientation of Surfaces Lesson 2020 — Flux Through a Surface: Definition and Setup
Normal(μ, σ²): The bell curve:; Lesson 2962 — MGFs of Common Distributions
Normality: Lesson 3040 — Assumptions and Diagnostics for t-Tests Lesson 3068 — Assumptions of Simple Linear Regression Lesson 3094 — One-Way ANOVA: The Model and Assumptions Lesson 3101 — Checking ANOVA Assumptions
Normality assumption: F tests can be misleading if your data is heavily skewed.; Lesson 3051 — Applications and Limitations
normalization: .; Lesson 1249 — Unit Vectors and Normalization Lesson 2054 — Unit Vectors and Normalization Lesson 2842 — Verifying Valid PMFs and PDFs
Normalize: **u** = **v**/||**v**|| = ⟨v₁/||**v**||, v₂/||**v**||, .; Lesson 1849 — Directional Derivatives in Non-Unit Directions Lesson 2224 — Gram-Schmidt for Subspaces
Normalize (optional): e = u / ǁu ǁ; Lesson 2226 — Gram-Schmidt in Inner Product Spaces
not: work for subtraction or division.; Lesson 36 — The Commutative Property Lesson 173 — Like Terms Lesson 177 — Order of Operations with Variables Lesson 310 — What is a Polynomial?Lesson 547 — Common Function Notation Mistakes Lesson 697 — Line Segments and Rays Lesson 787 — The Converse of the Pythagorean Theorem Lesson 858 — Area of Parallelograms (+21 more)
Not a partition: `{{1, 2}, {2, 3}, {4}}`; Lesson 2554 — Partitions of a Set Lesson 2780 — Partitions of the Sample Space
Not arithmetic: 1, 2, 4, 8, 16, .; Lesson 651 — Definition of Arithmetic Sequences Lesson 656 — Determining if a Sequence is Arithmetic
Not causation: High correlation doesn't imply X causes Y; Lesson 2956 — Examples and Applications of Correlation
not commutative: AB ≠ BA in general.; Lesson 1285 — Matrix Multiplication is Not Commutative Lesson 1287 — The Distributive Property for Matrices Lesson 1307 — Using Inverses to Solve Matrix Equations Lesson 2267 — Composition of Linear Transformations
Not differentiable: at x = 0 (the sharp corner creates a discontinuity in slope); Lesson 594 — The Absolute Value Function
Not in standard form: Lesson 314 — Standard Form of a Polynomial
Not Independent: Drawing two cards without replacement.; Lesson 2802 — Definition of Independent Events Lesson 2966 — Linear Combinations of Random Variables
Not injective: `g(x) = x²` on real numbers.; Lesson 2560 — Injective Functions (One-to-One)
Not intuition: Even if something "feels obvious," mathematics demands explicit logical steps; Lesson 2490 — What Does It Mean to Prove Something?
Not invertible: Lesson 1297 — Using Determinants to Test Invertibility
Not just examples: Showing that something works for 5, 10, or even a million cases doesn't prove it's true for *all* cases; Lesson 2490 — What Does It Mean to Prove Something?
Not persuasion: Unlike a debate, a proof isn't about convincing through rhetoric—it's about demonstrating logical necessity; Lesson 2490 — What Does It Mean to Prove Something?
NOT random experiments: Lesson 2771 — What is a Random Experiment?
Not separable: Lesson 2308 — What is a Separable Differential Equation?
Not simply connected: A washer (ring), the plane with the origin removed, a torus—these have "holes" that loops can wrap around; Lesson 1990 — Simply Connected Domains and Why They Matter
Not symmetric: Lesson 2543 — Symmetric Relations
not unique: .; Lesson 1536 — What is an Antiderivative?Lesson 2716 — Bézout's Identity
Notation: R₁ ↔ R₂ means "swap row 1 and row 2"; Lesson 1310 — Elementary Row Operations Lesson 2777 — Operations on Events: Union and Intersection
Note the radius: of the circle; Lesson 842 — Sector Area Formula
Notice: We integrate with respect to **x first** (inner integral), then **y** (outer integral).; Lesson 1903 — Type II Regions: Horizontally Simple Regions
Notice the pattern: The exponent 4 is exactly double the exponent 2.; Lesson 372 — Factoring Higher-Degree Trinomials Using Substitution
Now: apply the natural logarithm; Lesson 633 — Exponential Equations Requiring Algebraic Manipulation
Now subtract: to eliminate x:; Lesson 287 — Multiplying Both Equations to Enable Elimination
Now subtract ones: 12 - 8 = 4; Lesson 16 — Subtraction with Regrouping (Borrowing)
nth partial sum: , denoted `S `, is simply the sum of the first n terms of the series.; Lesson 680 — Partial Sums and Series Limits Lesson 1714 — Sequence of Partial Sums
nth powers: , like:; Lesson 1745 — The Root Test: Statement and Motivation Lesson 1746 — Applying the Root Test Lesson 1747 — Comparing Ratio and Root Tests
nth term: is given by:; Lesson 653 — The Explicit Formula for Arithmetic Sequences Lesson 1703 — Definition of a Sequence and Notation
Nuclear physics: uses exponential distributions to model decay times.; Lesson 2903 — Applications of the Exponential Distribution
Null hypothesis: H₀: μ = μ₀; Lesson 3026 — Two-Tailed z-Tests
Null hypothesis (H₀): The default assumption or "status quo.; Lesson 3020 — Introduction to Hypothesis Testing Lesson 3021 — Null and Alternative Hypotheses
null space: (or *kernel*) of a matrix **A** is the set of all vectors **x** that satisfy **Ax = 0**—the homogeneous system.; Lesson 2081 — Column Space and Null Space Lesson 2096 — Bases for Important Subspaces Lesson 2130 — The Null Space Lesson 2134 — Orthogonality of Fundamental Subspaces Lesson 2183 — Roots of the Characteristic Polynomial as Eigenvalues Lesson 2261 — The Kernel (Null Space) of a Linear Transformation Lesson 2283 — Left and Right Singular Vectors
Null Space, N(A): Lesson 2132 — The Four Fundamental Subspaces
nullity(A): = dimension of the null space (the solution space of **Ax** = **0**); Lesson 2095 — Rank-Nullity Theorem Lesson 2133 — The Rank-Nullity Theorem
Number being subtracted: (how many you remove); Lesson 11 — Understanding Subtraction as Taking Away
Number comes first: Write `5x`, not `x5`; Lesson 178 — Reading and Interpreting Algebraic Notation
Number line: An open circle at -2, an open circle at 4, with the region between them shaded; Lesson 231 — Solving 'And' Compound Inequalities
Number of attempts: until first success: {1, 2, 3, .; Lesson 2823 — Discrete Random Variables: Definition and Examples
Number of defective items: in a batch: {0, 1, 2, .; Lesson 2823 — Discrete Random Variables: Definition and Examples
Number of emails received: in a day: {0, 1, 2, 3, .; Lesson 2823 — Discrete Random Variables: Definition and Examples
Number of heads: when flipping 3 coins: Can be 0, 1, 2, or 3 (finite); Lesson 2823 — Discrete Random Variables: Definition and Examples
Number problems: consecutive integers, products; Lesson 400 — Applications and Word Problems
numerator: (top number) tells you *how many parts you have*; Lesson 59 — What is a Fraction?Lesson 442 — What is a Rational Expression?Lesson 532 — Evaluating Expressions with Fractional Exponents Lesson 1222 — Dividing Complex Numbers in Polar Form Lesson 1338 — Combining Multiple Limit Laws Lesson 1406 — Combining Product and Quotient Rules Lesson 2793 — Computing Conditional Probabilities from Tables Lesson 3096 — The F-Statistic and F-Distribution
Numerator result: (12 + 5)·1 = 17; Lesson 1338 — Combining Multiple Limit Laws
Numerical methods: Finite differences, finite elements for irregular domains; Lesson 2455 — Limitations and Extensions of the Method
Numerically stable: for positive definite systems; Lesson 2295 — The Cholesky Decomposition

O

O(n³) complexity: , making it efficient for moderately-sized systems but increasingly expensive as *n* grows large— understanding operation counts guides practical solver selection.; Lesson 2115 — Computational Complexity and Efficiency
Objective Function: A linear equation that represents what you want to optimize (maximize or minimize).; Lesson 304 — Introduction to Linear Programming Lesson 305 — The Objective Function Lesson 1505 — Introduction to Optimization Lesson 1516 — Applied Optimization Strategy
oblique cylinder: (tilted) has the same volume as an upright cylinder with the same base and height; Lesson 897 — Introduction to Cavalieri's Principle Lesson 899 — Cavalieri's Principle with Cylinders
Obtuse: 90° < angle < 180° (wide and open); Lesson 704 — Classifying Angles by Measure Lesson 731 — Classifying Triangles by Angles Lesson 789 — Acute and Obtuse Triangle Tests Lesson 1125 — The Ambiguous Case with Obtuse Angles
obtuse angle: is between 90° and 180°; Lesson 703 — Measuring Angles and Angle Notation Lesson 704 — Classifying Angles by Measure
Obtuse Angle Oversight: Lesson 1128 — Common Errors and Verification in SSA
Obtuse Triangle: Lesson 731 — Classifying Triangles by Angles Lesson 788 — Using the Converse to Identify Right Triangles Lesson 789 — Acute and Obtuse Triangle Tests
Obtuse Triangle Test: Lesson 789 — Acute and Obtuse Triangle Tests
Obtuse triangles: When the base is adjacent to the obtuse angle, the height lands outside the triangle on the base's extension; Lesson 1112 — Area Formula Using Base and Height
odd: .; Lesson 138 — Exponents with Negative Bases Lesson 569 — Even and Odd Functions Lesson 1048 — Even- Odd Identities Lesson 1776 — Maclaurin Series for sin(x) and cos(x)Lesson 2603 — The Binomial Theorem for Negative Binomials Lesson 2606 — Binomial Coefficients Modulo 2 and Sierpiński's Triangle
Odd cycles: (n odd): Two colors create a conflict when you return to the start—adjacent vertices would need the same color.; Lesson 2696 — Coloring Special Graphs
Odd exponent: → **Negative result**; Lesson 138 — Exponents with Negative Bases
Odd extension: Mirror the function *antisymmetrically* (flip it upside down).; Lesson 2462 — Half-Range Expansions
Odd function: (origin symmetry): f(-x) = -f(x).; Lesson 1517 — Domain, Intercepts, and Symmetry
odd functions: have origin symmetry (f(-x) = -f(x)), like f(x) = x³.; Lesson 587 — Even and Odd Functions Under Transformation Lesson 1561 — Definite Integrals of Even and Odd Functions Lesson 2461 — Even and Odd Functions: Cosine and Sine Series
Odd integers: An integer *n* is odd if there exists an integer *k* such that *n = 2k + 1*.; Lesson 2492 — Direct Proof: Working with Definitions
Odd power of tangent: Factor out `tan x sec x` and convert remaining tangents to secants; Lesson 1610 — Strategy Selection for Trigonometric Integrals
Odd power present: Factor out one copy and use the Pythagorean identity to convert the rest; Lesson 1610 — Strategy Selection for Trigonometric Integrals
On the boundary: When |x - c| = R, you must check each endpoint separately—anything can happen; Lesson 1763 — Radius of Convergence
one: Lesson 626 — Logarithms of 1 and the Base Lesson 635 — Using Properties to Combine Logarithms Lesson 696 — Lines: Infinite Straight Paths Lesson 809 — Finding Individual Interior Angles in Regular Polygons Lesson 817 — Proving Quadrilaterals Are Parallelograms Lesson 1234 — Applications: Fractional Powers and Euler's Formula Lesson 2484 — Negating Quantified Statements Lesson 2543 — Symmetric Relations (+3 more)
one acute angle: , you can solve the entire right triangle using this three-step process:; Lesson 961 — Solving When Given the Hypotenuse and One Acute Angle Lesson 962 — Solving When Given One Leg and One Acute Angle
One curved lateral surface: (the side that wraps around); Lesson 888 — Surface Area of Cylinders
one point: → **one solution**; Lesson 272 — Systems and Geometric Reasoning with Parallel/Perpendicular Lines Lesson 293 — Visualizing Three-Variable Systems Lesson 432 — Using the Discriminant to Predict Intercepts Lesson 700 — Intersection of Lines and Planes
One sample, test median: Sign Test or Wilcoxon Signed-Rank Test; Lesson 3111 — Choosing and Interpreting Nonparametric Tests
One secant, one tangent: Use the arc cut by the secant and the arc from the tangent's point of contact.; Lesson 841 — Angles Formed by Secants and Tangents Outside
One solution: Most equations you've solved give you a single value, like `3x + 5 = 14` giving you `x = 3`.; Lesson 206 — Recognizing Identity and Contradiction Equations Lesson 272 — Systems and Geometric Reasoning with Parallel/Perpendicular Lines Lesson 920 — Intersection of Lines and Circles Lesson 1125 — The Ambiguous Case with Obtuse Angles
One triangle: The opposite side exactly reaches (perpendicular case) or the given angle is obtuse/right; Lesson 1096 — The SSA Configuration Introduction Lesson 1122 — The Ambiguous Case with Acute Angles Lesson 1123 — Using the Height to Determine Triangle Existence
One triangle face: = ½ × 6 × 5 = 15 cm²; Lesson 889 — Surface Area of Pyramids
One unique solution: (three planes meet at exactly one point); Lesson 292 — Introduction to Three-Variable Systems Lesson 2111 — Consistent and Inconsistent Systems
One vector: spans a **line** through the origin; Lesson 2062 — Geometric Span in R³: Lines, Planes, and All of R³
one way: the triangle can be constructed.; Lesson 745 — HL (Hypotenuse-Leg) Congruence Theorem Lesson 2592 — Combinations with All or None: C(n,0) and C(n,n)
One x-intercept: The vertex sits right on the x-axis—the parabola just touches it (discriminant = 0); Lesson 431 — x-Intercepts and Zeros of Quadratic Functions
one-sided limits: .; Lesson 1326 — Limits Approaching from Left and Right Lesson 1329 — Understanding One-Sided Limits
one-to-one function: (also called an *injective* function) is one where every output comes from exactly one input.; Lesson 562 — One-to-One Functions Lesson 2560 — Injective Functions (One-to-One)
One-Way ANOVA: (Analysis of Variance) solves this by testing all groups simultaneously.; Lesson 3049 — One-Way ANOVA F Test Introduction
Ones column: 4 + 5 = 9; Lesson 14 — Adding and Subtracting with Larger Numbers
Ones place: (the right digit) — tells you how many single items; Lesson 3 — Understanding Place Value: Ones and Tens Lesson 4 — Place Value: Hundreds Lesson 7 — Rounding to the Nearest Ten and Hundred
only: when:; Lesson 1506 — The Extreme Value Theorem Lesson 1534 — Common Mistakes and When Not to Use L'Hôpital's Rule Lesson 1752 — The Alternating Series Test (Leibniz Test)Lesson 2349 — The Method of Undetermined Coefficients Overview
only if: it's vertical (and vice versa); Lesson 270 — Special Cases: Horizontal and Vertical Lines Lesson 2679 — Euler's Theorem for Eulerian Circuits
only on x: (no y terms), then there exists an integrating factor μ(x), and:; Lesson 2331 — Finding Integrating Factors: Systematic Approach Lesson 2447 — Separating Variables in the Heat Equation
Only then: substitute values if needed; Lesson 177 — Order of Operations with Variables
Only when multiplying: You can regroup products freely:; Lesson 1286 — Matrix Multiplication is Associative
Open circle ( ○): Use when the endpoint is *not* included (< or >); Lesson 217 — Inequality Notation and the Number Line
Open endpoint: (< or >): Use an **open circle** ○ to show the point is excluded; Lesson 592 — Graphing Piecewise Functions
Open Intervals (a, b): Lesson 1353 — Continuity on an Interval
Open sentences: "x + 3 = 7" — Truth depends on what x is; it's not a statement until we specify x.; Lesson 2468 — Propositions and Truth Values
Opening: State the given information and what you're proving; Lesson 764 — Paragraph Proof Format
Operation: Addition (finding total); Lesson 82 — Word Problems Involving Fraction Operations
Operations: Lesson 2656 — Graph Representations: Adjacency List and Edge List
opposite: of what you see:; Lesson 407 — Finding the Vertex by Completing the Square Lesson 948 — Identifying Opposite, Adjacent, and Hypotenuse Lesson 951 — The Tangent Ratio (TOA)Lesson 953 — Finding Side Lengths Using Cosine Lesson 954 — Finding Side Lengths Using Tangent Lesson 956 — Choosing the Appropriate Trig Ratio Lesson 962 — Solving When Given One Leg and One Acute Angle Lesson 970 — Using SOH-CAH-TOA with Depression Angles (+3 more)
Opposite angles are congruent: Lesson 816 — Parallelograms: Definition and Properties
Opposite side: = height of the pole (what we want to find); Lesson 969 — Using SOH-CAH-TOA with Elevation Angles Lesson 977 — Finding Heights Using Angles of Elevation
Opposite sides are congruent: (equal in length); Lesson 816 — Parallelograms: Definition and Properties
Optical instruments: Designing reflectors that concentrate light or sound; Lesson 1165 — Reflective Property of Ellipses
Optimal Solution: The point in the feasible region where your objective function reaches its maximum or minimum value.; Lesson 304 — Introduction to Linear Programming
Optimizing: means selecting the strategy that gets you to the answer most efficiently.; Lesson 874 — Optimizing Composite Figure Problems
Option 1: ∫∫ f(x,y) dx dy — integrate with respect to x first (inner integral), then y; Lesson 1895 — Order of Integration
Option 2: ∫∫ f(x,y) dy dx — integrate with respect to y first (inner integral), then x; Lesson 1895 — Order of Integration
Option A: Use implicit differentiation on the constraint to find dy/dx, then substitute into your objective function's derivative; Lesson 1514 — Optimization with Implicit Constraints
Option A: Point-Slope Form: Lesson 259 — Writing Equations from Two Points
Option B: Cleverly manipulate the constraint algebraically before substituting (sometimes squaring both sides or using trig identities helps); Lesson 1514 — Optimization with Implicit Constraints
Option B: Slope-Intercept Form: Lesson 259 — Writing Equations from Two Points
order: , and negatives reverse order on the number line.; Lesson 219 — The Multiplication/Division Sign Rule Lesson 943 — Rotational Symmetry Lesson 1271 — What is a Matrix? Dimensions and Notation Lesson 2301 — Order and Degree of ODEs Lesson 2409 — Classification: Order and Linearity Lesson 3111 — Choosing and Interpreting Nonparametric Tests
Order matters: with subtraction and division!; Lesson 188 — Translating Words into Mathematical Expressions Lesson 377 — Identifying the Common Binomial Factor Lesson 558 — Function Composition Basics Lesson 2101 — Properties of Matrix Multiplication Lesson 2537 — Properties of Cartesian Products Lesson 2594 — Distinguishing When to Use Combinations vs Permutations
ordered pair: is a way to describe a specific location using two numbers written in parentheses with a comma between them: `(x, y)`.; Lesson 236 — Understanding Ordered Pairs and Coordinates Lesson 240 — Reading Coordinates from a Graph
ordered triple: to represent a vector:; Lesson 1238 — Component Form of Vectors in 3D Lesson 1792 — The 3D Coordinate System and Distance Formula
Ordinary: if both $p(x)$ and $q(x)$ are analytic (have convergent power series) at $x_0$; Lesson 2401 — Regular Singular Points and the Method of Frobenius
Ordinary Differential Equations (ODEs): involve derivatives with respect to a **single variable**.; Lesson 2300 — What is a Differential Equation?Lesson 2408 — What is a Partial Differential Equation?
ordinary induction: and **strong induction** prove statements for all integers n ≥ some base case.; Lesson 2512 — Review: Ordinary vs. Strong Induction Lesson 2520 — Choosing the Right Proof Technique
Ore's Theorem: provides exactly that.; Lesson 2686 — Ore's Theorem
Organize data: in a contingency table where rows represent populations and columns represent categories; Lesson 3045 — Chi-Squared Test for Homogeneity
Orient: it correctly (inward or outward as specified); Lesson 2020 — Flux Through a Surface: Definition and Setup
Orient each piece outward: Lesson 2042 — The Divergence Theorem for General Regions
Orientation: – which direction the figure faces; Lesson 921 — Introduction to Geometric Transformations Lesson 1172 — Hyperbolas with Horizontal vs Vertical Transverse Axes Lesson 1995 — Orientation and Simple Closed Curves Lesson 2017 — Orientation of Surfaces
origin symmetry: rotating it 180° around the origin leaves it unchanged.; Lesson 569 — Even and Odd Functions Lesson 1001 — Symmetry Properties of the Unit Circle
original: denominator zero are still forbidden, even after simplification.; Lesson 450 — Writing Simplified Expressions with Domain Restrictions Lesson 1177 — Conjugate Hyperbolas
Original point: (x, y); Lesson 935 — Dilations Centered at the Origin
orthogonal: (a fancy word for perpendicular), their dot product equals zero.; Lesson 1257 — Orthogonal Vectors and the Dot Product Lesson 2206 — Angle Between Vectors and Orthogonality Lesson 2209 — Orthogonal and Orthonormal Vectors Lesson 2223 — Normalizing to Obtain an Orthonormal Basis Lesson 2270 — Symmetric Matrices and Their Properties Lesson 2457 — Orthogonality of Trigonometric Functions
orthogonal basis: for a vector space is a basis where every pair of distinct vectors has inner product zero.; Lesson 2211 — Orthogonal Bases Lesson 2213 — Coordinates with Respect to Orthogonal Bases Lesson 2230 — Orthogonal Projection onto a Subspace
orthogonal complement: of **W**, denoted **W ⊥** (read "W perp"), is the set of *all* vectors in **V** that are orthogonal to *every* vector in **W**.; Lesson 2215 — Orthogonal Complements Lesson 2216 — Properties of Orthogonal Complements
Orthogonal diagonalization: is a premium upgrade: we find **P** whose columns are *orthonormal* eigenvectors, making **P** an orthogonal matrix.; Lesson 2271 — Orthogonal Diagonalization Lesson 2272 — The Spectral Theorem for Symmetric Matrices
Orthogonal eigenfunctions: (like perpendicular vectors, but for functions); Lesson 2405 — Eigenvalue Problems and Sturm-Liouville Theory
Orthogonal eigenvectors: Eigenvectors corresponding to *distinct* eigenvalues of a symmetric matrix are orthogonal.; Lesson 2272 — The Spectral Theorem for Symmetric Matrices
Orthogonal matrices: Eigenvalues reveal rotation structure; stability in numerical algorithms; Lesson 2176 — Eigenvalues of Special Matrices: Symmetric and Orthogonal
orthogonal matrix: satisfies QᵀQ = I, meaning its columns are orthonormal.; Lesson 2176 — Eigenvalues of Special Matrices: Symmetric and Orthogonal Lesson 2241 — Orthogonal Matrices and Their Properties
orthogonal projection: works similarly: given a vector **v** and a line defined by a direction vector **u**, we find the "shadow" of **v** that falls directly onto that line.; Lesson 2229 — Orthogonal Projection onto a Line Lesson 2234 — The Least Squares Problem Lesson 2236 — Least Squares via Orthogonal Projection
Orthogonality: = different measurement outcomes correspond to distinguishable states; Lesson 2279 — Spectral Theorem in Quantum Mechanics and Statistics Lesson 2421 — The General Series Solution Lesson 2422 — Determining Coefficients from Initial Conditions
Orthogonally diagonalize: A = PDP^T where P contains orthonormal eigenvectors; Lesson 2277 — Principal Axes and Quadratic Forms
Orthonormal: (each vector has length 1 and they're perpendicular); Lesson 2088 — Standard Bases Lesson 2209 — Orthogonal and Orthonormal Vectors Lesson 2213 — Coordinates with Respect to Orthogonal Bases Lesson 2218 — Standard Orthonormal Bases in Euclidean Space Lesson 2221 — Gram-Schmidt in ℝ²: First Example Lesson 2222 — Gram-Schmidt in ℝ³: Complete Example Lesson 2223 — Normalizing to Obtain an Orthonormal Basis Lesson 2274 — Eigenspaces and Multiplicity for Symmetric Matrices
orthonormal basis: the gold standard.; Lesson 2211 — Orthogonal Bases Lesson 2212 — Orthonormal Bases Lesson 2214 — Coordinates with Respect to Orthonormal Bases Lesson 2219 — Motivation for Orthonormal Bases Lesson 2230 — Orthogonal Projection onto a Subspace
Orthonormality: = components are uncorrelated; Lesson 2279 — Spectral Theorem in Quantum Mechanics and Statistics
Oscillation: (bouncing up and down); Lesson 1704 — Visualizing Sequences: Terms and Graphs
Other angle: 180° - 90° - 35° = 55°; Lesson 962 — Solving When Given One Leg and One Acute Angle
Other divisors: For divisibility by 2, 5, or 10, we only need the last digit because 10 ≡ 0 modulo these values.; Lesson 2740 — Applications to Divisibility Tests
Out-degree: the number of edges *leaving* (pointing away from) a vertex; Lesson 2649 — Directed Graphs (Digraphs)Lesson 2650 — Degree of a Vertex and the Handshaking Lemma
Outer: Multiply the outer terms: `x · 5 = 5x`; Lesson 326 — Multiplying Two Binomials Using FOIL Lesson 335 — Square of a Binomial
Outer boundary: traverse counterclockwise (standard positive orientation); Lesson 1999 — Green's Theorem for Regions with Holes
Outer curve: r = g₂(θ) (farther from the origin); Lesson 1918 — Regions Bounded by Polar Curves
Outer function: f: takes g(x) as its input; Lesson 1409 — The Chain Rule Statement Lesson 1433 — Chain Rule with Exponential Functions
Outer integral: Take that result and integrate with respect to x from a to b.; Lesson 1894 — Fubini's Theorem for Rectangles Lesson 1899 — Double Integrals of Non-Separable Functions Lesson 1902 — Type I Regions: Vertically Simple Regions
Outer operation: Taking the square root → so `f(x) = √x`; Lesson 561 — Decomposing Functions
outer radius: R(x) is the distance from the axis to the outer curve; Lesson 1657 — Washer Method: Introduction and Setup Lesson 1658 — Washer Method: Identifying Outer and Inner Radii Lesson 1659 — Washer Method: Rotation About the x-axis
Outer radius R(x): The function that is *higher up* (larger y-value) from the x-axis; Lesson 1658 — Washer Method: Identifying Outer and Inner Radii
Outer radius R(y): The function that is *farther right* (larger x-value) from the y-axis; Lesson 1658 — Washer Method: Identifying Outer and Inner Radii
Outer variable (usually z): What are the vertical limits?; Lesson 1936 — Setting Up Triple Integrals in Cylindrical Coordinates
Outermost integral: Integrate with respect to z; Lesson 1923 — Setting Up Triple Integrals over Rectangular Boxes
outliers: , we use fences:; Lesson 2991 — Box Plots and Outlier Detection Lesson 2992 — Histograms and Frequency Distributions Lesson 2994 — Comparing Distributions Lesson 3089 — Outliers in Regression
output: (function value) represents another real-world quantity that depends on the input; Lesson 546 — Function Evaluation in Context Lesson 547 — Common Function Notation Mistakes
Outside: (exterior to) the parallel lines; Lesson 716 — Alternate Exterior Angles Lesson 841 — Angles Formed by Secants and Tangents Outside Lesson 1411 — The Outside-Inside Method
Outside derivative: `d/dx[arcsin(u)] = 1/√(1 - u²)` where `u = 3x`; Lesson 1444 — Chain Rule with Inverse Trigonometric Functions
Outside the radius: If |x - c| > R, the series diverges; Lesson 1763 — Radius of Convergence
Outward orientation: Standard convention; flux > 0 means net outward flow; Lesson 2023 — Flux Through Closed Surfaces Lesson 2037 — Closed Surfaces and Orientation
Over intersection: A × (B ∩ C) = (A × B) ∩ (A × C); Lesson 2537 — Properties of Cartesian Products
Over union: A × (B ∪ C) = (A × B) ∪ (A × C); Lesson 2537 — Properties of Cartesian Products
Overdamped: (Δ > 0, two distinct real roots); Lesson 2347 — Applications: Free Oscillations and Damped Motion
Overlapping events: Drawing a red card OR a face card from a deck (some cards are both); Lesson 2785 — The General Addition Rule

P

p < 0: , the parabola opens **downward** (focus below vertex); Lesson 1145 — Standard Form: Vertex at Origin, Vertical Axis Lesson 1146 — Standard Form: Vertex at Origin, Horizontal Axis Lesson 1148 — Parabolas with Vertex at (h, k): Vertical Axis Lesson 1149 — Parabolas with Vertex at (h, k): Horizontal Axis
P = A(A ᵀA) ⁻¹A ᵀ: depends only on the subspace (through A), not on the specific vector b.; Lesson 2231 — The Projection Matrix
P = ρgh: , where ρ (rho) is the fluid's density, g is gravitational acceleration, and h is the depth below the surface.; Lesson 1699 — Fluid Pressure and Force on Submerged Surfaces
p > 0: , the parabola opens **upward** (focus above vertex); Lesson 1145 — Standard Form: Vertex at Origin, Vertical Axis Lesson 1146 — Standard Form: Vertex at Origin, Horizontal Axis Lesson 1148 — Parabolas with Vertex at (h, k): Vertical Axis Lesson 1149 — Parabolas with Vertex at (h, k): Horizontal Axis
P_B: changes coordinates from basis B to standard; Lesson 2266 — Change of Basis for Linear Transformations
P_C^{-1}: changes coordinates from standard to basis C; Lesson 2266 — Change of Basis for Linear Transformations
P_new: = **P_old** + **v**·t; Lesson 1265 — Velocity and Displacement Vectors
P_old: + **v**·t; Lesson 1265 — Velocity and Displacement Vectors
p-series: is an infinite series of the form:; Lesson 1725 — The p-Series: Definition Lesson 1726 — Convergence of p-Series Lesson 1728 — Comparing Geometric and p-Series Lesson 1735 — Choosing Comparison Series Effectively Lesson 1744 — When the Ratio Test is Inconclusive Lesson 1750 — Strategy: Choosing the Right Convergence Test
p-value: is the probability of observing a test statistic as extreme as (or more extreme than) what you actually got, *assuming the null hypothesis is true*.; Lesson 3022 — Test Statistics and P-values Lesson 3077 — Hypothesis Testing for Individual Coefficients Lesson 3097 — The ANOVA Table and Hypothesis Testing
p-value approach: , calculate P(|Z| ≥ |z_observed|) = 2·P(Z ≥ |z_observed|).; Lesson 3026 — Two-Tailed z-Tests Lesson 3027 — One-Tailed z-Tests
P(1 + r)ⁿ: Lesson 687 — Compound Interest and Financial Growth
P(2): Lesson 351 — The Remainder Theorem
P(A ∩ B): The probability that *both* A and B occur together (the intersection); Lesson 2792 — The Conditional Probability Formula Lesson 2799 — Common Misconceptions in Conditional Probability
P(A): or sometimes **2^A**.; Lesson 2531 — Introduction to Power Sets
P(A) ≤ P(B): .; Lesson 2787 — Monotonicity and Set Relationships
P(A) ≥ 0: for every event A.; Lesson 2781 — The Three Axioms of Probability
P(A|B): reads as "the probability of A *given* B" or "the probability of A *conditioned on* B.; Lesson 2792 — The Conditional Probability Formula Lesson 2799 — Common Misconceptions in Conditional Probability
P(B): The probability that B occurs—this must be greater than zero (we can't condition on impossible events); Lesson 2792 — The Conditional Probability Formula
P(B) > 0: .; Lesson 2792 — The Conditional Probability Formula
P(c): the value of the polynomial when you substitute **c** for **x**.; Lesson 351 — The Remainder Theorem
P(c) = R: .; Lesson 351 — The Remainder Theorem
P(E): Normalizing constant — total probability of seeing this evidence; Lesson 2817 — Prior and Posterior Probabilities
P(E|H): Likelihood — how likely this evidence is if hypothesis H is true; Lesson 2817 — Prior and Posterior Probabilities
P(H): Prior — what we believed before; Lesson 2817 — Prior and Posterior Probabilities
P(H|E): Posterior — our updated belief after seeing evidence E; Lesson 2817 — Prior and Posterior Probabilities
P(L, K) = A·L^α·K^β: Lesson 1810 — Applications: Modeling with Multivariable Functions
P(n,r): or sometimes Pᵣ.; Lesson 2580 — Permutations of n Objects Taken r at a Time
P(n,r) = n!/(n-r): , it's time to build computational fluency.; Lesson 2581 — Computing Permutation Values
P(S) = 1: Lesson 2781 — The Three Axioms of Probability
P(t): = population at time *t*; Lesson 645 — Logistic Growth Models
P(X < k): becomes P(Z < k − 0.; Lesson 2914 — Continuity Correction Lesson 2983 — The Continuity Correction
P(X = a): for discrete X: F(a) - F(a ); Lesson 2847 — Finding Probabilities Using the CDF
P(X = c): = 0 for any specific value c (because it's a single point, with zero width); Lesson 2838 — Computing Probabilities Using PDFs
P(X = k): becomes P(k − 0.; Lesson 2914 — Continuity Correction Lesson 2983 — The Continuity Correction
P(X > k): becomes P(Z > k + 0.; Lesson 2914 — Continuity Correction
P(X ≤ b): = ∫[from -∞ to b] f(x) dx; Lesson 2838 — Computing Probabilities Using PDFs
P(X ≤ k): becomes P(Z < k + 0.; Lesson 2914 — Continuity Correction Lesson 2983 — The Continuity Correction
P(X ≥ a): = ∫[from a to ∞] f(x) dx; Lesson 2838 — Computing Probabilities Using PDFs
P(X ≥ k): becomes P(Z > k − 0.; Lesson 2914 — Continuity Correction Lesson 2983 — The Continuity Correction
P(x, y): is the *horizontal component function* (how much the vector points in the x-direction); Lesson 1964 — Component Functions of Vector Fields
P(x): by a binomial of the form **(x - c)**, the **remainder** you get is exactly equal to **P(c)**—the value of the polynomial when you substitute **c** for **x**.; Lesson 351 — The Remainder Theorem Lesson 2479 — Predicates and Propositional Functions
P(Z > z): Use the complement rule: P(Z > z) = 1 − P(Z ≤ z); Lesson 2908 — Using the Standard Normal Table
P(Z ≤ z): the probability that a standard normal variable is less than or equal to a specific z-score.; Lesson 2908 — Using the Standard Normal Table
P⁻¹: (**A** - λ**I**)**P**); Lesson 2177 — Similar Matrices and Eigenvalue Invariance Lesson 2189 — What is Diagonalization?Lesson 2192 — Verifying Diagonalization Lesson 2255 — Change of Basis for Linear Transformations
P⁻¹ = P ᵀ: (its inverse equals its transpose), so we get the elegant form:; Lesson 2271 — Orthogonal Diagonalization
P⁻¹AP: λ**I**); Lesson 2177 — Similar Matrices and Eigenvalue Invariance
P⁻¹AP = D: .; Lesson 2191 — Constructing P and D from Eigenvectors
P₀(x) = f(a): just the height at *a*; Lesson 1772 — Taylor Polynomials: Approximating Functions Locally
P₁: to **P₂** is simply:; Lesson 1265 — Velocity and Displacement Vectors
P₁(x): adds the tangent line (slope); Lesson 1772 — Taylor Polynomials: Approximating Functions Locally
P₂: is simply:; Lesson 1265 — Velocity and Displacement Vectors
P² = P: (projecting twice is the same as projecting once); Lesson 2231 — The Projection Matrix
P₂(x): adds curvature (concavity); Lesson 1772 — Taylor Polynomials: Approximating Functions Locally
P₃(x): captures how curvature changes; Lesson 1772 — Taylor Polynomials: Approximating Functions Locally
PA PB: Lesson 848 — Properties of Tangents from an External Point
Padding: Never encrypt raw messages—always use padding schemes (like OAEP) to prevent mathematical attacks.; Lesson 2770 — Security and Practical Considerations
Painting a room: You might need the surface area of four walls but *not* the ceiling or floor.; Lesson 895 — Word Problems Involving Surface Area
Pair up equations: take equations 1 & 2, then equations 1 & 3 (or 2 & 3); Lesson 294 — Solving Three-Variable Systems by Elimination
Paired data: (e.; Lesson 3105 — Wilcoxon Signed-Rank Test
Pairwise Independence: Every *pair* of events is independent.; Lesson 2807 — Pairwise vs. Mutual Independence Lesson 2944 — Pairwise vs Mutual Independence
parabola: is the graph of any quadratic function.; Lesson 422 — Parabola Basics: Shape and Orientation Lesson 441 — Bridges, Cables, and Engineering Applications Lesson 1130 — Geometric Definitions of Conics Lesson 1131 — The General Second-Degree Equation Lesson 1132 — Using the Discriminant to Classify Conics Lesson 1133 — Standard Position and Orientation Lesson 1179 — Eccentricity: Measuring the Shape of Conics Lesson 1182 — Focus-Directrix Definition of Conics
Parabolas: limited on one end, unlimited on the other; Lesson 554 — Range from Graphs Lesson 1135 — Overview of Conic Applications
Parabolas and lines: Typically two intersection points; Lesson 1646 — Finding Intersection Points
Parabolic reflectors: (satellite dishes, headlights, telescope mirrors) exploit the parabola's reflective property: signals from the focus bounce off the curve and travel parallel to the axis, or vice versa.; Lesson 1187 — Real-World Conic Applications Review
Paradoxes: "This statement is false.; Lesson 2468 — Propositions and Truth Values
parallel: because both have slope = 3.; Lesson 265 — Slopes of Parallel Lines Lesson 269 — Determining Parallel, Perpendicular, or Neither Lesson 276 — Graphing Systems with Parallel and Coincident Lines Lesson 290 — Special Cases: No Solution and Infinite Solutions Lesson 715 — Alternate Interior Angles Lesson 1129 — What Are Conic Sections?Lesson 1242 — Equal Vectors and Parallel Vectors Lesson 1486 — Statement and Geometric Interpretation of the MVT (+1 more)
parallel lines: behave exactly like those tracks: they run in the same direction and never intersect.; Lesson 265 — Slopes of Parallel Lines Lesson 271 — Applications of Parallel and Perpendicular Lines Lesson 272 — Systems and Geometric Reasoning with Parallel/Perpendicular Lines Lesson 700 — Intersection of Lines and Planes Lesson 715 — Alternate Interior Angles Lesson 826 — Coordinate Geometry of Quadrilaterals Lesson 915 — Parallel and Perpendicular Lines in Coordinate Geometry
Parallel, evenly spaced lines: suggest constant slope; Lesson 1800 — Level Curves and Contour Maps
parallelogram: is a quadrilateral (four-sided polygon) where both pairs of opposite sides are parallel.; Lesson 816 — Parallelograms: Definition and Properties Lesson 825 — The Quadrilateral Family Tree Lesson 2058 — Geometric Interpretation of Linear Combinations
Parallelogram Law: states that for any vectors **u** and **v**:; Lesson 2208 — Parallelogram Law and Polarization Identity
parallelogram rule: .; Lesson 1211 — Geometric Interpretation of Addition Lesson 2048 — Vector Addition
Parallelograms: (opposite sides equal); Lesson 826 — Coordinate Geometry of Quadrilaterals Lesson 944 — Point Symmetry
parameter: (like "walk t meters from the corner") to describe all possible positions.; Lesson 1315 — Systems with Infinitely Many Solutions Lesson 2998 — What is Point Estimation?
Parameter p: The directed distance from the vertex to the focus; Lesson 1146 — Standard Form: Vertex at Origin, Horizontal Axis
Parametric assumptions are violated: (heteroscedasticity, non-independence patterns in residuals); Lesson 3103 — Introduction to Nonparametric Methods
Parametric solution: Lesson 2112 — Free Variables and Parametric Solutions
Parametrize: the curve appropriately; Lesson 1982 — Practice with Line Integrals in 2D and 3D Lesson 2020 — Flux Through a Surface: Definition and Setup
Parametrize each segment: Use separate parametric equations for each smooth piece; Lesson 1979 — Line Integrals Over Piecewise Smooth Curves
Parametrize the curve: C using a vector function **r**(t) for t ∈ [a, b]; Lesson 1977 — Computing Vector Line Integrals via Parametrization
Parametrize your surface: Express it as **r**(u, v) = 〈 x(u,v), y(u,v), z(u,v)〉; Lesson 2018 — Surface Integrals of Scalar Functions
Parentheses: first (if any); Lesson 81 — Operations with Multiple Fractions Lesson 178 — Reading and Interpreting Algebraic Notation Lesson 513 — Mixed Operations with Radicals
Parentheses ( ): , **brackets [ ]**, and **braces { }** all do the same job: they create a "do me first" zone.; Lesson 32 — Parentheses and Grouping Symbols
Parentheses are your friend: When substituting, always put the input value in parentheses, especially with negatives.; Lesson 540 — Evaluating Functions with Numeric Inputs
Parentheses first: `4 - 1 = 3`; Lesson 183 — Order of Operations in Evaluation
Parentheses matter: When substituting a fraction, wrap it in parentheses, especially with exponents or negatives; Lesson 185 — Evaluating Expressions with Fractions
Parentheses/Brackets first: – Handle anything inside grouping symbols; Lesson 47 — Mixed Operations with Negative Numbers
Part 1: Integration followed by differentiation gets you back where you started:; Lesson 1577 — Connecting Both Parts of the FTC
Part 2: Differentiation followed by integration (over an interval) also recovers the original:; Lesson 1577 — Connecting Both Parts of the FTC
part-to-part: relationships.; Lesson 105 — What is a Ratio?Lesson 108 — Part-to-Part vs Part-to-Whole Ratios
Part-to-Part Ratios: compare one part of a group directly to another part of the same group.; Lesson 108 — Part-to-Part vs Part-to-Whole Ratios
Part-to-Whole: Red cars to all cars = **8 : 20** (or simplified: **2 : 5**); Lesson 108 — Part-to-Part vs Part-to-Whole Ratios
Part-to-Whole Ratios: compare one part of a group to the entire group.; Lesson 108 — Part-to-Part vs Part-to-Whole Ratios
Partial derivatives: formalize this idea.; Lesson 1821 — Definition of Partial Derivatives Lesson 1833 — Equation of a Tangent Plane Lesson 2300 — What is a Differential Equation?Lesson 2346 — Existence and Uniqueness for Second-Order ODEs Lesson 2408 — What is a Partial Differential Equation?
Partial Differential Equations (PDEs): involve **partial derivatives** with respect to **multiple variables**.; Lesson 2300 — What is a Differential Equation?Lesson 2408 — What is a Partial Differential Equation?
Partial fraction decomposition: is like reversing the process of adding fractions.; Lesson 1621 — Introduction to Partial Fraction Decomposition Lesson 2393 — Inverse Laplace Transforms and Partial Fractions Lesson 2645 — Extracting Coefficients and Partial Fractions
partial sums: the sum of just the first *n* terms.; Lesson 679 — Infinite Series: Convergence vs Divergence Lesson 680 — Partial Sums and Series Limits Lesson 1713 — Definition of an Infinite Series
particular solution: is one specific member of that family—obtained by applying **initial conditions** (or boundary conditions) that fix the values of the arbitrary constants.; Lesson 2303 — General vs Particular Solutions Lesson 2304 — Initial Value Problems Lesson 2311 — Initial Value Problems with Separable ODEs Lesson 2324 — Initial Value Problems for Linear ODEs Lesson 2351 — Particular Solutions for Exponential Functions Lesson 2635 — General Solution of Non-Homogeneous Recurrences
partition: of [a, b] using points:; Lesson 1551 — Riemann Sums with Unequal Subintervals Lesson 2552 — Equivalence Classes Lesson 2554 — Partitions of a Set Lesson 2780 — Partitions of the Sample Space
Partitions your objects: cleanly (every object goes into exactly one hole); Lesson 2576 — Constructing Pigeonholes Strategically
Pascal's Triangle: The entry in row *n*, position *k* equals $\binom{n}{k}$; Lesson 2599 — Binomial Coefficients: Definition and Notation
Path: A trail in which *no vertex is repeated* either (except possibly the first and last in a closed path).; Lesson 2657 — Walks, Trails, and Paths
Path probabilities: are found by multiplying along branches from root to leaf; Lesson 2796 — Sequential Experiments and Tree Diagrams
Path testing: still works: try approaching along different curves (straight lines, spirals, etc.; Lesson 1820 — Limits and Continuity in Three or More Variables
Pattern: `(x, y)` becomes `(x, -y)`; Lesson 1001 — Symmetry Properties of the Unit Circle Lesson 1438 — Higher-Order Derivatives of Exponential and Logarithmic Functions
Pattern 1: If you see *f(g(x)) · g'(x)*, let *u = g(x)*.; Lesson 1589 — Practice Problems and Strategy
Pattern 1: Chain reactions: Lesson 712 — Solving Multi-Step Angle Problems
Pattern 2: When an expression appears both inside and outside a function (like *x* in *x·cos(x²)*), try making the repeated expression *u*.; Lesson 1589 — Practice Problems and Strategy
Pattern 3: For products where one factor "almost" differentiates to the other, choose *u* as the more complex expression and check if adjusting constants works.; Lesson 1589 — Practice Problems and Strategy
Pattern 4: If multiple substitutions seem possible, choose the one that simplifies the integrand most dramatically—usually the "deepest" nested function.; Lesson 1589 — Practice Problems and Strategy
Pattern for primes: If *p* is prime, then φ(p) = p − 1, because every smaller positive integer is coprime to a prime.; Lesson 2746 — Euler's Totient Function φ(n)
Pattern identified: Difference of squares → factors as (x − 5)(x + 5); Lesson 391 — Mixed Practice: Identifying Special Forms
Pattern recognition: Immediately shows if lines are parallel (no solution) or coincident (infinite solutions); Lesson 281 — Comparing Graphing and Substitution Methods Lesson 1589 — Practice Problems and Strategy Lesson 1610 — Strategy Selection for Trigonometric Integrals
Patterns or funneling: suggest heteroscedasticity (unequal variances); Lesson 3101 — Checking ANOVA Assumptions
Pause: before multiplying; Lesson 342 — Using Special Products to Simplify Expressions
PDF analogy: Height of a smooth curve tells you *density*—where probability is concentrated; Lesson 2836 — Introduction to Probability Density Functions (PDFs)
PDP ⁻¹: where **D** is diagonal.; Lesson 2190 — The Diagonalization Theorem Lesson 2192 — Verifying Diagonalization
Pearson correlation coefficient: , usually denoted ρ (rho) for populations or *r* for samples, is simply **standardized covariance**.; Lesson 2951 — Correlation Coefficient: Definition
Peel off sec²(x): from the even power of secant; Lesson 1603 — Powers of Tangent and Secant: Even Powers of Secant
PEMDAS: (common in the U.; Lesson 31 — Introduction to Order of Operations (PEMDAS/BODMAS)Lesson 183 — Order of Operations in Evaluation
PEMDAS/BODMAS: (the order of operations you learned earlier) while remembering the sign rules for each operation.; Lesson 47 — Mixed Operations with Negative Numbers
Pendulum swing: The path of a pendulum bob traces a circular arc.; Lesson 1682 — Applications: Arc Length in Physics and Geometry
Percentages: typically range from `0` to `100` (or `0` to `1` as decimals); Lesson 557 — Real-World Context and Restricted Domains
percentile: is the value below which a certain percentage of the data falls.; Lesson 2910 — Finding Percentiles and Critical Values Lesson 2990 — Percentiles and Quartiles
Perfect cubes: 1, 8, 27, 64, 125, 216.; Lesson 501 — Higher-Order Radicals: Cube Roots and Beyond
Perfect fourth powers: 1, 16, 81, 256.; Lesson 501 — Higher-Order Radicals: Cube Roots and Beyond
perfect matching: covers every single vertex in the graph—everyone gets paired.; Lesson 2699 — Matchings in Graphs: Definitions and Basic Properties Lesson 2700 — Hall's Marriage Theorem
perfect square: is a number you get by squaring a whole number.; Lesson 151 — Introduction to Square Roots Lesson 496 — Perfect Squares and Their Square Roots
perfect square factors: (1, 4, 9, 16, 25, 36, 49, 64, 100.; Lesson 154 — The Product Property of Radicals Lesson 782 — Simplifying Radical Answers
Perfect Square Recognition: When you see `√(1 + (dy/dx)²)`, try to express what's under the radical as a perfect square.; Lesson 1680 — Simplifying Arc Length Integrals
perfect square trinomial: is what you get when you square a binomial.; Lesson 384 — Perfect Square Trinomials: Recognition Lesson 399 — Solving Special Forms: Perfect Square Trinomials Lesson 401 — Perfect Square Trinomials Review
perfect squares: (and positive); Lesson 337 — Perfect Square Trinomials Lesson 1677 — Arc Length with Radical and Rational Functions
Perform fraction arithmetic: carefully at each step; Lesson 185 — Evaluating Expressions with Fractions
Perimeter: is the distance around the outside of a shape.; Lesson 193 — Perimeter and Area Word Problems Lesson 760 — Perimeter and Area Ratios of Similar Triangles Lesson 856 — Perimeter of Polygons
Perimeter constraint: 2L + 2W = P (for a rectangle); Lesson 435 — Area and Perimeter Optimization Problems
Perimeter of a rectangle: P = 2L + 2W; Lesson 1478 — Related Rates with Rectangles and Squares
period: of the function.; Lesson 570 — Periodic Functions Lesson 1014 — Period of Sine and Cosine Functions Lesson 1018 — Vertical Shift: y = sin(x) + D and the Midline Lesson 1019 — Combined Transformations: y = A·sin(B(x - C)) + D Lesson 1024 — Transformations of Tangent: Period and Vertical Stretch Lesson 1026 — Graphing the Cotangent Function Lesson 1028 — Graphing the Secant Function Lesson 1030 — Graphing the Cosecant Function (+6 more)
Period (2π/B): How long one complete cycle takes (e.; Lesson 1022 — Applications: Modeling Periodic Phenomena
Period (from B): How long one complete cycle takes; Lesson 1042 — Applications: Modeling Periodic Phenomena
Period = 2π/B: One complete cycle takes this many x-units; Lesson 1019 — Combined Transformations: y = A·sin(B(x - C)) + D
Period = π/B: Lesson 1024 — Transformations of Tangent: Period and Vertical Stretch
Period Change (B): The standard cosecant has period 2π (same as sine).; Lesson 1031 — Transformations of Cosecant
periodic function: is a function that repeats its output values over and over again at regular intervals as you move along the x-axis.; Lesson 570 — Periodic Functions Lesson 2456 — Periodic Functions and Fourier's Idea
permutation: is an arrangement of objects in a specific order.; Lesson 2578 — What is a Permutation?Lesson 2588 — Introduction to Combinations
Permutations: Use when the **order matters**.; Lesson 2594 — Distinguishing When to Use Combinations vs Permutations
Permutations (order matters): AB, BA, AC, CA, BC, CB → 6 arrangements; Lesson 2588 — Introduction to Combinations
Permute the unrestricted objects: in the remaining positions using standard permutation formulas; Lesson 2582 — Permutations with Restrictions: Fixed Positions
perpendicular: (they intersect at a 90° angle), their slopes have a special mathematical relationship: they are **negative reciprocals** of each other.; Lesson 267 — Slopes of Perpendicular Lines Lesson 269 — Determining Parallel, Perpendicular, or Neither Lesson 711 — Perpendicular Lines and Right Angles Lesson 927 — Reflections Across Arbitrary Lines Lesson 1129 — What Are Conic Sections?Lesson 1143 — Tangent Lines to Circles Lesson 1670 — Choosing Between Shell and Washer Methods Lesson 1846 — Maximum and Minimum Rates of Change
Perpendicular Diagonals: Lesson 824 — Kites: Properties and Diagonal Relationships
Perpendicular lines: have slopes that are negative reciprocals (their product is -1) and meet at right angles.; Lesson 271 — Applications of Parallel and Perpendicular Lines Lesson 272 — Systems and Geometric Reasoning with Parallel/Perpendicular Lines Lesson 826 — Coordinate Geometry of Quadrilaterals Lesson 915 — Parallel and Perpendicular Lines in Coordinate Geometry
Perpendicular property: Tangent ⊥ radius at point of contact; Lesson 847 — Tangent Lines to Circles
Perpendicular to level curves: Gradient arrows always cross contour lines at right angles; Lesson 1850 — Gradient Fields and Visualization
Perturbation methods: Approximate solutions when problems are "nearly separable"; Lesson 2455 — Limitations and Extensions of the Method
PF/PD = e: Lesson 1182 — Focus-Directrix Definition of Conics
Phase angle: $\delta = \arctan(c_2/c_1)$ shifts the wave; Lesson 2369 — Free Undamped Harmonic Motion
phase shift: .; Lesson 1017 — Phase Shift: Horizontal Translation y = sin(x - C)Lesson 1019 — Combined Transformations: y = A·sin(B(x - C)) + D Lesson 1020 — Comparing Sine and Cosine: The Phase Relationship Lesson 1037 — Phase Shift: Horizontal Translation
Phase Shift (C): When the cycle starts (e.; Lesson 1022 — Applications: Modeling Periodic Phenomena Lesson 1031 — Transformations of Cosecant Lesson 1042 — Applications: Modeling Periodic Phenomena
Phase Shift = C: The entire graph slides C units right (positive C) or left (negative C); Lesson 1019 — Combined Transformations: y = A·sin(B(x - C)) + D
Phone calls: to a call center; Lesson 2893 — Applications of the Poisson Distribution
Physical dimensions: (length, height, volume) must be **positive**; Lesson 557 — Real-World Context and Restricted Domains
Physical Interpretation: Think of steady-state heat distribution.; Lesson 2444 — Mean Value Property and Applications
Physical intuition: Drop ink in water—the way it gradually disperses follows the heat equation.; Lesson 2410 — The Three Classical PDEs
Physical law: Heat flows from hot to cold at a rate proportional to the temperature gradient.; Lesson 2415 — Physical Interpretation and Modeling
Physical or Geometric Reasoning: Lesson 1880 — Identifying Maxima and Minima from Critical Points
Physics: Decomposing motion equations into simpler components; Lesson 354 — Applications of Polynomial Division Lesson 1962 — Radius of Gyration and Applications
Physics and Engineering: The area between velocity curves of two objects shows the relative distance traveled.; Lesson 1652 — Applied Problems: Area Between Curves
Pick a nearby point: `a` where f(a) and f'(a) are easy to compute; Lesson 1470 — Estimating Function Values with Linear Approximation
Pick two points: on the line that land on grid intersections (these are easiest to work with); Lesson 248 — Finding Slope from a Graph
Pick your point: where you want the instantaneous rate (call it x = a); Lesson 1374 — Estimating Instantaneous Rates Numerically
pie charts: when emphasizing proportions of a whole, especially with fewer categories (2–5 work best).; Lesson 2995 — Categorical Data: Bar Charts and Pie Charts Lesson 2997 — Choosing Appropriate Summaries and Displays
Piecewise continuous: f has at most finitely many discontinuities; Lesson 2460 — Convergence of Fourier Series
Pigeonhole Principle: is deceptively simple: if you have **n + 1 or more items** to distribute among **n containers** (pigeonholes), then at least one container must contain **two or more items**.; Lesson 2570 — The Pigeonhole Principle: Statement and Intuition
pivot: ) is to the right of the leading entry in the row above it; Lesson 1311 — Row Reduction to Echelon Form Lesson 2109 — Pivot Positions and Leading Entries Lesson 2116 — Row Echelon Form: Definition and Properties Lesson 2117 — Leading Entries and Pivot Positions
pivot position: .; Lesson 2109 — Pivot Positions and Leading Entries Lesson 2117 — Leading Entries and Pivot Positions
pivot positions: and their **leading entries**.; Lesson 2109 — Pivot Positions and Leading Entries Lesson 2126 — Definition of Matrix Rank
Pivot variables: correspond to columns that contain a leading 1 (a pivot position).; Lesson 2123 — Pivot and Free Variables from RREF
Place the decimal point: in your answer space directly above it; Lesson 90 — Dividing Decimals by Whole Numbers
Place the restricted objects: in their required positions (there's only 1 way to do this per restriction); Lesson 2582 — Permutations with Restrictions: Fixed Positions
Placeholder terms: If your dividend is missing a degree (like going from x³ straight to x with no x² term), write *0x²* as a placeholder; Lesson 346 — Dividing by Binomials Using Long Division
planar graph: is a graph that can be drawn in the plane (on a flat surface like paper) such that no two edges cross each other except at their endpoints (vertices).; Lesson 2689 — Planar Graphs and the Definition of Planarity Lesson 2693 — K₅ and K₃,₃ are Non-Planar
plane: is a perfectly flat, two-dimensional surface that extends infinitely in all directions.; Lesson 699 — Planes: Flat Surfaces Extending Infinitely Lesson 1207 — Complex Numbers as Points in the Plane Lesson 1807 — Common Surfaces: Planes, Paraboloids, and Saddles Lesson 2059 — The Span of a Set of Vectors Lesson 2062 — Geometric Span in R³: Lines, Planes, and All of R³
Planes through the origin: the span of two linearly independent vectors; Lesson 2080 — Subspaces of R^2 and R^3
Planes through the z-axis: The equation **θ = c** describes a half-plane extending from the z-axis at a fixed angle in the xy- plane, like a slice of pie standing upright.; Lesson 1939 — Graphing Surfaces in Spherical Coordinates
Planetary orbits: follow elliptical paths with the sun at one focus, as Kepler discovered—a natural consequence of gravitational physics.; Lesson 1187 — Real-World Conic Applications Review
Plot the center: (h, k) as a point on your coordinate plane; Lesson 1137 — Graphing Circles from Standard Form Lesson 1171 — Graphing Hyperbolas Centered at the Origin
Plot the point: (x,y,z) in 3D space; Lesson 1806 — Graphing Surfaces in 3D
Plug in known values: and solve for the unknown rate; Lesson 1482 — Distance and Position Related Rates Lesson 1484 — Water Tank and Fluid Flow Problems
Plug into the formula: Lesson 414 — Solving Quadratics When a ≠ 1 Lesson 1399 — The Product Rule Formula Lesson 1470 — Estimating Function Values with Linear Approximation
plus sign: (+) means "combine" or "put together"; Lesson 9 — Understanding Addition as Combining Lesson 386 — Sum of Squares (Why It Doesn't Factor)Lesson 1069 — Half Angle Formula for Cosine
PMF analogy: Height of bars tells you exact probabilities; Lesson 2836 — Introduction to Probability Density Functions (PDFs)
Point A: or simply **A**; Lesson 695 — Points: The Building Blocks of Geometry
point estimate: ) from sample data to approximate an unknown population parameter.; Lesson 2998 — What is Point Estimation?Lesson 3012 — Margin of Error and Interval Width
Point estimation: is the process of calculating a single number (called a **point estimate**) from sample data to approximate an unknown population parameter.; Lesson 2998 — What is Point Estimation?
point of tangency: or **point of contact**.; Lesson 847 — Tangent Lines to Circles Lesson 849 — Tangent-Radius Perpendicularity Theorem
Point symmetry: is when a figure looks exactly the same after being rotated 180° around a central point.; Lesson 944 — Point Symmetry
Point-slope: `y - y₁ = m(x - x₁)`; Lesson 261 — Converting Between Forms of Linear Equations
point-slope form: shines.; Lesson 257 — Point-Slope Form: y - y₁ = m(x - x₁)Lesson 258 — Writing Equations from a Point and Slope Lesson 264 — Choosing the Best Form for a Problem Lesson 266 — Writing Equations of Parallel Lines Lesson 912 — Point-Slope Form and Writing Line Equations Lesson 1466 — Finding Equations of Tangent Lines Using the Derivative
Point: (√2/2, -√2/2): Lesson 997 — Fourth Quadrant Points
Pointwise convergence: asks: at each individual point x, does the partial sum approach f(x)?; Lesson 2460 — Convergence of Fourier Series
Poisson distribution: Counts the *number of events* occurring in a fixed time interval; Lesson 2902 — Relationship Between Exponential and Poisson
Poisson(λ): The classic rare-event distribution:; Lesson 2962 — MGFs of Common Distributions
Poissons: is Poisson; Lesson 2957 — Sums of Independent Random Variables
Polar: Simply `0 ≤ r ≤ a` and `0 ≤ θ ≤ 2π` (clean and direct); Lesson 1913 — Why Use Polar Coordinates for Integration Lesson 2454 — Separation in Other Coordinate Systems
polar coordinates: for circular laminas (then `x² + y² = r²`).; Lesson 1961 — Computing Moments of Inertia Lesson 2433 — The Two-Dimensional Wave Equation
Polar direction (φ): Moving through angle `dφ` traces an arc of length `ρ dφ`; Lesson 1940 — Volume Elements in Spherical Coordinates
Polar/spherical coordinates: can simplify radial symmetry; Lesson 1820 — Limits and Continuity in Three or More Variables
Polarization Identity: goes the other way—it lets you *recover* the inner product from just the norm:; Lesson 2208 — Parallelogram Law and Polarization Identity
polynomial: is a mathematical expression made up of one or more **terms** added (or subtracted) together.; Lesson 310 — What is a Polynomial?Lesson 311 — Identifying Polynomials vs Non-Polynomials Lesson 316 — Classifying Polynomials by Number of Terms Lesson 318 — What is a Polynomial? Terms and Like Terms Lesson 325 — Multiplying a Monomial by a Polynomial
Polynomial division: (using long division or synthetic division); Lesson 351 — The Remainder Theorem
Polynomial expansion: The coefficient of $x^k y^{n-k}$ in $(x+y)^n$ is $\binom{n}{k}$; Lesson 2599 — Binomial Coefficients: Definition and Notation
Polynomial forcing: If **f**(t) = [t², 3t]ᵀ, guess **x_p** = **a**t² + **b**t + **c** (vectors of constants).; Lesson 2386 — Non-Homogeneous Systems: Undetermined Coefficients
polynomial functions: completely.; Lesson 1336 — The Power and Root Laws Lesson 1818 — Continuity on a Region
Polynomial inside a power: (2x - 5)³ → u = 2x - 5; Lesson 1580 — Identifying the Inner Function u
polynomial long division: to divide the numerator by the denominator.; Lesson 490 — Slant (Oblique) Asymptotes Lesson 1524 — Slant (Oblique) Asymptotes Lesson 1622 — Proper vs Improper Rational Functions
Polynomial resonance: If your trial is a polynomial and constants (or polynomials) are in the homogeneous solution, multiply by *x*.; Lesson 2354 — The Modification Rule for Resonance
Polynomial Spaces: P_n: Lesson 2076 — Examples of Vector Spaces
Polynomials: Functions like f(x) = 3x² - 5x + 7 are continuous *everywhere* (for all real numbers).; Lesson 1354 — Continuity of Elementary Functions Lesson 1813 — Computing Limits Using Direct Substitution
Polynomials become negligible: High powers of n in the ratio often simplify or vanish as n → ∞.; Lesson 1743 — Applying the Ratio Test to Common Series
Pool and rank: Combine all observations from all groups and assign ranks (1 = smallest, n = largest).; Lesson 3107 — Kruskal-Wallis Test
Population Growth: A bacteria colony doubles every hour.; Lesson 666 — Growth and Decay Applications Lesson 1376 — Applications: Velocity and Other Rates Lesson 1437 — Applications: Exponential Growth and Decay Rates Lesson 1504 — Applications: Real-World Monotonicity and Concavity Lesson 2374 — Exponential Growth and Decay Models
Population model: zeros = when population reaches zero (extinction); Lesson 577 — Zeros and Intercepts
Population standard deviation (σ): Often estimated from prior studies or a pilot sample; Lesson 3013 — Choosing Sample Size for Desired Margin of Error
Population variability (σ): More spread in your population means more uncertainty, thus wider intervals.; Lesson 3012 — Margin of Error and Interval Width
Population variance known: (rare): Use z-tests; Lesson 3032 — Why t-Tests? When the Population Variance is Unknown
Population variance unknown: Use t-tests; Lesson 3032 — Why t-Tests? When the Population Variance is Unknown
Portfolio optimization: Measures risk correlations between assets; Lesson 2299 — Covariance Matrices and Applications
Position: – where the figure sits in the plane; Lesson 921 — Introduction to Geometric Transformations Lesson 1428 — Applications: Rates of Change in Oscillatory Motion Lesson 1545 — Applications: Position from Velocity, Velocity from Acceleration Lesson 2994 — Comparing Distributions
positive: .; Lesson 45 — Multiplying Positive and Negative Numbers Lesson 138 — Exponents with Negative Bases Lesson 218 — Solving One-Step Inequalities Lesson 246 — Positive, Negative, Zero, and Undefined Slopes Lesson 250 — Slope and Direction of Lines Lesson 333 — Multiplying Polynomials with Negative Coefficients Lesson 362 — Factoring Trinomials with Leading Coefficient 1 Lesson 366 — Factoring Trinomials with Negative Coefficients (+12 more)
Positive (counterclockwise) orientation: means:; Lesson 1995 — Orientation and Simple Closed Curves
Positive angle: rotate counterclockwise; Lesson 999 — Negative Angles and Clockwise Rotation
Positive case: x = 7; Lesson 227 — Solving Basic Absolute Value Equations
Positive covariance: X and Y tend to increase together; Lesson 2947 — Covariance: Definition and Interpretation
positive definite: if **x**ᵀA**x** > 0 for *all* nonzero vectors **x**.; Lesson 2292 — Positive Definite, Negative Definite, and Indefinite Matrices Lesson 2293 — Eigenvalue Criterion for Definiteness Lesson 2294 — Principal Minors and Sylvester's Criterion
positive diagonal entries: , then QR decomposition is **unique**.; Lesson 2244 — Uniqueness and Existence of QR Decomposition Lesson 2295 — The Cholesky Decomposition
Positive divergence (div: F** > 0):** The field is **spreading out** — more fluid is leaving than arriving.; Lesson 2006 — Interpreting Divergence Physically
Positive exponents: = multiply the base by itself; Lesson 135 — Negative Integer Exponents Lesson 141 — Mixed Practice with Integer Exponents
Positive flux: field flows *out* through the surface (in the direction of **n**); Lesson 2019 — Vector Fields and Surface Integrals
Positive integer exponents: Multiply the base by itself that many times; Lesson 601 — Evaluating Exponential Expressions
Positive numbers: (1, 2, 3, .; Lesson 40 — Representing Negative Numbers on the Number Line Lesson 43 — Adding Positive and Negative Numbers
Positive orientation: by convention, often the "outward" normal (pointing away from enclosed volume); Lesson 2017 — Orientation of Surfaces Lesson 2027 — Orientation of Surfaces and Boundary Curves Lesson 2029 — Parametrizing Boundary Curves
Positive Predictive Value (PPV): P(Disease Present | Positive Test) — what we actually want to know!; Lesson 2818 — Medical Testing and Screening Problems
Positive real parts: solutions grow → **unstable**; Lesson 2278 — Applications to Differential Equations and Dynamics
Positive residual: Your model *under*-predicted (actual value was higher than predicted); Lesson 3083 — What Are Residuals?
Positive result: Lesson 138 — Exponents with Negative Bases
Positive Semidefinite: All eigenvalues are **non-negative** (λ ≥ 0), at least one is zero; Lesson 2293 — Eigenvalue Criterion for Definiteness
positive semidefinite matrix: is a symmetric matrix **A** where the quadratic form **x^T A x ≥ 0** for *all* vectors **x**.; Lesson 2297 — Positive Semidefinite Matrices Lesson 2299 — Covariance Matrices and Applications
Positive slope: Line goes uphill from left to right (rise is positive); Lesson 244 — Rise Over Run: Understanding Slope Geometrically Lesson 246 — Positive, Negative, Zero, and Undefined Slopes
Positive terms: Each b > 0; Lesson 1753 — Verifying Conditions for the Alternating Series Test
positive to negative: , the function was rising then started falling — you've found a **local maximum** (a hilltop); Lesson 1496 — The First Derivative Test for Local Extrema Lesson 1502 — Comparing First and Second Derivative Tests
Positive λ: Both gradients point in the same general direction (local maximum along the constraint); Lesson 1876 — The Lagrange Multiplier λ
Positivity: Lesson 2200 — Definition of Inner Product Spaces Lesson 2202 — Inner Products on Function Spaces
Post-hoc power analysis: tells you: "Given the effect size we observed, what was our actual power?; Lesson 3060 — Power Analysis and Study Design
Posterior: Your updated belief after combining the prior with the likelihood.; Lesson 3112 — Prior, Likelihood, and Posterior: Bayes' Theorem Revisited Lesson 3114 — Conjugate Priors: Beta-Binomial Model Lesson 3115 — Conjugate Priors: Normal-Normal Model
posterior distribution: (after observing data) is also normal.; Lesson 3115 — Conjugate Priors: Normal-Normal Model Lesson 3120 — Predictive Distributions: Prior and Posterior Predictive
posterior probability: ).; Lesson 2817 — Prior and Posterior Probabilities Lesson 2820 — Sequential Updating with Bayes' Theorem
Postulate 5: Line-Plane Intersection: Lesson 701 — Postulates of Points, Lines, and Planes
Postulates: Lesson 766 — Common Reasons in Proofs
Potential Energy: often requires integration too.; Lesson 1702 — Other Physical Applications: Energy and Probability Lesson 2432 — Energy Conservation in Wave Motion
Power of a Power: – Simplify any exponents raised to exponents first; Lesson 149 — Combining Multiple Exponent Rules
Power of a Product: `(xy)^(a/b) = x^(a/b) · y^(a/b)`; Lesson 533 — Simplifying Expressions with Fractional Exponents
Power of a Product/Quotient: – Distribute exponents across multiplication or division inside parentheses; Lesson 149 — Combining Multiple Exponent Rules
Power Rule: `(x^(a/b))^(c/d) = x^((a/b)·(c/d))`; Lesson 533 — Simplifying Expressions with Fractional Exponents Lesson 624 — Combining Logarithm Properties Lesson 631 — Using Logarithms to Solve Exponential Equations Lesson 636 — Solving Logarithmic Equations with Multiple Logs Lesson 1389 — The Power Rule for Positive Integer Exponents Lesson 1412 — Chain Rule with Radicals Lesson 1464 — Combining Logarithm Properties Before Differentiating Lesson 1708 — Limit Laws for Sequences
Power Rule for Antiderivatives: reverses this process.; Lesson 1538 — Power Rule for Antiderivatives
power series: is an infinite series where each term contains a power of a variable (usually *x*).; Lesson 1761 — What is a Power Series?Lesson 1768 — Integration of Power Series Lesson 2399 — Power Series Solutions: Introduction and Ordinary Points
power set: of a set is the collection of *all possible subsets* of that set.; Lesson 2531 — Introduction to Power Sets Lesson 2532 — Computing Power Sets
Powers: If *a ≡ b (mod n)*, then *a^k ≡ b^k (mod n)* for any positive integer *k*; Lesson 2739 — Properties of Congruence Relations
Powers and factorials mixed: When you see \(n^n\) or expressions where the exponent depends on n, Root Test often untangles the complexity better than Ratio Test.; Lesson 1746 — Applying the Root Test
Practical power: Orthogonality underlies decompositions, projections, and many algorithms you'll encounter later; Lesson 2206 — Angle Between Vectors and Orthogonality
Practical significance: asks: *How large is the effect, and does it matter?; Lesson 3041 — Effect Size and Practical Significance in t-Tests
Practical workflow: Lesson 1791 — Error Analysis in Numerical Computation
Practically: , neither makes sense:; Lesson 557 — Real-World Context and Restricted Domains
Practice the reverse thinking: When you see 45 ÷ 9, ask yourself "9 times what equals 45?; Lesson 25 — Division Facts and Basic Division Fluency
Precise: (efficiency means tightest confidence intervals); Lesson 3007 — Properties and Advantages of MLEs
Precision of our estimate: (width of the interval—narrower is more precise); Lesson 3009 — Introduction to Confidence Intervals
Prevalence: P(Disease Present) — the prior probability, how common the disease is in the population; Lesson 2818 — Medical Testing and Screening Problems Lesson 2819 — False Positives and Base Rate Fallacy
Price: $3 per pound (dollars ÷ pounds); Lesson 109 — Introduction to Rates
prime: they cannot be factored into simpler polynomial factors using integers.; Lesson 370 — Prime Trinomials and When Factoring is Not Possible Lesson 373 — Applications and Strategy Selection for Trinomial Factoring Lesson 386 — Sum of Squares (Why It Doesn't Factor)Lesson 2745 — Modular Inverses via Fermat's Little Theorem Lesson 2751 — Problem Solving with Fermat's and Euler's Theorems
Prime factorization: is the process of finding which prime numbers multiply together to make your original number.; Lesson 57 — Prime Factorization Lesson 155 — Simplifying Square Roots Lesson 2725 — Prime Factorization: Finding the Decomposition
prime number: is a whole number greater than 1 that has exactly two factors: **1 and itself**.; Lesson 55 — Introduction to Prime Numbers Lesson 2720 — Prime Numbers: Definition and Basic Properties
Prime numbers: have *exactly two factors*: 1 and themselves (like 2, 3, 5, 7, 11…).; Lesson 56 — Composite Numbers
principal argument: is usually taken in the range **(-π, π]** or **[0, 2π)**; Lesson 1210 — Argument of a Complex Number Lesson 1218 — Principal Argument and Angle Conventions
Principal Component Analysis (PCA): Finds directions of maximum variance by diagonalizing the covariance matrix; Lesson 2299 — Covariance Matrices and Applications
principal square root: .; Lesson 151 — Introduction to Square Roots Lesson 495 — Understanding Radicals and the Square Root Symbol
Prior: Your initial belief about a parameter before seeing data.; Lesson 3112 — Prior, Likelihood, and Posterior: Bayes' Theorem Revisited Lesson 3114 — Conjugate Priors: Beta-Binomial Model Lesson 3115 — Conjugate Priors: Normal-Normal Model
Prior elicitation: is the process of translating expert knowledge, historical data, or domain information into a mathematical prior distribution.; Lesson 3118 — Prior Elicitation and Informative vs. Noninformative Priors
prior probability: ), and then observe new evidence, you can use Bayes' Theorem to calculate an updated, improved belief (your **posterior probability**).; Lesson 2817 — Prior and Posterior Probabilities Lesson 2820 — Sequential Updating with Bayes' Theorem
private key: ); Lesson 2761 — Public-Key Cryptography Concept Lesson 2769 — RSA Example with Small Primes
Pro tip: You can use *either* point—you'll get the same final equation both ways!; Lesson 914 — Equations from Two Points Lesson 973 — Combined Problems with Distance and Height Lesson 1038 — Determining Phase Shift from Equations Lesson 1482 — Distance and Position Related Rates Lesson 2087 — Verifying a Set is a Basis Lesson 2178 — Computing Eigenvalues for 3×3 Matrices
Probability Connection: Notice that E[I_A] = 1·P(A) + 0·P(A^c) = P(A).; Lesson 2830 — Indicator Random Variables
probability density function (PDF): f(x) describes the likelihood of outcomes.; Lesson 1702 — Other Physical Applications: Energy and Probability Lesson 2836 — Introduction to Probability Density Functions (PDFs)Lesson 2837 — Properties of Probability Density Functions Lesson 2839 — Graphical Interpretation of PDFs Lesson 2855 — Definition of Expectation for Continuous Random Variables Lesson 2895 — PDF and CDF of the Uniform Distribution
Probability Mass Function (PMF): is a function that tells you the probability that a discrete random variable equals each specific value in its range.; Lesson 2832 — Introduction to Probability Mass Functions (PMFs)Lesson 2833 — Properties of Probability Mass Functions
Problem: 23 students need to form teams of 4.; Lesson 28 — Division with Remainders Lesson 117 — Unit Rates in Proportions Lesson 189 — Writing Equations from Word Problems Lesson 655 — Finding the First Term from Given Information Lesson 1878 — Solving Two-Variable Constrained Problems Lesson 2575 — Pigeonhole Principle in Geometry Lesson 2582 — Permutations with Restrictions: Fixed Positions Lesson 2593 — Combinations in Real-World Problems (+4 more)
Process: Lesson 675 — Nested Summations and Double Sums
Product: If f(x) and g(x) are continuous at x = a, then f(x) · g(x) is continuous at a.; Lesson 1355 — Continuity of Combinations of Functions Lesson 1436 — Combining Product, Quotient, and Exponential Rules Lesson 1819 — Properties and Composition of Continuous Functions Lesson 2642 — Using Generating Functions to Count Distributions
Product Property: (√(a · b) = √a · √b), radicals have a similar rule for division:; Lesson 500 — Quotient Property of Radicals Lesson 504 — Combining Simplification Techniques
Product Property of Radicals: states that the square root of a product equals the product of the square roots:; Lesson 154 — The Product Property of Radicals Lesson 155 — Simplifying Square Roots Lesson 159 — Multiplying Radical Expressions Lesson 497 — Product Property of Radicals Lesson 499 — Simplifying Square Roots with Variables Lesson 509 — Multiplying Radicals Using the Product Property
Product Rule: `x^(a/b) · x^(c/d) = x^(a/b + c/d)`; Lesson 533 — Simplifying Expressions with Fractional Exponents Lesson 534 — Multiplying and Dividing with Fractional Exponents Lesson 621 — The Product Rule for Logarithms Lesson 624 — Combining Logarithm Properties Lesson 636 — Solving Logarithmic Equations with Multiple Logs Lesson 1399 — The Product Rule Formula Lesson 1400 — Applying the Product Rule to Polynomials Lesson 1405 — Choosing Between Product and Quotient Rules (+4 more)
Product rule first: `dy/dx = [(3x + 1)⁴]' · (2x² - 5)³ + (3x + 1)⁴ · [(2x² - 5)³]'`; Lesson 1415 — Chain Rule with Product and Quotient Rules
Product rule test: Does P(A ∩ B) = P(A) · P(B)?; Lesson 2804 — Testing for Independence
Product/Quotient Rules: – Combine bases that are multiplying or dividing; Lesson 149 — Combining Multiple Exponent Rules
Products: that telescope nicely when you form the ratio a_{n+1}/a_n; Lesson 1747 — Comparing Ratio and Root Tests
Products with Different Arguments: Lesson 1610 — Strategy Selection for Trigonometric Integrals
Professional mathematics: – Published solutions and formal proofs use rationalized forms as standard notation.; Lesson 521 — Applications and When Not to Rationalize
Profit: `P(x) = R(x) - C(x)` where `C(x)` is your costs; Lesson 436 — Revenue and Profit Maximization
Profit function: x-intercept = break-even point (profit = $0); Lesson 577 — Zeros and Intercepts
Project and reduce: Transform data via **X̃V** and keep only the top *k* columns for dimensionality reduction; Lesson 2289 — Principal Component Analysis (PCA) and SVD
Project any vector: For any vector b, just compute Pb; Lesson 2231 — The Projection Matrix
Project the region: onto the appropriate coordinate plane; Lesson 1928 — Changing the Order of Integration in Triple Integrals
Projectile motion: Solving `h = -16t² + vt + h₀` for time; Lesson 373 — Applications and Strategy Selection for Trinomial Factoring Lesson 400 — Applications and Word Problems Lesson 420 — Applications: Projectile Motion and Area Problems Lesson 1682 — Applications: Arc Length in Physics and Geometry
Proof: Suppose {**v**₁, **v**₂, .; Lesson 2210 — Orthogonal Sets of Vectors Lesson 2490 — What Does It Mean to Prove Something?Lesson 2493 — Direct Proof Examples and Practice Lesson 2573 — Pigeonhole Principle in Number Theory Lesson 2575 — Pigeonhole Principle in Geometry Lesson 2749 — Proof of Euler's Theorem Lesson 2803 — Independence vs. Disjoint Events
Proof by Contradiction: Lesson 767 — Proof by Contradiction Lesson 2497 — Proof by Contradiction: Basic Idea Lesson 2722 — Infinitude of Primes: Euclid's Proof Lesson 2726 — Uniqueness of Prime Factorization: The Proof Lesson 3024 — The Logic of Hypothesis Testing
Proof sketch: Lesson 2701 — Augmenting Paths and Berge's Theorem
propagated error: .; Lesson 1473 — Applications of Differentials in Measurement Lesson 1839 — Applications to Measurement and Error Propagation
Proper rational function: The degree (highest power) of the numerator is *less than* the degree of the denominator.; Lesson 1622 — Proper vs Improper Rational Functions
Proper Subspaces: Any subspace that's strictly between these two extremes is called a **proper subspace**.; Lesson 2079 — Trivial and Proper Subspaces Lesson 2224 — Gram-Schmidt for Subspaces
Properties: Lesson 766 — Common Reasons in Proofs
Property: When you multiply any matrix **A** by the identity matrix (of compatible dimensions), you get **A** back unchanged:; Lesson 2103 — Special Matrices: Identity and Zero Lesson 2671 — Binary Trees and Complete Trees
proportion: is a special kind of equation that states two ratios are equal to each other.; Lesson 114 — What is a Proportion?Lesson 120 — Maps and Scale Drawings Lesson 215 — Ratios and Proportions Leading to Linear Equations
Prove: Lesson 762 — Introduction to Geometric Proof Lesson 767 — Proof by Contradiction Lesson 768 — Direct Proof Techniques Lesson 770 — Proving Theorems About Lines and Angles Lesson 771 — Proving Properties of Parallel Lines Lesson 1790 — Applications to Common Functions Lesson 2491 — Direct Proof: Basic Structure Lesson 2493 — Direct Proof Examples and Practice
Prove triangle congruence: Use SSS, SAS, ASA, AAS, or HL; Lesson 747 — CPCTC: Corresponding Parts of Congruent Triangles
Proving Irrationality: Lesson 2498 — Proof by Contradiction: Common Patterns
Proving non-planarity: If V - E + F ≠ 2, the graph cannot be planar; Lesson 2691 — Euler's Formula for Planar Graphs
Proving Nonexistence: Lesson 2498 — Proof by Contradiction: Common Patterns
Proving Uniqueness: Lesson 2498 — Proof by Contradiction: Common Patterns
Pᵀ = P: (it's symmetric); Lesson 2231 — The Projection Matrix
public exponent: .; Lesson 2764 — Choosing the Public Exponent e Lesson 2766 — RSA Encryption
public key: ); Lesson 2761 — Public-Key Cryptography Concept Lesson 2769 — RSA Example with Small Primes
Pull out constants: using the constant multiple rule; Lesson 1542 — Antiderivatives of Polynomial Functions
PyMC, NIMBLE: Flexible, programmable frameworks; Lesson 3121 — Computational Tools: Introduction to MCMC and Bayesian Software
Pyramid structures: Objects decreasing by 1 per layer from bottom to top; Lesson 693 — Stacking and Construction Problems
Pythagorean identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ; Lesson 1050 — Simplifying Trigonometric Expressions Lesson 1090 — Equations Involving Multiple Functions
Pythagorean Identity: because it comes directly from the Pythagorean Theorem applied to the unit circle.; Lesson 1045 — The Pythagorean Identity Lesson 1440 — Derivative of Arcsine Using Implicit Differentiation
Pythagorean identity for sine: since sin²(θ) + cos²(θ) = 1, when you substitute x = a sin(θ), the radical simplifies to a cos(θ).; Lesson 1612 — The Three Standard Forms
Pythagorean theorem: tells us that in a right triangle, a² + b² = c², where c is the hypotenuse.; Lesson 437 — Geometry: Pythagorean Applications Lesson 855 — Applications of Tangents and Chords Lesson 958 — Given Two Sides: Finding the Third Side Lesson 963 — Solving When Given Both Legs Lesson 1479 — Related Rates with Triangles
Pythagorean Theorem in disguise: !; Lesson 903 — Distance Formula in the Coordinate Plane
Pythagorean triple: is a set of three positive integers (whole numbers) that satisfy the Pythagorean Theorem: a² + b² = c².; Lesson 783 — Introduction to Pythagorean Triples
Pythagorean triples: are sets of three whole numbers that satisfy the Pythagorean Theorem.; Lesson 784 — Verifying Pythagorean Triples

Q

Q ᵀ: gives:; Lesson 2228 — Applications: QR Factorization Preview Lesson 2247 — QR and Least Squares Problems
Q-Q plot: ) gives you an immediate visual assessment.; Lesson 3086 — Normal Probability Plots (Q-Q Plots)
Q-Q plots: (normal probability plots) or formal tests like Shapiro-Wilk to check if residuals follow a normal distribution.; Lesson 3101 — Checking ANOVA Assumptions
Q(x, y): is the *vertical component function* (how much the vector points in the y-direction); Lesson 1964 — Component Functions of Vector Fields Lesson 2479 — Predicates and Propositional Functions
Q^T: preserves norms (think of **Q** as a rigid rotation—it doesn't stretch or compress).; Lesson 2237 — Least Squares with QR Decomposition Lesson 2246 — Solving Linear Systems Using QR
Q^T = Q^(-1): .; Lesson 2241 — Orthogonal Matrices and Their Properties
Q^T b: , then solve the triangular system **Rx̂ = Q^T b**.; Lesson 2237 — Least Squares with QR Decomposition Lesson 2246 — Solving Linear Systems Using QR
Q^T Q = I: Lesson 2241 — Orthogonal Matrices and Their Properties Lesson 2246 — Solving Linear Systems Using QR
Q1: first quartile; Lesson 2990 — Percentiles and Quartiles
Q1 (First Quartile): = 25th percentile: 25% of data falls below this value; Lesson 2990 — Percentiles and Quartiles
Q2 (Second Quartile): = 50th percentile: this is the median; Lesson 2990 — Percentiles and Quartiles
Q3: third quartile; Lesson 2990 — Percentiles and Quartiles
Q3 (Third Quartile): = 75th percentile: 75% of data falls below this value; Lesson 2990 — Percentiles and Quartiles
QR: (where **Q** is orthogonal and **R** is upper triangular), two natural questions arise: Does this decomposition always exist?; Lesson 2244 — Uniqueness and Existence of QR Decomposition
QR decomposition: expresses a matrix A as the product of an **orthogonal matrix Q** (whose columns are orthonormal) and an **upper triangular matrix R**.; Lesson 2242 — QR Decomposition via Gram-Schmidt Lesson 2243 — Computing QR: Step-by-Step Algorithm Lesson 2248 — Computational Advantages of QR Decomposition
QR method: condition number stays ≈ κ(A); Lesson 2248 — Computational Advantages of QR Decomposition
QRx = b: .; Lesson 2228 — Applications: QR Factorization Preview Lesson 2246 — Solving Linear Systems Using QR
Quadrant: II (since 90° < 150° < 180°); Lesson 1005 — Evaluating Trig Functions for Angles Greater than 90°Lesson 1210 — Argument of a Complex Number
Quadrant I: Upper-right (where both x and y are positive); Lesson 238 — The Four Quadrants Lesson 1003 — Reference Angles Lesson 1217 — Finding the Argument (Angle)
Quadrant II: Upper-left (where x is negative and y is positive); Lesson 238 — The Four Quadrants Lesson 1003 — Reference Angles Lesson 1005 — Evaluating Trig Functions for Angles Greater than 90°Lesson 1217 — Finding the Argument (Angle)
Quadrant III: Lower-left (where both x and y are negative); Lesson 238 — The Four Quadrants Lesson 1003 — Reference Angles Lesson 1005 — Evaluating Trig Functions for Angles Greater than 90°Lesson 1217 — Finding the Argument (Angle)
Quadrant IV: Lower-right (where x is positive and y is negative); Lesson 238 — The Four Quadrants Lesson 1003 — Reference Angles Lesson 1005 — Evaluating Trig Functions for Angles Greater than 90°Lesson 1217 — Finding the Argument (Angle)
Quadrant rules: Which trig functions are positive or negative in each quadrant; Lesson 1005 — Evaluating Trig Functions for Angles Greater than 90°
Quadrants I and II: (0° to 180°, or 0 to π radians): sine is **positive** → use **+**; Lesson 1068 — Half Angle Formula for Sine
Quadrants III and IV: (180° to 360°, or π to 2π radians): sine is **negative** → use **−**; Lesson 1068 — Half Angle Formula for Sine
quadratic equation: that we must solve.; Lesson 437 — Geometry: Pythagorean Applications Lesson 482 — Rational Equations Leading to Quadratics
Quadratic Family: The parent function is `f(x) = x²`, which creates a U-shaped parabola centered at the origin.; Lesson 588 — Transformations and Function Families
quadratic form: is an expression like Q(**x**) = x₁² + 4x₁x₂ + 3x₂², which involves squared and cross-product terms of variables.; Lesson 2277 — Principal Axes and Quadratic Forms Lesson 2291 — Quadratic Forms and Their Matrix Representation Lesson 2296 — Diagonalization of Quadratic Forms
quadratic formula: !; Lesson 408 — Deriving the Quadratic Formula Lesson 412 — Understanding the Structure of the Quadratic Formula
Quadratic functions: like `f(x) = x²` have a limited range.; Lesson 553 — Finding Range from Equations Lesson 1382 — Computing Derivatives from the Definition: Quadratic Functions
quadrilateral: (4 sides) splits into 2 triangles → (4 - 2) × 180° = **360°**; Lesson 808 — Interior Angle Sum Formula for Polygons Lesson 815 — Defining Quadrilaterals and Basic Properties
Quality control: If two independent machines each work properly with probability 0.; Lesson 2805 — Independence and the Multiplication Rule Lesson 3051 — Applications and Limitations
Quantify error: in practical approximations; Lesson 1790 — Applications to Common Functions
Quantile-Quantile plot: or **Q-Q plot**) gives you an immediate visual assessment.; Lesson 3086 — Normal Probability Plots (Q-Q Plots)
Quantities: (people, objects) must be **whole numbers** and **non-negative**; Lesson 557 — Real-World Context and Restricted Domains
Quantum mechanics: (Schrödinger equation); Lesson 2408 — What is a Partial Differential Equation?
Quartiles: are specific percentiles that divide data into four equal parts:; Lesson 2990 — Percentiles and Quartiles
Quasilinear PDE: Linear in the highest-order derivatives, but lower-order terms or coefficients may depend on u or its lower derivatives.; Lesson 2409 — Classification: Order and Linearity
Questions: "Is x > 0?; Lesson 2468 — Propositions and Truth Values
Quick check: Does `h² + k² - F > 0`?; Lesson 1141 — Finding Circle Properties from General Form
Quick estimates: Great when you need an approximate answer fast; Lesson 281 — Comparing Graphing and Substitution Methods
Quick estimation: If you observe the mean number of occurrences, you immediately know the variance; Lesson 2891 — Mean and Variance of the Poisson Distribution
Quick test: Three or more points are collinear if you can draw exactly one straight line that passes through all of them.; Lesson 698 — Collinear and Coplanar Points
Quick values: D(2) = 1, D(3) = 2, D(4) = 9, D(5) = 44; Lesson 2586 — Derangements
Quick verification: If you graph both a function and its supposed inverse and they're not mirror images across y = x, something went wrong; Lesson 565 — Graphs of Inverse Functions
Quotient: The answer you get from the division; Lesson 344 — Introduction to Polynomial Division Lesson 490 — Slant (Oblique) Asymptotes Lesson 502 — Simplifying nth Roots with Variables Lesson 1355 — Continuity of Combinations of Functions Lesson 1436 — Combining Product, Quotient, and Exponential Rules Lesson 1819 — Properties and Composition of Continuous Functions
quotient identities: reveal a beautiful relationship between the six trigonometric functions.; Lesson 1044 — Quotient Identities Lesson 1050 — Simplifying Trigonometric Expressions Lesson 1090 — Equations Involving Multiple Functions
Quotient Law: Split into numerator limit ÷ denominator limit; Lesson 1338 — Combining Multiple Limit Laws
Quotient property first: √(50x⁵y⁴/2x) = √(25x⁴y⁴); Lesson 504 — Combining Simplification Techniques
Quotient Property of Radicals: tells us that the square root of a fraction equals the square root of the numerator divided by the square root of the denominator:; Lesson 156 — The Quotient Property of Radicals
Quotient Rule: `x^(a/b) ÷ x^(c/d) = x^(a/b - c/d)`; Lesson 533 — Simplifying Expressions with Fractional Exponents Lesson 534 — Multiplying and Dividing with Fractional Exponents Lesson 624 — Combining Logarithm Properties Lesson 636 — Solving Logarithmic Equations with Multiple Logs Lesson 1405 — Choosing Between Product and Quotient Rules Lesson 1423 — Derivatives of cot(x), sec(x), and csc(x)Lesson 1446 — Products and Quotients Involving Inverse Trig Functions Lesson 1453 — Implicit Differentiation with Products and Quotients (+2 more)
Quotient Rule for Exponents: , which says when you divide powers with the same base, you subtract the exponents:; Lesson 147 — Zero Exponent Rule Lesson 148 — Negative Exponent Rule
Quotient Rule for Logarithms: Lesson 622 — The Quotient Rule for Logarithms

R

R = ||u ||: (the normalizing factors); Lesson 2243 — Computing QR: Step-by-Step Algorithm
R = ∞: The series converges for all real numbers x; Lesson 1763 — Radius of Convergence
R = 0: The series only converges at x = c (just the center point); Lesson 1763 — Radius of Convergence
R = 1/ℓ: .; Lesson 1764 — Finding Radius of Convergence Using the Ratio Test
r dr dθ dz: (not just dr dθ dz).; Lesson 1937 — Evaluating Triple Integrals Using Cylindrical Coordinates Lesson 1956 — Computing Total Mass with Triple Integrals
R is NOT reflexive: because 3 doesn't relate to itself.; Lesson 2542 — Reflexive Relations
R is reflexive: Lesson 2542 — Reflexive Relations
R(x): = outer radius = distance from x-axis to outer curve; Lesson 1659 — Washer Method: Rotation About the x-axis
R(y): is the radius function (distance from the y-axis to the curve, expressed in terms of y); Lesson 1655 — Disk Method: Rotation About the y-axis Lesson 1660 — Washer Method: Rotation About the y-axis
R^m: (the output space); Lesson 2132 — The Four Fundamental Subspaces
R^n: consists of all ordered lists of n real numbers.; Lesson 2076 — Examples of Vector Spaces Lesson 2132 — The Four Fundamental Subspaces
R^n: The Classic: Lesson 2076 — Examples of Vector Spaces
r²: .; Lesson 917 — Finding Center and Radius from Circle Equations Lesson 2061 — Geometric Span in R²: Lines and Planes Lesson 2086 — What is a Basis?Lesson 2088 — Standard Bases Lesson 2091 — Dimension of a Vector Space Lesson 2093 — The Basis Theorem Lesson 3065 — The Coefficient of Determination (R²)
R² = 0: The predictor explains *none* of the variance; the regression line does no better than simply using the mean of y.; Lesson 3065 — The Coefficient of Determination (R²)
R² = 0.75: The model explains 75% of the variance in y; 25% remains unexplained (residual variance).; Lesson 3065 — The Coefficient of Determination (R²)
R² = 1: The regression line perfectly predicts every observation.; Lesson 3065 — The Coefficient of Determination (R²)
R³: `{(1,0,0), (0,1,0), (0,0,1)}` is the standard basis.; Lesson 2086 — What is a Basis?Lesson 2088 — Standard Bases Lesson 2091 — Dimension of a Vector Space Lesson 2093 — The Basis Theorem
Radial: Arrows point away from/toward a center; magnitude often grows with distance; Lesson 1967 — Radial and Rotational Vector Fields
Radial direction (ρ): The thickness is simply `dρ`; Lesson 1940 — Volume Elements in Spherical Coordinates
Radial equation: Lesson 2441 — Laplace's Equation in Polar Coordinates
Radial limits: r goes from 0 to f(θ); Lesson 1919 — Computing Areas Using Polar Integrals
radians: Lesson 837 — Arc Length Formula Lesson 842 — Sector Area Formula Lesson 987 — Arc Length Using Radians Lesson 988 — Sector Area Using Radians Lesson 1210 — Argument of a Complex Number
radical: √25 = 5.; Lesson 151 — Introduction to Square Roots Lesson 495 — Understanding Radicals and the Square Root Symbol
radical equation: is any equation where the variable appears inside a radical symbol (like a square root, cube root, or other root).; Lesson 522 — Introduction to Radical Equations Lesson 524 — Solving Simple Radical Equations
Radicals: (like √ or ∛): identify domain restrictions and where the function might have cusps or vertical tangents; Lesson 1526 — Sketching Complex Functions
radicand: the number under the symbol (in √25, the radicand is 25); Lesson 495 — Understanding Radicals and the Square Root Symbol Lesson 550 — Domain Restrictions from Square Roots
Radioactive decay: If a substance decays as `A(t) = 100e^(-0.; Lesson 1437 — Applications: Exponential Growth and Decay Rates Lesson 2327 — Applications: RC Circuits and Decay Lesson 2374 — Exponential Growth and Decay Models Lesson 2893 — Applications of the Poisson Distribution
radius: is a line segment connecting the center to any point on the circle.; Lesson 827 — Parts of a Circle: Center, Radius, Diameter, and Chord Lesson 834 — Distance from Center to Chord Lesson 891 — Surface Area of Spheres Lesson 983 — What is a Radian?Lesson 991 — Defining the Unit Circle Lesson 1141 — Finding Circle Properties from General Form Lesson 1654 — Disk Method: Rotation About the x-axis Lesson 1655 — Disk Method: Rotation About the y-axis (+9 more)
radius of convergence: R that creates a "zone of convergence" around the center point c.; Lesson 1763 — Radius of Convergence Lesson 1764 — Finding Radius of Convergence Using the Ratio Test Lesson 1765 — Finding Radius of Convergence Using the Root Test Lesson 1766 — Interval of Convergence
Radius of each disk: The distance from the axis to the curve at that point; Lesson 1653 — Volume of Revolution: Introduction and Disk Method Concept
Raise both sides: to that reciprocal power; Lesson 537 — Solving Equations with Fractional Exponents
Ramp Slopes: Lesson 981 — Applications in Construction and Engineering
Ramsey Theory: extends this idea profoundly: it asks "how large must a structure be before some specific pattern *must* appear, no matter how you try to avoid it?; Lesson 2577 — Advanced Pigeonhole Arguments and Ramsey Theory
Ramsey's Theorem: In any two-coloring (say, red and blue) of the edges of a sufficiently large complete graph, there must exist a monochromatic complete subgraph of a specified size.; Lesson 2577 — Advanced Pigeonhole Arguments and Ramsey Theory
Random Arrivals: Suppose a bus arrives uniformly between 2:00 PM and 2:10 PM.; Lesson 2897 — Applications of the Uniform Distribution
Random experiments: Lesson 2771 — What is a Random Experiment?
Random Number Generators: Computer random number generators typically produce values uniformly on [0, 1].; Lesson 2897 — Applications of the Uniform Distribution
Random scatter: around zero (good!; Lesson 3101 — Checking ANOVA Assumptions
Randomization: Randomly assigning subjects to groups reduces bias and supports the independence assumption; Lesson 3102 — ANOVA Applications and Experimental Design
Range: From vertex y-value to infinity (up if a > 0, down if a < 0); Lesson 433 — Key Features Summary and Complete Graphing Lesson 548 — What Are Domain and Range?Lesson 553 — Finding Range from Equations Lesson 554 — Range from Graphs Lesson 556 — Domain and Range of Piecewise Functions Lesson 566 — Domain and Range of Inverse Functions Lesson 578 — Vertical Shifts of Functions Lesson 594 — The Absolute Value Function (+14 more)
Range of R: = {a, c} — only these elements from B are actually paired; Lesson 2540 — Domain, Codomain, and Range of Relations
Range(T) = W: Lesson 2262 — The Range (Image) of a Linear Transformation
rank: of that matrix—the number of pivot positions (leading 1's in row echelon form)—tells you exactly how many vectors in your set are linearly independent.; Lesson 2070 — Linear Independence and the Rank of a Matrix Lesson 2092 — Dimension of Subspaces Lesson 2126 — Definition of Matrix Rank Lesson 2127 — Computing Rank via Row Reduction Lesson 2128 — The Column Space Lesson 2286 — SVD and the Matrix Rank Lesson 3105 — Wilcoxon Signed-Rank Test
Rank by importance: The singular values (diagonal of **Σ**) tell you the importance of each component; Lesson 2289 — Principal Component Analysis (PCA) and SVD
Rank each variable separately: The smallest value gets rank 1, next gets rank 2, and so on.; Lesson 3109 — Spearman's Rank Correlation
Rank equals trace: The rank of P (dimension of the subspace) equals its trace (sum of diagonal entries); Lesson 2232 — Properties of Projection Matrices
Rank-Nullity Theorem: states that for any matrix **A** with *n* columns:; Lesson 2095 — Rank-Nullity Theorem Lesson 2263 — Rank-Nullity Theorem for Linear Transformations
rank(A): = dimension of the column space (how many linearly independent columns); Lesson 2095 — Rank-Nullity Theorem Lesson 2133 — The Rank-Nullity Theorem
rate: is a special type of ratio that compares two quantities that have *different units*.; Lesson 109 — Introduction to Rates Lesson 439 — Work and Mixture Rate Problems
Rate (β): Controls how quickly events occur.; Lesson 2917 — The Gamma Distribution: Definition and Parameters
rate in: minus the **rate out**:; Lesson 2326 — Applications: Mixing Problems Lesson 2377 — Mixing Problems and Compartment Models
rate of change: equals the **current value**.; Lesson 613 — The Derivative Property: Why e is Special Lesson 2317 — Applications: Mixing and Velocity Problems Lesson 2326 — Applications: Mixing Problems Lesson 3064 — Interpreting Slope and Intercept
rate out: Lesson 2326 — Applications: Mixing Problems Lesson 2377 — Mixing Problems and Compartment Models
Rate problems: "A pool fills at 50 gallons/minute.; Lesson 884 — Volume Word Problems and Applications
ratio: is a way to compare two quantities by showing how much of one thing there is compared to another.; Lesson 105 — What is a Ratio?Lesson 983 — What is a Radian?Lesson 1946 — The Jacobian Determinant
Ratio method: Divide one equation by another to eliminate λ; Lesson 1878 — Solving Two-Variable Constrained Problems
Ratio Test: asks: "What happens when I divide the next term by the current term?; Lesson 1742 — The Ratio Test: Statement and Logic Lesson 1747 — Comparing Ratio and Root Tests Lesson 1748 — Series with Factorials and Exponentials Lesson 1750 — Strategy: Choosing the Right Convergence Test Lesson 1759 — Choosing the Right Test for Alternating Series
rational expression: is simply a fraction where both the numerator (top) and denominator (bottom) are polynomials.; Lesson 442 — What is a Rational Expression?Lesson 476 — Introduction to Rational Equations
rational expressions: these are fractions involving variables that you manipulate and simplify.; Lesson 476 — Introduction to Rational Equations Lesson 1526 — Sketching Complex Functions
rational function: is simply a function that can be written as the quotient (division) of two polynomials.; Lesson 485 — Introduction to Rational Functions Lesson 1384 — Computing Derivatives from the Definition: Rational Functions Lesson 1622 — Proper vs Improper Rational Functions
Rational functions: like `f(x) = 1/x` can take almost any value except where horizontal asymptotes block them—this one never equals zero, so its range is all real numbers except 0.; Lesson 553 — Finding Range from Equations Lesson 1343 — Limits at Infinity and Horizontal Asymptotes Lesson 1621 — Introduction to Partial Fraction Decomposition Lesson 1750 — Strategy: Choosing the Right Convergence Test Lesson 1813 — Computing Limits Using Direct Substitution Lesson 1818 — Continuity on a Region
Rational numbers: A number *r* is rational if there exist integers *p* and *q* with *q ≠ 0* such that *r = p/q*.; Lesson 2492 — Direct Proof: Working with Definitions Lesson 2558 — Applications and Quotient Sets
Rationalization: is the process of eliminating imaginary parts from the denominator, and then expressing the result in **standard form**: *a + bi*, where *a* and *b* are real numbers.; Lesson 1206 — Rationalization and Standard Form
Rationalize denominators: by multiplying by the conjugate when dividing; Lesson 1202 — Simplifying Complex Expressions
Rationalizing: Lesson 1341 — Algebraic Manipulation for 0/0
Ray AB: Starts at A, passes through B, and continues forever beyond B.; Lesson 697 — Line Segments and Rays
Ray BA: Starts at B, passes through A, and continues forever beyond A.; Lesson 697 — Line Segments and Rays
RC circuit: ), the voltage across the capacitor decreases over time according to a first-order linear ODE.; Lesson 2327 — Applications: RC Circuits and Decay
Reach an impossibility: (a contradiction); Lesson 2497 — Proof by Contradiction: Basic Idea
Reach perfectly: when *a* is long enough or angle *A* is large enough (one triangle); Lesson 1121 — Introduction to the Ambiguous Case (SSA)
Read: the coordinates (x, y); Lesson 1007 — Negative Angles and Trigonometric Values
Read and Understand: Lesson 1516 — Applied Optimization Strategy
Read and Visualize: Lesson 982 — Mixed Application Problems and Problem-Solving Strategies
Read carefully: and identify what's proportional; Lesson 123 — Real-World Proportion Problems Lesson 529 — Word Problems Involving Radical Equations Lesson 1652 — Applied Problems: Area Between Curves
Read the coordinates: (x, y) = (cos θ, sin θ); Lesson 1010 — Using the Unit Circle to Evaluate Trig Functions
Read the problem carefully: – identify what's changing and what drives the change; Lesson 2335 — Applications of Exact and Bernoulli Equations
Real consequences: In medicine, a false positive might mean prescribing unnecessary treatment; in quality control, it might mean discarding good products.; Lesson 3052 — Type I Error: False Positives
Real eigenvalues: For any symmetric matrix **A**, all eigenvalues are real (you proved this earlier).; Lesson 2272 — The Spectral Theorem for Symmetric Matrices
real number: Lesson 1196 — Complex Conjugates Lesson 1199 — Complex Conjugates
real part: (an ordinary real number); Lesson 1190 — Standard Form of Complex Numbers Lesson 1191 — Identifying Real and Imaginary Parts Lesson 1192 — Plotting Complex Numbers on the Complex Plane Lesson 1207 — Complex Numbers as Points in the Plane Lesson 1208 — The Argand Diagram
Real-world analogies: Lesson 2661 — Cut Vertices and Cut Edges
Real-world measurements: – If you're calculating the length of lumber needed (`12/√5` feet), a contractor wants `5.; Lesson 521 — Applications and When Not to Rationalize
Real-world uses: Lesson 103 — Percentages Greater Than 100% and Less Than 1%
reason: justifies why that statement is valid—it might be a definition (like "definition of midpoint"), a postulate (like "SSS Congruence"), a theorem you've already learned (like "Vertical Angles Theorem"), or algebraic properties.; Lesson 762 — Introduction to Geometric Proof Lesson 766 — Common Reasons in Proofs
reasons: (why each statement is true).; Lesson 748 — Writing Two-Column Congruence Proofs Lesson 763 — Two-Column Proof Format Lesson 765 — Flow Chart Proof Format
Recall: Lesson 1424 — Derivatives of Trig Functions with Coefficients and Sums
Recipes and Cooking: Adjusting ingredient amounts when changing serving sizes.; Lesson 123 — Real-World Proportion Problems
reciprocal: is what you get when you flip a fraction upside down.; Lesson 77 — Reciprocals and Division Setup Lesson 135 — Negative Integer Exponents
Reciprocal identities: sin θ = 1/csc θ, cos θ = 1/sec θ, tan θ = 1/cot θ; Lesson 1050 — Simplifying Trigonometric Expressions Lesson 1090 — Equations Involving Multiple Functions
Reciprocal property: If X ~ F(d₁, d₂), then 1/X ~ F(d₂, d₁); Lesson 2926 — Properties and Applications of the F-Distribution
recognize: when to use them in real problems.; Lesson 342 — Using Special Products to Simplify Expressions Lesson 482 — Rational Equations Leading to Quadratics
Recognize patterns: – Is it a trinomial?; Lesson 381 — Applications and Mixed Practice
Recognize the pattern: Both x² and 16 are perfect squares (16 = 4²); Lesson 398 — Solving Special Forms: Difference of Squares Lesson 1079 — Simplifying Expressions with Product and Sum Formulas
Recognize the resulting sums: as G(x) or shifted/scaled versions of it; Lesson 2643 — Solving Recurrence Relations with Generating Functions
Recognize the right triangle: (the ground, wall, and ladder form one); Lesson 974 — Setting Up Right Triangle Problems from Word Descriptions
Recommendation systems: Compress user-rating matrices; Lesson 2288 — Low-Rank Approximation and Data Compression
rectangle: is a parallelogram with an extra requirement: all four angles must be right angles (90°).; Lesson 818 — Rectangles: Properties and Characteristics Lesson 825 — The Quadrilateral Family Tree Lesson 856 — Perimeter of Polygons Lesson 857 — Area of Rectangles and Squares Lesson 942 — Line Symmetry (Reflectional Symmetry)
Rectangles and rhombuses: inherit this property too; Lesson 826 — Coordinate Geometry of Quadrilaterals
Rectangular: `−√(a² − x²) ≤ y ≤ √(a² − x²)` (square roots everywhere!; Lesson 1913 — Why Use Polar Coordinates for Integration Lesson 1943 — Choosing the Best Coordinate System
Recurrence: F(n) = F(n-1) + F(n-2) for n ≥ 2; Lesson 2627 — What is a Recurrence Relation?
recurrence relation: is an equation that defines a sequence by describing how to compute each term from one or more earlier terms in the sequence.; Lesson 2627 — What is a Recurrence Relation?Lesson 2628 — Initial Conditions and Solutions Lesson 2916 — Introduction to the Gamma Function
Recurse: treat *u* as the new starting point and repeat; Lesson 2673 — Finding Spanning Trees: Depth-First Search
Recursive formula: a_n = 2 × a_(n-1), with a_1 = 3; Lesson 664 — Recursive Formulas for Geometric Sequences
Red flags: Lesson 3084 — Residual Plots: Fitted Values vs. Residuals
Redescribe the region: from the other variable's viewpoint; Lesson 1905 — Changing the Order of Integration
Reduce early and often: Since (a · b) mod n = [(a mod n) · (b mod n)] mod n, we can take the modulus after each multiplication; Lesson 2734 — Powers in Modular Arithmetic
Reduced (Thin) QR: **Q** is *m × n* with orthonormal columns, **R** is *n × n* upper triangular; Lesson 2245 — QR Decomposition for Square vs Rectangular Matrices
Reduced QR: is more computationally efficient and commonly used in applications like least squares, since the extra columns in full **Q** multiply zeros in **R** anyway.; Lesson 2245 — QR Decomposition for Square vs Rectangular Matrices
reduced row echelon form: ), where:; Lesson 2105 — Introduction to Gaussian Elimination Lesson 2120 — Reduced Row Echelon Form: Definition
Reducible fractions: – If you have `6/12` or `15/10`, reduce it.; Lesson 520 — Simplifying After Rationalizing
Reduction: Lesson 119 — Scale Factors
Reduction formulas: are recursive formulas that express an integral involving tanⁿ(x) or secⁿ(x) in terms of a similar integral with a *lower power* (n-2 typically).; Lesson 1604 — Reduction Formulas for Tangent and Secant
reference angle: is the acute angle (between 0° and 90°) formed between the terminal side of your angle and the x- axis.; Lesson 994 — Reference Angles and the First Quadrant Lesson 997 — Fourth Quadrant Points Lesson 1003 — Reference Angles Lesson 1005 — Evaluating Trig Functions for Angles Greater than 90°
Reflecting across the x-axis: Lesson 242 — Reflections and Symmetry in the Coordinate Plane
Reflecting across the y-axis: Lesson 242 — Reflections and Symmetry in the Coordinate Plane
Reflection: Flipping a figure over a line, like looking at your image in a mirror.; Lesson 724 — Congruence Transformations Lesson 925 — Reflections Across the Axes
Reflections: (across x-axis or y-axis); Lesson 584 — Combining Multiple Transformations Lesson 589 — Inverse Relationships in Transformations Lesson 939 — Introduction to Rigid Motions (Isometries)Lesson 941 — Congruence Through Rigid Motions Lesson 1196 — Complex Conjugates
Reflections across the x-axis: preserve symmetry type.; Lesson 587 — Even and Odd Functions Under Transformation
Reflections across the y-axis: preserve evenness (already symmetric about y-axis) and preserve oddness (flipping both sides maintains origin symmetry).; Lesson 587 — Even and Odd Functions Under Transformation
reflective property of ellipses: .; Lesson 1165 — Reflective Property of Ellipses Lesson 1166 — Applications of Ellipses
reflexive: if and only if:; Lesson 2542 — Reflexive Relations Lesson 2546 — Combining Properties: Partial Orders Lesson 2549 — Inverse Relations Lesson 2550 — Definition of Equivalence Relations Lesson 2551 — Verifying Equivalence Relations Lesson 2556 — Partitions Induce Equivalence Relations Lesson 2557 — Congruence Modulo n Lesson 2731 — Equivalence Classes in Modular Arithmetic
Reflexivity: Lesson 2550 — Definition of Equivalence Relations Lesson 2709 — Divisibility: Definition and Basic Properties
Region 1: (from *a* to *b*): Integrate |*f(x)* − *g(x)*|; Lesson 1650 — Area Enclosed by Intersecting Curves
Region 2: (from *b* to *c*): Integrate |*g(x)* − *f(x)*| (they switched!; Lesson 1650 — Area Enclosed by Intersecting Curves
Register allocation: Assign computer registers to variables in a program; Lesson 2695 — Graph Coloring and the Chromatic Number
regression: , gives you a mathematical model to predict values or understand the relationship.; Lesson 649 — Fitting Exponential and Logarithmic Models to Data Lesson 3066 — Correlation vs Regression
Regroup terms: Lesson 1419 — Derivative of sin(x) from First Principles
Regular hexagon: (n = 6): Each exterior angle = 360° ÷ 6 = **60°**; Lesson 812 — Finding Individual Exterior Angles in Regular Polygons Lesson 856 — Perimeter of Polygons
Regular hexagons: have point symmetry at their center (among other rotational symmetries); Lesson 944 — Point Symmetry
Regular octagon: (n = 8): Each exterior angle = 360° ÷ 8 = **45°**; Lesson 812 — Finding Individual Exterior Angles in Regular Polygons Lesson 814 — Determining the Number of Sides from Angle Information
regular polygon: , all sides are equal length AND all interior angles are equal measure.; Lesson 809 — Finding Individual Interior Angles in Regular Polygons Lesson 812 — Finding Individual Exterior Angles in Regular Polygons Lesson 863 — Area of Regular Polygons Lesson 945 — Symmetry in Regular Polygons
regular polygons: , if you know one exterior angle, use:; Lesson 814 — Determining the Number of Sides from Angle Information Lesson 856 — Perimeter of Polygons Lesson 946 — Applications of Symmetry in Design and Nature
Regular singular point: if $(x - x_0)p(x)$ and $(x - x_0)^2q(x)$ are both analytic at $x_0$; Lesson 2401 — Regular Singular Points and the Method of Frobenius Lesson 2402 — Frobenius Method: Finding Indicial Equations
Reject any solution: that:; Lesson 481 — Checking for Extraneous Solutions
reject H₀: (conclude there's sufficient evidence for H₁) or **fail to reject H₀** (insufficient evidence to abandon the null hypothesis).; Lesson 3020 — Introduction to Hypothesis Testing Lesson 3024 — The Logic of Hypothesis Testing Lesson 3097 — The ANOVA Table and Hypothesis Testing
Related rates preview: Two changing quantities connected by an equation (you'll explore this deeper later).; Lesson 1418 — Chain Rule Applications and Problem-Solving
relation: between sets A and B is simply a subset of `A × B`.; Lesson 2538 — Applications and Higher-Order Products Lesson 2539 — Definition of a Relation Lesson 2549 — Inverse Relations
Relations: "Every student has taken at least one course":; Lesson 2486 — Nested Quantifiers in Mathematics
relative frequency: k/n as our estimate; Lesson 2975 — Applications: Estimating Probabilities and Means Lesson 2992 — Histograms and Frequency Distributions
relatively prime: or **coprime**; Lesson 2711 — Greatest Common Divisor (GCD): Definition Lesson 2718 — Relatively Prime Integers Lesson 2746 — Euler's Totient Function φ(n)
Reliable: (consistency ensures accuracy with large samples); Lesson 3007 — Properties and Advantages of MLEs
remainder: is what's "left over" after division.; Lesson 28 — Division with Remainders Lesson 344 — Introduction to Polynomial Division Lesson 351 — The Remainder Theorem Lesson 502 — Simplifying nth Roots with Variables Lesson 1201 — Powers of i Lesson 1733 — Estimating Remainders with the Integral Test Lesson 1784 — Taylor's Theorem Statement and Motivation
Remainder 0: → i^n = 1; Lesson 1194 — Powers of i
Remainder 1: → i^n = i; Lesson 1194 — Powers of i
Remainder 2: → i^n = −1; Lesson 1194 — Powers of i
Remainder 3: → i^n = −i; Lesson 1194 — Powers of i
remainder term: that captures exactly what you're missing.; Lesson 1784 — Taylor's Theorem Statement and Motivation Lesson 1789 — Proving Convergence with the Remainder Term
Remaining balance: after \(k\) payments is the present value of remaining payments, also a geometric series calculation.; Lesson 689 — Loan Amortization
Remarkable property: Mean equals variance!; Lesson 2862 — Expectation and Variance in Common Distributions
Remember: You might need to use alternate interior angles when angles of depression create congruent angles in different positions within your diagram.; Lesson 972 — Multi-Step Elevation and Depression Problems Lesson 1393 — Combining Basic Rules for Polynomials Lesson 1455 — Second Derivatives by Implicit Differentiation Lesson 2170 — Finding Eigenvectors from Eigenvalues
Remember both signs: x = ±√c; Lesson 393 — Solving x² = c by Square Roots
Remember the reciprocal rule: negative exponents flip the base; Lesson 141 — Mixed Practice with Integer Exponents
Remember the sign rule: when you multiply or divide both sides by a negative number, you must flip the inequality sign.; Lesson 220 — Solving Multi-Step Linear Inequalities
Remember the sign-flip rule: if you multiply or divide by a negative number; Lesson 221 — Inequalities with Variables on Both Sides
Remove genuine outliers: (after careful investigation—never automatically!; Lesson 3092 — Model Refinement Based on Diagnostics
Remove one redundant vector: (typically one that can be expressed in terms of the others); Lesson 2089 — Finding a Basis from a Spanning Set
Remove redundant constraints: Drop the dependent constraint(s) from your system; Lesson 1889 — Dealing with Redundant Constraints
Repackage: using definitions to match the conclusion; Lesson 2493 — Direct Proof Examples and Practice
Repeat: until you have a simple fraction; Lesson 474 — Nested Complex Fractions Lesson 1311 — Row Reduction to Echelon Form Lesson 1312 — Back Substitution from Echelon Form Lesson 2089 — Finding a Basis from a Spanning Set Lesson 2090 — Extending to a Basis Lesson 2108 — Forward Elimination Process Lesson 2676 — Minimum Spanning Trees: Prim's Algorithm Lesson 2702 — The Hungarian Algorithm for Maximum Bipartite Matching (+1 more)
Repeat for each piece: until you reach simple limits you can evaluate directly; Lesson 1338 — Combining Multiple Limit Laws
Repeat upward: Continue substituting and solving until you reach the top row.; Lesson 2110 — Back Substitution
Repeat with different x-values: Choose x = -1 to eliminate B and solve for A; Lesson 1624 — Solving for Coefficients: Substitution Method
Repeatability: – The experiment can be performed multiple times under the same conditions; Lesson 2771 — What is a Random Experiment?
repeated root: use `y_p = Ax²e^(rx)`; Lesson 2351 — Particular Solutions for Exponential Functions Lesson 2631 — Repeated Roots in the Characteristic Equation
repeated roots: .; Lesson 2184 — Algebraic Multiplicity of Eigenvalues Lesson 2632 — Higher-Order Linear Homogeneous Recurrences
Repeated squaring: Express the exponent in binary to minimize multiplications; Lesson 2734 — Powers in Modular Arithmetic
Repeating When Not Indeterminate: Lesson 1534 — Common Mistakes and When Not to Use L'Hôpital's Rule
Replace: √(a²-x²) with a cos θ; Lesson 1613 — Substitution for √(a²-x²): Using x = a sin θ
Replace dA: with the polar area element: r dr dθ; Lesson 1915 — Setting Up Double Integrals in Polar Coordinates
Replace dx with du: using the derivative relationship; Lesson 1579 — What is u-Substitution?
Replace f(x) with y: to make the equation easier to work with; Lesson 563 — Finding Inverse Functions Algebraically
Replace the infinite limit: with a variable *t*; Lesson 1634 — Type 1 Improper Integrals: Infinite Upper Limit
Replace x and y: with their polar equivalents: x = r cos θ and y = r sin θ; Lesson 1915 — Setting Up Double Integrals in Polar Coordinates
Replication: Multiple observations per group increase power and allow variance estimation; Lesson 3102 — ANOVA Applications and Experimental Design
residual: (error): the difference between the actual $y_i$ and the predicted value $mx_i + b$.; Lesson 2238 — Applications: Linear Regression Lesson 3083 — What Are Residuals?
Residual error: Lesson 3100 — Two-Way ANOVA With Interaction
Residual size: how poorly the model fits that point; Lesson 3088 — Cook's Distance and Influence Measures
Resonance: occurs when the driving frequency `ω` nearly equals the system's natural frequency `ω₀ = √(k/m)`.; Lesson 2371 — Forced Harmonic Motion and Resonance
Resonance warning: If any part of your guess solves the homogeneous equation, multiply the *entire guess* by t (or t² if needed).; Lesson 2353 — Products of Basic Functions
Respect order: "5 less than a number" means *x - 5*, not *5 - x*; Lesson 172 — Writing Expressions from Word Problems
Respect the hierarchy: even when variables are present; Lesson 177 — Order of Operations with Variables
Restrict to Level Curves: Lesson 1868 — Inconclusive Cases and Further Analysis
Restricted values: are the specific numbers that would make the denominator zero—and we *must* exclude them from the domain (the set of allowable inputs).; Lesson 444 — Finding Restricted Values Lesson 530 — Mixed Radical and Rational Equations
Restrictions: x ≠ 2, x ≠ -2; Lesson 448 — Simplifying with Trinomial Factors
Result: 6 × 10⁹; Lesson 165 — Multiplying Numbers in Scientific Notation Lesson 322 — Subtracting Polynomials: Distributing the Negative Lesson 455 — Dividing and Simplifying in One Step Lesson 456 — Mixed Operations: Multiplication and Division Together Lesson 502 — Simplifying nth Roots with Variables Lesson 995 — Second Quadrant Points Lesson 1219 — Converting from Polar to Rectangular Form Lesson 1221 — Multiplying Complex Numbers in Polar Form (+11 more)
Result F(s): Encodes information about *f(t)* in a form where derivatives become multiplication; Lesson 2388 — Definition of the Laplace Transform
Result: 2/3: Lesson 65 — Simplifying Fractions
Result: 3/5: Lesson 65 — Simplifying Fractions
Revenue/profit models: Finding break-even points; Lesson 373 — Applications and Strategy Selection for Trinomial Factoring
Review your multiplication tables: (0–10).; Lesson 25 — Division Facts and Basic Division Fluency
Rewrite: the complex fraction as a division problem; Lesson 80 — Complex Fractions Lesson 233 — Absolute Value Inequalities of the Form |x| < a Lesson 358 — Factoring Out the GCF from Polynomials Lesson 372 — Factoring Higher-Degree Trinomials Using Substitution Lesson 448 — Simplifying with Trinomial Factors Lesson 530 — Mixed Radical and Rational Equations Lesson 1412 — Chain Rule with Radicals Lesson 1600 — Powers of Sine and Cosine: Odd Powers (+1 more)
Rewrite before deciding: Sometimes rewriting (like expanding products or splitting fractions) makes differentiation easier.; Lesson 1407 — Common Errors and Problem-Solving Strategies
Rewrite each rational: as an equivalent expression with the LCD; Lesson 467 — Adding and Subtracting Three or More Rationals
Rewrite first: when:; Lesson 1405 — Choosing Between Product and Quotient Rules
Rewrite the bounds: for R in terms of r and θ; Lesson 1915 — Setting Up Double Integrals in Polar Coordinates
Rewrite the curves: as x = f(y) and x = g(y); Lesson 1648 — Area Between Curves: Horizontal Slicing
Rewrite the exponent: *m^(ed) = m^(1 + k·φ(n)) = m · m^(k·φ(n)) = m · (m^φ(n))^k*; Lesson 2768 — Why RSA Works: Euler's Theorem
Rewrite the integral: ∫[g(a) to g(b)] (expression in u) du; Lesson 1586 — Definite Integrals and Changing Limits
Rewrite the middle term: (bx) using those two numbers; Lesson 363 — The AC Method for General Trinomials
Rewrite the solution vector: by factoring out the parameters; Lesson 2064 — Applications: Expressing Solutions as Linear Combinations
Rewrite using logarithms: For exponential differences; Lesson 1532 — Indeterminate Differences: ∞ - ∞
Rewrite your curve: as x = f(y) instead of y = f(x); Lesson 1655 — Disk Method: Rotation About the y-axis
rhombus: is a special type of parallelogram where **all four sides are congruent** (equal in length).; Lesson 819 — Rhombuses: Properties and Characteristics Lesson 825 — The Quadrilateral Family Tree
Rhombuses: (all four sides equal); Lesson 826 — Coordinate Geometry of Quadrilaterals
Richter scale: measures earthquake magnitude:; Lesson 647 — pH and the Richter Scale
Riemann sum: Σ(i=1 to n) f(xᵢ)·Δx = Σ(i=1 to n) (2i/n)·(2/n) = (4/n²)·Σ(i=1 to n) i; Lesson 1556 — Computing Simple Integrals Using Riemann Sum Limits
Riemann sums: (left, right, midpoint rectangles); Lesson 1681 — Numerical Approximation of Arc Length
right: of zero; Lesson 40 — Representing Negative Numbers on the Number Line Lesson 41 — Comparing and Ordering Negative Numbers Lesson 150 — Common Mistakes with Exponent Rules Lesson 163 — Converting from Standard Form to Scientific Notation Lesson 164 — Converting from Scientific Notation to Standard Form Lesson 239 — Plotting Points on the Coordinate Plane Lesson 547 — Common Function Notation Mistakes Lesson 579 — Horizontal Shifts of Functions (+11 more)
right 3: units (the `x - 3` inside); Lesson 584 — Combining Multiple Transformations Lesson 1038 — Determining Phase Shift from Equations
right angle: measures exactly 90° (like the corner of a square); Lesson 703 — Measuring Angles and Angle Notation Lesson 704 — Classifying Angles by Measure Lesson 738 — Medians, Altitudes, and Angle Bisectors Lesson 847 — Tangent Lines to Circles
Right column (Reasons): Justifications like definitions, postulates, theorems, or given information; Lesson 763 — Two-Column Proof Format
Right distributive: `(A + B)C = AC + BC`; Lesson 1287 — The Distributive Property for Matrices
Right Rectangles: Use the *right* endpoint; Lesson 1546 — Approximating Area Under a Curve with Rectangles
Right Riemann Sums: work almost identically, except now we use the **right endpoint** of each subinterval to set the height of each rectangle.; Lesson 1548 — Right Riemann Sums
Right side: Already in that form; Lesson 1071 — Verifying Identities with Double and Half Angle Formulas Lesson 2001 — Connection to Curl in Two Dimensions Lesson 2026 — Introduction to Stokes' Theorem Lesson 2038 — Statement of the Divergence Theorem Lesson 2043 — Physical Interpretation: Conservation Laws
Right singular vectors (V): The first r columns of **V** span the **row space** of **A**.; Lesson 2283 — Left and Right Singular Vectors
Right sum: Uses the right edge of each interval; Lesson 1550 — Comparing Left, Right, and Midpoint Approximations
Right Triangle: Lesson 731 — Classifying Triangles by Angles Lesson 788 — Using the Converse to Identify Right Triangles Lesson 839 — Angles Inscribed in a Semicircle Lesson 978 — Finding Distances Using Angles of Depression Lesson 1110 — The Law of Cosines and the Pythagorean Theorem
Right-hand limit: lim(x→ a ) f(x) — the superscript plus means "from the right"; Lesson 1326 — Limits Approaching from Left and Right Lesson 1329 — Understanding One-Sided Limits Lesson 1331 — Evaluating One-Sided Limits Graphically Lesson 1333 — One-Sided Limits at Jump Discontinuities
Right-skewed: Most probability mass is near zero; long tail extends right; Lesson 2926 — Properties and Applications of the F-Distribution Lesson 2993 — Shape of Distributions: Symmetry and Skewness Lesson 3047 — The F Distribution
Right-skewed (positively skewed): A long tail stretches to the right.; Lesson 2993 — Shape of Distributions: Symmetry and Skewness
Right-skewed shape: The distribution is asymmetric, skewing right, especially for small *k*; Lesson 2922 — Properties and Applications of Chi-Squared
rigid motion: or **isometry**) is a movement that slides, flips, or turns a figure without changing its dimensions.; Lesson 724 — Congruence Transformations Lesson 924 — Properties of Translations
Rise: = the vertical change (how far up or down); Lesson 244 — Rise Over Run: Understanding Slope Geometrically Lesson 245 — Calculating Slope from Two Points Lesson 248 — Finding Slope from a Graph
rise over run: or `(y₂ - y₁)/(x₂ - x₁)`.; Lesson 251 — Average Rate of Change vs Constant Rate Lesson 253 — Finding Missing Coordinates Using Slope Lesson 1369 — Understanding Average Rate of Change
Rⁿ: , the standard basis vectors are **e₁, e₂, .; Lesson 2088 — Standard Bases
Roadway and track design: Engineers need the precise length of curved highway ramps or racetracks for material estimates and speed calculations.; Lesson 1682 — Applications: Arc Length in Physics and Geometry
Robin (Type III): Specifies a linear combination; Lesson 2404 — Boundary Value Problems: Introduction and Types
Robust interpretation: Tau directly measures the probability that two randomly chosen pairs are in the same order; Lesson 3110 — Kendall's Tau
Robust regression methods: that downweight extreme points; Lesson 3092 — Model Refinement Based on Diagnostics
Robustness: Valid conclusions even with messy, non-normal data; Lesson 3103 — Introduction to Nonparametric Methods
Rolle's Theorem: states: If a function *f* is continuous on [a, b], differentiable on (a, b), and **f(a) = f(b)**, then there exists at least one point *c* in (a, b) where **f'(c) = 0**.; Lesson 1487 — Rolle's Theorem as a Special Case
Rolling a six-sided die: Lesson 2772 — Sample Spaces: The Set of All Possible Outcomes
Rolling two dice: The result of the first die doesn't affect the second.; Lesson 2805 — Independence and the Multiplication Rule
Roof Pitches: Lesson 981 — Applications in Construction and Engineering
Root Test: examines the nth root of the absolute value of each term.; Lesson 1745 — The Root Test: Statement and Motivation Lesson 1747 — Comparing Ratio and Root Tests Lesson 1750 — Strategy: Choosing the Right Convergence Test Lesson 1759 — Choosing the Right Test for Alternating Series Lesson 1765 — Finding Radius of Convergence Using the Root Test
Rose curves: (like r = sin(3θ)): Flower-petal patterns.; Lesson 1919 — Computing Areas Using Polar Integrals
Rotate about the origin: Apply the standard rotation rules you already know (90°, 180°, 270°, etc.; Lesson 932 — Rotations About Points Other Than the Origin
Rotate each segment: to create a frustum; Lesson 1684 — Deriving the Surface Area Formula
Rotation: Turning a figure around a fixed point by a certain angle.; Lesson 724 — Congruence Transformations Lesson 930 — Introduction to Rotations in the Coordinate Plane Lesson 1224 — Geometric Interpretation of Polar Operations Lesson 2009 — Interpreting Curl Physically Lesson 2034 — Physical Interpretations: Circulation and Rotation
Rotational: Arrows form circular or spiral patterns; perpendicular to position vectors; Lesson 1967 — Radial and Rotational Vector Fields
Rotations: – Turning the figure around a fixed point; Lesson 939 — Introduction to Rigid Motions (Isometries)Lesson 941 — Congruence Through Rigid Motions
Round trips: (different rates or conditions); Lesson 484 — Distance-Rate-Time and Mixture Problems
Round-Off Errors: When you round measurements to the nearest unit, the rounding error is approximately uniform on [-0.; Lesson 2897 — Applications of the Uniform Distribution
Round-off errors accumulate: during the many arithmetic operations required; Lesson 2147 — Computational Considerations and Singular Matrices
Rounding: works great for multiplication; Lesson 30 — Estimating Products and Quotients
Route Planning: Postal workers, garbage collectors, and street sweepers need routes that cover every street without wasteful repetition.; Lesson 2682 — Applications of Eulerian Paths: The Königsberg Bridge Problem
Row Addition: Add a multiple of one row to another row; Lesson 2106 — Elementary Row Operations Lesson 2141 — Row Operations and Matrix Inverses
row echelon form: (REF) when it has a staircase-like structure:; Lesson 1311 — Row Reduction to Echelon Form Lesson 1312 — Back Substitution from Echelon Form Lesson 2105 — Introduction to Gaussian Elimination Lesson 2116 — Row Echelon Form: Definition and Properties Lesson 2120 — Reduced Row Echelon Form: Definition
Row reduce: the augmented matrix [A | 0] to reduced row echelon form (RREF); Lesson 2130 — The Null Space Lesson 2170 — Finding Eigenvectors from Eigenvalues Lesson 2252 — Computing Transition Matrices
Row Replacement (Addition): Lesson 2118 — Elementary Row Operations
Row Scaling: Multiply an entire row by a nonzero constant; Lesson 2106 — Elementary Row Operations Lesson 2141 — Row Operations and Matrix Inverses
Row Scaling (Multiplication): Lesson 2118 — Elementary Row Operations
Row Space: (C(A ᵀ)): All linear combinations of A's rows; lives in ℝⁿ; Lesson 2096 — Bases for Important Subspaces Lesson 2129 — The Row Space Lesson 2134 — Orthogonality of Fundamental Subspaces Lesson 2283 — Left and Right Singular Vectors
Row space view: The rank also equals the dimension of the span of the matrix's rows.; Lesson 2126 — Definition of Matrix Rank
Row Space, C(A^T): Lesson 2132 — The Four Fundamental Subspaces
Row Sums: Add all numbers in row *n*, and you get 2ⁿ.; Lesson 2598 — Pascal's Triangle: Construction and Patterns
Row Swap: Exchange two rows; Lesson 2106 — Elementary Row Operations Lesson 2141 — Row Operations and Matrix Inverses
Row Swapping (Interchange): Lesson 2118 — Elementary Row Operations
row vector: (or row matrix) has exactly **one row** but can have multiple columns.; Lesson 1272 — Types of Matrices: Square, Row, Column, and Zero Lesson 2097 — Matrix Notation and Dimensions
Row-reduce: Use row operations to reach reduced row echelon form; Lesson 2069 — The Matrix Test for Linear Independence
rows × columns: (read "rows by columns").; Lesson 1271 — What is a Matrix? Dimensions and Notation Lesson 2097 — Matrix Notation and Dimensions
RSA modulus: .; Lesson 2762 — The RSA Setup: Choosing Primes
rule: that makes them work.; Lesson 148 — Negative Exponent Rule Lesson 925 — Reflections Across the Axes Lesson 931 — Rotations of 90°, 180°, and 270° About the Origin Lesson 1295 — Determinants of Special Matrices
Rule of thumb: For moderately skewed distributions, n ≥ 30 is often sufficient.; Lesson 2981 — Sample Size and Normal Approximation Lesson 2997 — Choosing Appropriate Summaries and Displays Lesson 3080 — Multicollinearity: Detection and Consequences
Ruler Postulate: it establishes a one-to-one correspondence between points on a line and real numbers.; Lesson 702 — Distance and the Ruler Postulate
Run: = the horizontal change (how far left or right); Lesson 244 — Rise Over Run: Understanding Slope Geometrically Lesson 245 — Calculating Slope from Two Points Lesson 248 — Finding Slope from a Graph
Runs: Long stretches of consecutive positive residuals followed by negative ones (or vice versa); Lesson 3090 — Checking for Independence: Residual Sequence Plots
Rx = c: by back substitution; Lesson 2246 — Solving Linear Systems Using QR Lesson 2247 — QR and Least Squares Problems
Rx = Q ᵀb: Lesson 2228 — Applications: QR Factorization Preview Lesson 2247 — QR and Least Squares Problems
Rx = Q^T b: Lesson 2246 — Solving Linear Systems Using QR Lesson 2248 — Computational Advantages of QR Decomposition
Rx̂ = Q^T b: Lesson 2237 — Least Squares with QR Decomposition

S

S = 2π∫ y√(1+(dy/dx)²)dx: , combining circular motion (2πy) with the curve's arc length element.; Lesson 1685 — Surface Area Formula for Revolution Around x-axis
S is decreasing: Survival probability decreases over time; Lesson 2851 — Applications: Reliability and Survival Functions
S-shaped (sigmoid) curve: slow start, rapid middle growth, then leveling off at $K$.; Lesson 2375 — Logistic Growth Model
S-shaped curve: = heavy or light tails; Lesson 3086 — Normal Probability Plots (Q-Q Plots)
S(∞) = 0: Eventually everything fails; Lesson 2851 — Applications: Reliability and Survival Functions
S(0) = 1: Everything starts working (100% survival at time 0); Lesson 2851 — Applications: Reliability and Survival Functions
S(x): = P(X > x) = probability of survival beyond time x; Lesson 2851 — Applications: Reliability and Survival Functions
saddle point: , the function curves up in some directions and down in others—like sitting on a horse's saddle.; Lesson 1863 — Classifying Critical Points: Local Extrema and Saddle Points Lesson 1866 — The Second Derivative Test: Statement and Conditions Lesson 1867 — Applying the Second Derivative Test
Salary with Fixed Raises: If you start at $30,000 per year and receive a $2,000 raise annually, your yearly salaries form an arithmetic sequence: 30000, 32000, 34000, 36000.; Lesson 659 — Applications of Arithmetic Sequences
same base: , you keep the base and **add the exponents**.; Lesson 142 — Product Rule for Exponents Lesson 143 — Quotient Rule for Exponents Lesson 609 — Solving Simple Exponential Equations Lesson 630 — Solving Simple Exponential Equations Lesson 2634 — Finding Particular Solutions
same degree: (same highest exponent), the horizontal asymptote is simply the ratio of their leading coefficients.; Lesson 489 — Finding Horizontal Asymptotes by Leading Coefficients Lesson 2634 — Finding Particular Solutions
same denominator: (the bottom number), comparing them becomes incredibly simple.; Lesson 66 — Comparing Fractions with Like Denominators Lesson 67 — Comparing Fractions with Unlike Denominators Lesson 459 — Adding Rationals with Like Denominators
same index: (like square root, cube root, etc.; Lesson 505 — Adding and Subtracting Like Radicals Lesson 510 — Dividing Radicals with the Same Index Lesson 511 — Dividing Radicals Using the Quotient Property
same line: (not just parallel lines).; Lesson 265 — Slopes of Parallel Lines Lesson 290 — Special Cases: No Solution and Infinite Solutions Lesson 298 — Dependent Three-Variable Systems
same point: on the unit circle, they have **identical coordinates**.; Lesson 1006 — Coterminal Angles and Periodicity Lesson 1007 — Negative Angles and Trigonometric Values
same sign: (both positive OR both negative), your answer is always **positive**.; Lesson 45 — Multiplying Positive and Negative Numbers Lesson 362 — Factoring Trinomials with Leading Coefficient 1
same slope: as the original line with **point-slope form** using your new point.; Lesson 266 — Writing Equations of Parallel Lines Lesson 269 — Determining Parallel, Perpendicular, or Neither
Same vertex: Both angles must meet at the exact same point.; Lesson 705 — Adjacent Angles
Same-side interior angles: are supplementary (sum to 180°) when lines are parallel; Lesson 718 — Using Angle Relationships to Find Measures
sample mean: is simply the average of a collection of random observations drawn from a population.; Lesson 2968 — Understanding Sample Means and Averages Lesson 2970 — Weak Law of Large Numbers: Statement Lesson 2984 — CLT for Sample Sums Lesson 3000 — Sample Mean and Variance as Estimators
Sample size: Larger samples make it easier to detect real effects (lower β); Lesson 3053 — Type II Error: False Negatives Lesson 3054 — The Relationship Between Type I and Type II Errors
Sample size (n): Larger samples give more precise estimates.; Lesson 3012 — Margin of Error and Interval Width
Sample size matters intensely: With small samples, chi-squared tests produce unreliable p-values when expected frequencies drop below 5.; Lesson 3051 — Applications and Limitations
Sample sizes are small: (so you can't rely on the Central Limit Theorem); Lesson 3103 — Introduction to Nonparametric Methods Lesson 3111 — Choosing and Interpreting Nonparametric Tests
sample space: is the complete set of all possible outcomes of a random experiment.; Lesson 2772 — Sample Spaces: The Set of All Possible Outcomes Lesson 2773 — Finite vs Infinite Sample Spaces Lesson 2774 — Events as Subsets of the Sample Space
Sampling without replacement: from a finite population; Lesson 2884 — Introduction to the Hypergeometric Distribution
SAS: , △ ABD ≅ △ ACD.; Lesson 750 — Congruence in Isosceles and Equilateral Triangles Lesson 1102 — Introduction to the Law of Cosines Lesson 1106 — Choosing Between Law of Sines and Law of Cosines
SAS Congruence: (which you learned earlier): instead of the sides being *equal*, they must be *proportional* with the same ratio.; Lesson 754 — SAS Similarity Criterion
SAS Congruence Postulate: states that if two sides and the **included angle** (the angle formed between those two sides) of one triangle are congruent to two sides and the included angle of another triangle, then the triangles themselves are congruent.; Lesson 742 — SAS (Side-Angle-Side) Congruence Postulate
SAS Similarity Criterion: (Side-Angle-Side Similarity) states that if two triangles have one angle that is congruent *and* the two sides that form (include) that angle are proportional with the same scale factor, then the triangles are similar.; Lesson 754 — SAS Similarity Criterion
SAS situations: (Side-Angle-Side) where you know two sides and the angle sandwiched between them.; Lesson 1113 — Area Formula Using Two Sides and Included Angle
Save one factor: of tan(x)sec(x) for your du; Lesson 1602 — Powers of Tangent and Secant: Odd Powers of Tangent
Saving Plans: You deposit $50 in January, $75 in February, $100 in March, adding $25 more each month.; Lesson 659 — Applications of Arithmetic Sequences
Scalar multiplication: multiply each matrix by its coefficient; Lesson 1278 — Linear Combinations of Matrices Lesson 1794 — Vectors in 3D Space Lesson 2099 — Scalar Multiplication of Matrices Lesson 2640 — Operations on Generating Functions: Addition and Scalar Multiplication
scalar projection: tells you exactly this: given two vectors, it measures how much of one vector extends in the direction of the other.; Lesson 1258 — Scalar Projection of a Vector Lesson 1259 — Vector Projection Formula
Scale consistently: You may need to shrink very large vectors so arrows don't overlap—just keep the direction and relative size; Lesson 1968 — Sketching Vector Fields by Hand
scale factor: is a number you multiply by to make something bigger or smaller while keeping its proportions the same.; Lesson 119 — Scale Factors Lesson 727 — Scale Factor Lesson 728 — Properties of Similar Figures Lesson 752 — Definition of Similar Triangles Lesson 756 — Scale Factor in Similar Triangles Lesson 894 — Similar Solids and Surface Area Lesson 934 — Introduction to Dilations and Scale Factor
Scale factor < 1: The figure gets *smaller* (reduction); Lesson 119 — Scale Factors
Scale factor = 1: The figure stays the *same size*; Lesson 119 — Scale Factors
Scale factor > 1: The figure gets *larger* (enlargement); Lesson 119 — Scale Factors
Scale the pivot row: so the pivot becomes exactly 1 (if it isn't already); Lesson 2121 — Converting REF to RREF
Scalene Triangle: Lesson 730 — Classifying Triangles by Sides Lesson 942 — Line Symmetry (Reflectional Symmetry)
Scaling: Multiply the moduli (distances from origin); Lesson 1224 — Geometric Interpretation of Polar Operations Lesson 2058 — Geometric Interpretation of Linear Combinations Lesson 2853 — Properties of Expectation: Linearity Lesson 2864 — Moment Generating Functions (MGFs)
Scaling along axes: The diagonal matrix **Σ** stretches or compresses along these aligned directions by the singular values; Lesson 2281 — Geometric Interpretation of SVD
Scaling amplifies spread quadratically: If you convert measurements from meters to centimeters (multiplying by 100), distances between values grow by factor 100.; Lesson 2859 — Properties of Variance: Scaling and Translation
Scaling Down: Divide each quantity by the same scale factor.; Lesson 118 — Scaling Up and Down
Scaling rule: E[aX + b] = aE[X] + b; Lesson 2853 — Properties of Expectation: Linearity
Scaling Up: Multiply each quantity by the same scale factor.; Lesson 118 — Scaling Up and Down
Scan the equation: for fractions or rational expressions; Lesson 549 — Finding Domain from Equations
Scan the matrix first: look for zeros, patterns, or near-triangular structure; Lesson 2158 — Practice: Mixed Determinant Computation
Scatter plots: show relationships between two variables and correlation strength; Lesson 2997 — Choosing Appropriate Summaries and Displays
Scenario 1: Selecting 3 students from a class of 10 to form a study group.; Lesson 2594 — Distinguishing When to Use Combinations vs Permutations Lesson 2604 — Pascal's Identity and Recursive Properties
Scenario 2: Awarding gold, silver, and bronze medals to 3 runners from 10 finalists.; Lesson 2594 — Distinguishing When to Use Combinations vs Permutations Lesson 2604 — Pascal's Identity and Recursive Properties
Scheduling: Assign time slots to exams so students taking multiple exams don't have conflicts; Lesson 2695 — Graph Coloring and the Chromatic Number Lesson 2752 — Systems of Linear Congruences
Search: for an augmenting path starting from any unmatched vertex in one partition; Lesson 2702 — The Hungarian Algorithm for Maximum Bipartite Matching Lesson 2706 — The Ford- Fulkerson Algorithm
Seating Arrangements: An auditorium has 20 seats in the first row, 24 in the second, 28 in the third, and so on.; Lesson 659 — Applications of Arithmetic Sequences
Secant: Asymptotes wherever cosine equals zero: **x = π/2 + nπ**; Lesson 1032 — Comparing and Analyzing All Six Trig Functions Lesson 1604 — Reduction Formulas for Tangent and Secant
Secant (sec): is the reciprocal of cosine: `sec(θ) = 1/cos(θ)`; Lesson 1009 — Reciprocal Trigonometric Functions: Secant, Cosecant, Cotangent
Secant & Cosecant: Unbounded; U-shaped curves with values ≥1 or ≤-1; Lesson 1032 — Comparing and Analyzing All Six Trig Functions
Secant and Cosecant: Like their reciprocals (cosine and sine), these have a period of **2π**.; Lesson 1032 — Comparing and Analyzing All Six Trig Functions
secant identity: saves us.; Lesson 1442 — Derivative of Arctangent Lesson 1612 — The Three Standard Forms
secant line: .; Lesson 1369 — Understanding Average Rate of Change Lesson 1373 — From Secant Lines to Tangent Lines Lesson 1377 — Average Rate of Change
Second: Reflect across the x-axis; Lesson 929 — Composition of Translations and Reflections Lesson 972 — Multi-Step Elevation and Depression Problems Lesson 1397 — Higher-Order Derivatives
Second column: Get a 1 in position (2,2) and zeros above and below it; Lesson 1305 — Row Reduction to Find Inverses of 3×3 Matrices
second derivative: .; Lesson 1397 — Higher-Order Derivatives Lesson 1497 — Introduction to Concavity Lesson 1498 — The Second Derivative and Concavity Lesson 1499 — Finding Intervals of Concavity Lesson 1501 — The Second Derivative Test for Local Extrema Lesson 1504 — Applications: Real-World Monotonicity and Concavity Lesson 1521 — Second Derivative and Concavity
Second Derivative Test: offers a shortcut: once you find a critical point c (where f'(c) = 0), you can evaluate the **second derivative** f''(c) to classify it immediately.; Lesson 1501 — The Second Derivative Test for Local Extrema Lesson 1502 — Comparing First and Second Derivative Tests Lesson 1515 — Optimization on Open Intervals Lesson 1523 — The Second Derivative Test for Extrema Lesson 1525 — Comprehensive Curve Sketching Algorithm Lesson 1863 — Classifying Critical Points: Local Extrema and Saddle Points Lesson 1866 — The Second Derivative Test: Statement and Conditions Lesson 1867 — Applying the Second Derivative Test
second factor: is a trinomial with three parts:; Lesson 339 — Sum and Difference of Cubes Lesson 1338 — Combining Multiple Limit Laws
Second group: `(4x² + 6x)`; Lesson 376 — Factoring Out the GCF from Each Group
Second leading principal minor: (full 2×2): det(A) = 2·3 − (−1)(−1) = 6 − 1 = 5 > 0; Lesson 2294 — Principal Minors and Sylvester's Criterion
Second moment: Lesson 2961 — Finding Moments from MGFs
second number: (y) is called the **y-coordinate** – it shows vertical position; Lesson 236 — Understanding Ordered Pairs and Coordinates Lesson 1221 — Multiplying Complex Numbers in Polar Form
second position: , you have **(n-1) choices** (any remaining book); Lesson 2579 — Counting Permutations of n Distinct Objects Lesson 2580 — Permutations of n Objects Taken r at a Time
Second relationship: "one is 8 more than the other" → x = y + 8; Lesson 282 — Application Problems Using Systems and Substitution
Second rotation: (or reflection): The matrix **U** rotates the scaled result into the output space; Lesson 2281 — Geometric Interpretation of SVD
Second term: ∫(-5x) dx = -5 · (x²/2) = -(5x²)/2; Lesson 1542 — Antiderivatives of Polynomial Functions
Second term times everything: Multiply *2* by each of the three terms: *2·x*, *2·1*, *2·4*; Lesson 329 — Multiplying Two Trinomials
Second-order PDE: Contains at least one second derivative like ∂²u/∂x² or ∂²u/∂x∂y; Lesson 2409 — Classification: Order and Linearity
Second-order tests: (when applicable): These can sometimes classify extrema, though they're trickier with constraints.; Lesson 1891 — Verifying and Interpreting Solutions
Second-stage branches: emanate from each first-stage outcome, labeled with *conditional* probabilities; Lesson 2796 — Sequential Experiments and Tree Diagrams
sector: of the circular pizza—it's the region bounded by two radii and the arc connecting them.; Lesson 842 — Sector Area Formula Lesson 843 — Segment Area Lesson 846 — Applications of Arcs and Sectors Lesson 864 — Area of Sectors and Arc Length Lesson 988 — Sector Area Using Radians Lesson 1917 — Integrating over Sectors and Wedges
sector area: is that same fraction of the circle's total area; Lesson 864 — Area of Sectors and Arc Length Lesson 865 — Area of Segments
Segment area: = 52.; Lesson 865 — Area of Segments
Seismology: Earthquake wave analysis through Earth's interior; Lesson 2434 — The Three-Dimensional Wave Equation
Select specific solutions: by choosing initial conditions (a point the curve must pass through); Lesson 2302 — Solutions and Solution Curves
semiperimeter: (half the perimeter):; Lesson 1115 — Heron's Formula: Introduction and Derivation Lesson 1117 — Area from Inradius and Semiperimeter
Sensitivity: (True Positive Rate): P(Positive Test | Disease Present) — how good the test is at detecting the disease when you have it; Lesson 2818 — Medical Testing and Screening Problems
separable: integrands like f(x,y) = x²y, which splits neatly into g(x)·h(y) = x² · y, letting you evaluate:; Lesson 1899 — Double Integrals of Non-Separable Functions Lesson 2334 — Special Cases: n=0 and n=1 in Bernoulli Equations Lesson 2336 — Strategy Guide: Choosing the Right Solution Method
separable equation: .; Lesson 2316 — Applications: Exponential Growth and Decay Lesson 2375 — Logistic Growth Model
Separate and integrate: the ODE to find the general solution (with constant *C*); Lesson 2311 — Initial Value Problems with Separable ODEs
Separate into standard form: Write the result as *a + bi*; Lesson 1206 — Rationalization and Standard Form
Separate the terms: using the sum/difference rule; Lesson 1542 — Antiderivatives of Polynomial Functions
separate variables: in a differential equation and rewrite it so each side can be integrated independently.; Lesson 2309 — The Separation of Variables Method Lesson 2431 — Fourier Series Solutions for Bounded Domains
separately: , not as a single quotient.; Lesson 1529 — Applying L'Hôpital's Rule to 0/0 Forms Lesson 1766 — Interval of Convergence
Separating logs: If you have `log(x²) - log(x) = 2`, use the power and quotient rules:; Lesson 629 — Applying Logarithm Properties to Solve Equations
separation constant: , often denoted `λ` or `-λ`).; Lesson 2445 — What Is Separation of Variables?Lesson 2452 — Separation of Variables for the Wave Equation
separation of variables: technique assumes that a solution to a PDE can be written as a *product* of functions, each depending on only one variable.; Lesson 2419 — The Method of Separation of Variables Lesson 2439 — Separation of Variables for Laplace's Equation Lesson 2447 — Separating Variables in the Heat Equation Lesson 2466 — Fourier Series and Partial Differential Equations
sequences: ordered lists of numbers like 2, 4, 6, 8, 10.; Lesson 677 — What is a Series?Lesson 694 — Signal Processing and Sampling Lesson 2574 — Pigeonhole Principle with Subsets and Sequences
sequential updating: or **Bayesian updating**.; Lesson 2820 — Sequential Updating with Bayes' Theorem Lesson 3114 — Conjugate Priors: Beta-Binomial Model
Series like Σ(2ⁿ/n!): The ratio gives 2/(n+1) → 0 < 1, so it converges; Lesson 1743 — Applying the Ratio Test to Common Series
Series like Σ(n²/3ⁿ): The ratio simplifies to (n+1)²/(3n²) → 1/3 < 1, convergence; Lesson 1743 — Applying the Ratio Test to Common Series
Series like Σ(nⁿ/n!): The ratio involves (n+1)ⁿ ¹·n!; Lesson 1743 — Applying the Ratio Test to Common Series
Series with nth powers: For \(\sum \left(\frac{2n+1}{3n-2}\right)^n\), taking the nth root immediately gives you \ (\frac{2n+1}{3n-2}\), which simplifies to \(L = \frac{2}{3} < 1\) → converges!; Lesson 1746 — Applying the Root Test
Set f'(x) = 0: and solve for x (these are critical points); Lesson 1507 — Critical Points in Optimization
Set form: "All elements satisfying P also satisfy Q"; Lesson 2489 — Structure of Mathematical Statements
Set the exponents equal: to each other; Lesson 609 — Solving Simple Exponential Equations
Set the functions equal: f(x) = g(x); Lesson 1646 — Finding Intersection Points
Set them equal: E[X] = m₁, E[X²] = m₂, and so on for k equations; Lesson 3003 — Method of Moments for Multiple Parameters
Set up: the expression from the word problem; Lesson 458 — Applications of Multiplying and Dividing Rationals Lesson 475 — Applications and Word Problems with Complex Fractions Lesson 747 — CPCTC: Corresponding Parts of Congruent Triangles Lesson 1305 — Row Reduction to Find Inverses of 3×3 Matrices Lesson 1597 — Cyclic Integration by Parts Lesson 1623 — Linear Factors: Distinct Denominators Lesson 1698 — Applications of Average Value Lesson 2040 — Applying the Theorem to a Sphere
Set up a proportion: Place a stick vertically in the ground and measure its height and shadow length; Lesson 761 — Applications and Proofs with Similar Triangles
Set up an equation: based on the problem's condition; Lesson 192 — Age Problems and Consecutive Integer Problems
Set up an inequality: radicand ≥ 0; Lesson 550 — Domain Restrictions from Square Roots
Set up and evaluate: the integral; Lesson 1695 — Pumping Liquids from Tanks
Set up coordinates: for both objects' positions; Lesson 1482 — Distance and Position Related Rates
Set up equations: based on those relationships; Lesson 712 — Solving Multi-Step Angle Problems
Set up hypotheses: Your null hypothesis typically states H₀: μ = μ₀ (some claimed value), while your alternative might be H₁: μ ≠ μ₀ (two-sided), H₁: μ > μ₀, or H₁: μ < μ₀; Lesson 3025 — One-Sample z-Test for Means Lesson 3048 — F Test for Equality of Variances
Set up limits: Establish integration bounds for your chosen system; Lesson 1956 — Computing Total Mass with Triple Integrals
Set up separate integrals: for each region; Lesson 1647 — Area with Multiple Regions Lesson 1650 — Area Enclosed by Intersecting Curves Lesson 1909 — Regions Requiring Multiple Integrals
Set up the constraint: Express the area requirement as an equation (e.; Lesson 1510 — Geometric Optimization: Minimizing Perimeter
Set up the division: just like you would with whole numbers; Lesson 90 — Dividing Decimals by Whole Numbers
Set up the equation: with the new scenario; Lesson 130 — Real-World Inverse Variation Problems Lesson 400 — Applications and Word Problems Lesson 2130 — The Null Space
Set up the inequality: `M · |x - c|^(n+1) / (n+1)!; Lesson 1787 — Determining Required Degree for Given Accuracy
Set up the integral: with respect to y:; Lesson 1648 — Area Between Curves: Horizontal Slicing Lesson 1652 — Applied Problems: Area Between Curves Lesson 1657 — Washer Method: Introduction and Setup Lesson 1932 — Applications: Density and Total Mass
Set up the system: $A\mathbf{c} \approx \mathbf{y}$ where $\mathbf{y}$ contains your data values; Lesson 2239 — Applications: Data Fitting and Approximation
Set up two columns: one for the polynomial part (which you'll differentiate repeatedly) and one for the exponential or trig part (which you'll integrate repeatedly).; Lesson 1596 — Tabular Integration
Set up your bounds: for r, θ, and z based on the region's geometry; Lesson 1937 — Evaluating Triple Integrals Using Cylindrical Coordinates
Set up your equation: – Substitute known values into the formula; Lesson 193 — Perimeter and Area Word Problems
Set up your proportion: with matching units; Lesson 123 — Real-World Proportion Problems
Setting up the integral: The shell method formula adapts as follows:; Lesson 1671 — Shell Method with Functions of y
Setup: ⅔ + ¾; Lesson 82 — Word Problems Involving Fraction Operations Lesson 2726 — Uniqueness of Prime Factorization: The Proof
Setup Phase: Lesson 2769 — RSA Example with Small Primes
Shape: The curve is always increasing, always concave up (curves upward), and passes through (0, 1).; Lesson 612 — Properties of e^x: Growth and Behavior Lesson 1018 — Vertical Shift: y = sin(x) + D and the Midline Lesson 1806 — Graphing Surfaces in 3D Lesson 2992 — Histograms and Frequency Distributions Lesson 2993 — Shape of Distributions: Symmetry and Skewness Lesson 2994 — Comparing Distributions Lesson 3042 — Chi-Squared Distribution Basics
Shape (α): Controls the "shape" of the distribution.; Lesson 2917 — The Gamma Distribution: Definition and Parameters
Shared boundaries: If you have a surface that's part of a closed boundary, you might use Stokes' on the open part and the Divergence Theorem on the entire closed region.; Lesson 2045 — Combining with Stokes' Theorem and Applications
Shared parts are key: A common side belongs to both triangles (reflexive property).; Lesson 749 — Proving Overlapping Triangles Congruent
Shared side: One ray or segment must be part of both angles.; Lesson 705 — Adjacent Angles
Sharing: means giving each friend the same number of cookies.; Lesson 23 — Understanding Division as Sharing and Grouping
shell method: uses this approach: you take a vertical (or horizontal) slice of your region, rotate it around an axis, and it sweeps out a hollow cylinder—a "shell.; Lesson 1664 — Introduction to the Shell Method Lesson 1665 — Setting Up a Shell Integral Lesson 1669 — Revolving Around Horizontal Lines (y = k)
Shortest side × √3: = medium side (opposite 60°); Lesson 795 — Finding Missing Sides in 30-60-90 Triangles
Shortest side × 2: = hypotenuse; Lesson 795 — Finding Missing Sides in 30-60-90 Triangles
Show your work step-by-step: Don't try to do everything mentally.; Lesson 540 — Evaluating Functions with Numeric Inputs
SI units: Force in newtons (N), distance in meters (m) → Work in joules (J); Lesson 1692 — Work Done by a Constant Force
Sides: The two rays that form the angle; Lesson 703 — Measuring Angles and Angle Notation Lesson 725 — Corresponding Parts of Congruent Figures
Sierpiński's Triangle: , one of mathematics' most famous fractals.; Lesson 2606 — Binomial Coefficients Modulo 2 and Sierpiński's Triangle
Sieve of Eratosthenes: for primes—crossing out multiples systematically.; Lesson 2623 — The Sieve Method in Practice
Sigma notation: (using the Greek letter Σ) gives us a compact way to write these series without listing every term.; Lesson 678 — Finite Series and Sigma Notation
sign: of a slope tells you exactly which way a line tilts on the coordinate plane.; Lesson 250 — Slope and Direction of Lines Lesson 422 — Parabola Basics: Shape and Orientation Lesson 3075 — Interpretation of Regression Coefficients
Sign doesn't matter: gcd(*a*, *b*) = gcd(|*a*|, |*b*|); Lesson 2711 — Greatest Common Divisor (GCD): Definition Lesson 3062 — The Least Squares Criterion
Sign errors: during multiplication and division; Lesson 184 — Evaluating with Negative Numbers
Sign in Quadrant II: sine is positive; Lesson 1005 — Evaluating Trig Functions for Angles Greater than 90°
signed area: .; Lesson 1557 — The Definite Integral as Signed Area Lesson 1573 — Area and Signed Area Lesson 1643 — Area Between a Curve and the x-axis
significance level: , denoted **α** (alpha), is the probability threshold you set *before* collecting data.; Lesson 3023 — Significance Levels and Critical Regions Lesson 3055 — Significance Level and Error Control
Significance level (α): typically 0.; Lesson 3035 — Calculating the t-Statistic and Finding Critical Values
significance level α: (alpha) is the probability threshold we set *before* conducting a test.; Lesson 3052 — Type I Error: False Positives Lesson 3053 — Type II Error: False Negatives
Signs matter: Lesson 384 — Perfect Square Trinomials: Recognition
similar: if they have the same shape but not necessarily the same size.; Lesson 121 — Similar Figures and Proportions Lesson 726 — Introduction to Similarity Lesson 728 — Properties of Similar Figures Lesson 729 — Distinguishing Congruence from Similarity Lesson 759 — Parallel Lines and Triangle Similarity Lesson 2177 — Similar Matrices and Eigenvalue Invariance Lesson 2187 — Similar Matrices Have the Same Characteristic Polynomial Lesson 2256 — Similarity Transformations
Similar figures: (~): same shape, but possibly different sizes; Lesson 726 — Introduction to Similarity
Similar in form: to make the inequality easy to verify; Lesson 1736 — Direct Comparison Test: Proving Convergence
Similar triangle ratios: Proportional sides when triangles maintain shape; Lesson 1479 — Related Rates with Triangles
Similar triangles: are different: they have the *same shape* but can be *different sizes*.; Lesson 752 — Definition of Similar Triangles Lesson 1481 — Ladder and Shadow Problems
Similarity: means two figures have the same shape but *different sizes*.; Lesson 729 — Distinguishing Congruence from Similarity
similarity transformation: .; Lesson 2255 — Change of Basis for Linear Transformations Lesson 2256 — Similarity Transformations
Simple case: Lesson 459 — Adding Rationals with Like Denominators
simple event: Lesson 2774 — Events as Subsets of the Sample Space Lesson 2775 — Simple Events vs Compound Events
Simple events: Lesson 2775 — Simple Events vs Compound Events
Simple Linear Arguments: Lesson 1610 — Strategy Selection for Trigonometric Integrals
Simpler algebra: Terms like (x - 0) just become x; Lesson 1774 — Maclaurin Series: Taylor Series at Zero
Simpler antiderivatives: If integrating with respect to `x` first gives you a messy expression, try `y` first instead; Lesson 1900 — Choosing Integration Order Strategically
Simplicity: They convert event-based questions into arithmetic.; Lesson 2830 — Indicator Random Variables
Simplification: Starting at n=0 or n=1 can make formulas cleaner; Lesson 1720 — Reindexing and Shifting Series
Simplified expression, restricted domain: Lesson 450 — Writing Simplified Expressions with Domain Restrictions
Simplified form: 3x² + 3x + 7; Lesson 332 — Simplifying Products by Combining Like Terms
Simplified norm calculations: ||x||² = sum of squared coordinates; Lesson 2211 — Orthogonal Bases
Simplified ratio: 3:4: Lesson 107 — Simplifying Ratios
Simplify: the resulting fraction if possible; Lesson 75 — Multiplying Fractions Lesson 76 — Multiplying Mixed Numbers Lesson 80 — Complex Fractions Lesson 185 — Evaluating Expressions with Fractions Lesson 209 — Equations with Fractions: Clearing Denominators Lesson 245 — Calculating Slope from Two Points Lesson 342 — Using Special Products to Simplify Expressions Lesson 421 — Verifying Solutions by Substitution (+21 more)
Simplify aggressively: before integrating; Lesson 1677 — Arc Length with Radical and Rational Functions
Simplify and rearrange: to match `(x - h)² = 4p(y - k)`; Lesson 1150 — Converting to Standard Form by Completing the Square
Simplify at each stage: before moving forward; Lesson 519 — Complex Denominators with Multiple Radicals
Simplify boundary descriptions: (from curves to constants); Lesson 1944 — Motivation for Change of Variables in Multiple Integrals
Simplify by factoring: Lesson 520 — Simplifying After Rationalizing
Simplify calculations: by breaking complex intervals into easier pieces; Lesson 1560 — Properties of Definite Integrals: Intervals Lesson 2907 — Standardization and Z-Scores
Simplify completely: using your reference triangle; Lesson 1619 — Combining Trig Substitution with Other Techniques
Simplify each piece: by extracting what comes out of the radical; Lesson 504 — Combining Simplification Techniques
Simplify each radical: using prime factorization or recognizing perfect squares; Lesson 507 — Simplifying Before Adding or Subtracting
simplify first: before you can see like radicals:; Lesson 158 — Adding and Subtracting Radicals Lesson 186 — Simplifying Before Evaluating
Simplify if needed: – Reduce the fraction to lowest terms; Lesson 69 — Adding Fractions with Like Denominators Lesson 70 — Subtracting Fractions with Like Denominators Lesson 1400 — Applying the Product Rule to Polynomials
Simplify it completely: using LCD or division methods you've learned; Lesson 474 — Nested Complex Fractions
Simplify powers of i: using the pattern i, -1, -i, 1; Lesson 1202 — Simplifying Complex Expressions
Simplify step-by-step: Lesson 414 — Solving Quadratics When a ≠ 1
Simplify the angle: Make sure nθ is in standard position (adjust by ±2π if needed); Lesson 1226 — Using De Moivre's Theorem to Compute Powers
Simplify the arithmetic: under the square root and in the denominator; Lesson 414 — Solving Quadratics When a ≠ 1
Simplify the constant outside: Distribute any changes and combine constants; Lesson 406 — Converting to Vertex Form
Simplify the denominator: into a single fraction; Lesson 471 — Simplifying Complex Fractions by Simplifying Numerator and Denominator
Simplify the numerator: Perform any necessary multiplication in the numerator; Lesson 462 — Building Equivalent Rationals Lesson 471 — Simplifying Complex Fractions by Simplifying Numerator and Denominator
Simplify the perfect square: by taking its square root outside; Lesson 782 — Simplifying Radical Answers
Simplify the result: if possible; Lesson 508 — Multiplying Radicals with the Same Index
Simplify using log properties: Break down products, quotients, and powers; Lesson 1459 — Logarithmic Differentiation: Basic Technique Lesson 1462 — Variable Bases and Variable Exponents
Simplify using other identities: reciprocal, quotient, or Pythagorean identities; Lesson 1056 — Verifying Identities with Sum and Difference Formulas
Simplifying: (6x³ - x² + 4) ÷ (2x + 1) converts a complex rational expression into quotient + remainder/divisor form, revealing its structure more clearly.; Lesson 354 — Applications of Polynomial Division
Simplifying (x⁵): Lesson 157 — Simplifying Higher-Order Radicals
Simplifying 54: Lesson 157 — Simplifying Higher-Order Radicals
Simplifying 80: Lesson 157 — Simplifying Higher-Order Radicals
Simplifying calculations: You only need to estimate one parameter (λ) to know both center and spread; Lesson 2891 — Mean and Variance of the Poisson Distribution
Simplifying Complex Fractions: Lesson 1341 — Algebraic Manipulation for 0/0
Simply connected: A solid disk, all of ℝ², a sphere's interior—no holes, no obstacles; Lesson 1990 — Simply Connected Domains and Why They Matter
simply connected domain: is a region with no "holes"—you can shrink any closed loop to a point without leaving the region.; Lesson 1987 — Testing for Conservative Fields: The Curl Test in ℝ³ Lesson 1990 — Simply Connected Domains and Why They Matter
Simpson's Rule: Lesson 1681 — Numerical Approximation of Arc Length
sin(x): and **cos(x)**:; Lesson 1014 — Period of Sine and Cosine Functions Lesson 1421 — Derivatives of sin(x) and cos(x): Pattern Recognition
sin(x) and cos(x): have period **2π**; Lesson 1083 — General Solutions Using Period Lesson 1584 — u-Substitution with Trigonometric Functions Lesson 1790 — Applications to Common Functions
Since: 2k² is an integer, n² has the form 2m (where m = 2k²); Lesson 2491 — Direct Proof: Basic Structure
Sine: = opposite side ÷ hypotenuse; Lesson 947 — Introduction to Trigonometric Ratios Lesson 949 — The Sine Ratio (SOH)Lesson 974 — Setting Up Right Triangle Problems from Word Descriptions Lesson 1427 — Higher-Order Derivatives of Trigonometric Functions
Sine (SOH): Use when working with **opposite** and **hypotenuse**; Lesson 956 — Choosing the Appropriate Trig Ratio
Sine & Cosine: Bounded between -1 and 1; smooth oscillation; Lesson 1032 — Comparing and Analyzing All Six Trig Functions
Sine and Cosine: Both have a period of **2π** (360°).; Lesson 1032 — Comparing and Analyzing All Six Trig Functions
Sine and cosine functions: These classic trigonometric functions repeat every `2π` units, so their period is `2π`; Lesson 570 — Periodic Functions
Sine is Sneaky: " (odd—it flips sign).; Lesson 1048 — Even-Odd Identities
Sine series: Extend oddly, compute only `b ` coefficients.; Lesson 2462 — Half-Range Expansions
Single value: If p(3) = 0.; Lesson 2834 — Computing Probabilities from PMFs
singular: .; Lesson 2138 — Definition of Matrix Inverse Lesson 2183 — Roots of the Characteristic Polynomial as Eigenvalues Lesson 2401 — Regular Singular Points and the Method of Frobenius
Singular (not invertible): det(A) = 0, no inverse exists; Lesson 1301 — When Does an Inverse Exist?
Singular continuous: Spread over uncountably many points, yet assigning zero probability to any interval (rare and pathological); Lesson 2831 — Mixed and Other Types of Random Variables
Singular Value Decomposition (SVD): is a factorization that breaks *any* matrix **A** (m×n) into three components:; Lesson 2280 — What is the Singular Value Decomposition?
Singular values: are the square roots of the eigenvalues of the matrix **A^T A** (or equivalently, **A^H A** for complex matrices).; Lesson 2282 — Singular Values and Their Meaning Lesson 2286 — SVD and the Matrix Rank
Sink (t): A special vertex where flow terminates (like a drain); Lesson 2704 — Networks and Flows: Definitions and Notation
Sinusoidal forcing: If **f**(t) = [cos(2t), sin(2t)]ᵀ, guess **x_p** = **a**cos(2t) + **b**sin(2t).; Lesson 2386 — Non-Homogeneous Systems: Undetermined Coefficients
Sinusoidal resonance: If your trial includes *cos(ωx)* or *sin(ωx)* and these appear in the complementary solution, multiply the entire trial by *x*.; Lesson 2354 — The Modification Rule for Resonance
size: of `a` affects how "wide" or "narrow" the parabola appears, but the **sign** of `a` alone determines the orientation.; Lesson 422 — Parabola Basics: Shape and Orientation Lesson 921 — Introduction to Geometric Transformations Lesson 1271 — What is a Matrix? Dimensions and Notation
Size formula: If |A| = m and |B| = n, then |A × B| = m × n.; Lesson 2535 — Computing Cartesian Products
Size matters: If you choose small primes like `p = 61` and `q = 53`, factoring `n = 3233` takes milliseconds.; Lesson 2770 — Security and Practical Considerations
Sketch: Draw the parabola opening toward the focus, symmetric about the axis.; Lesson 1151 — Graphing Parabolas from Standard Form Lesson 1652 — Applied Problems: Area Between Curves Lesson 1657 — Washer Method: Introduction and Setup
Sketch and label: – Draw the shape with given measurements (feet, meters, etc.; Lesson 873 — Composite Figures in Real-World Contexts
Sketch and Understand: Lesson 1673 — Applications and Problem-Solving Strategies
Sketch or visualize: the 3D region bounded by your surfaces; Lesson 1929 — Triple Integrals over General 3D Regions
Sketch the branches: between asymptotes, descending from left to right; Lesson 1027 — Transformations of Cotangent
Sketch the circle: connecting these four points smoothly; Lesson 1137 — Graphing Circles from Standard Form
Sketch the framework: Lesson 1040 — Graphing with All Transformations Combined
Sketch the hyperbola branches: starting at each vertex, curving toward the asymptotes; Lesson 1171 — Graphing Hyperbolas Centered at the Origin
Sketch the region: (even roughly) to visualize which intersection points are relevant; Lesson 1656 — Finding Limits of Integration for Disk Method Lesson 1665 — Setting Up a Shell Integral Lesson 1905 — Changing the Order of Integration Lesson 1928 — Changing the Order of Integration in Triple Integrals
Sketching inverses: You can sketch an approximate inverse graph by reflecting key points across y = x; Lesson 565 — Graphs of Inverse Functions
Skew < 0: Left-skewed (long left tail, mean < median); Lesson 2861 — Higher Moments: Skewness and Kurtosis
Skew = 0: Symmetric (like the normal distribution); Lesson 2861 — Higher Moments: Skewness and Kurtosis
Skew > 0: Right-skewed (long right tail, mean > median); Lesson 2861 — Higher Moments: Skewness and Kurtosis
Skew lines: They don't intersect and aren't parallel.; Lesson 700 — Intersection of Lines and Planes
Skew-symmetric: and **skew-Hermitian** matrices are normal; Lesson 2276 — The Spectral Theorem for Normal Matrices
Skewness: measures *asymmetry*: Does the distribution lean more to one side?; Lesson 2861 — Higher Moments: Skewness and Kurtosis Lesson 2863 — Higher Moments: Skewness and Kurtosis
Skewness < 0: Left-skewed (negative skew)—the left tail is longer, mass concentrates on the right; Lesson 2863 — Higher Moments: Skewness and Kurtosis
Skewness = 0: Symmetric distribution (like the normal distribution); Lesson 2863 — Higher Moments: Skewness and Kurtosis
Skewness > 0: Right-skewed (positive skew)—the right tail is longer, mass concentrates on the left; Lesson 2863 — Higher Moments: Skewness and Kurtosis
Skewness check: If the population is strongly skewed or has extreme outliers, you may need n ≥ 50 or even n ≥ 100; Lesson 2986 — Rule of Thumb for CLT Applicability
Slant asymptote exists: when: degree of P = degree of Q + 1; Lesson 1524 — Slant (Oblique) Asymptotes
slant asymptotes: that you've already studied.; Lesson 492 — End Behavior of Rational Functions Lesson 1525 — Comprehensive Curve Sketching Algorithm
Slanted triangle: Height might fall *outside* the triangle when using certain bases; Lesson 1112 — Area Formula Using Base and Height
Slice area: (sector area): How much pizza surface you're getting; Lesson 846 — Applications of Arcs and Sectors
Slope: tells you how steep that hill is by answering: "For every step I take forward (horizontally), how much do I go up or down (vertically)?; Lesson 244 — Rise Over Run: Understanding Slope Geometrically Lesson 248 — Finding Slope from a Graph Lesson 251 — Average Rate of Change vs Constant Rate Lesson 254 — Slope-Intercept Form: y = mx + b Lesson 255 — Graphing Lines from Slope-Intercept Form Lesson 951 — The Tangent Ratio (TOA)Lesson 3063 — Calculating the Regression Coefficients Lesson 3064 — Interpreting Slope and Intercept
Slope-intercept: `y = mx + b`; Lesson 261 — Converting Between Forms of Linear Equations
slope-intercept form: is a special way to write linear equations that makes important features instantly visible.; Lesson 254 — Slope-Intercept Form: y = mx + b Lesson 264 — Choosing the Best Form for a Problem Lesson 911 — Slope-Intercept Form in the Coordinate Plane Lesson 912 — Point-Slope Form and Writing Line Equations
Small `d` attack: If your private exponent is too small, the Wiener attack can recover it.; Lesson 2770 — Security and Practical Considerations
smaller: .; Lesson 219 — The Multiplication/Division Sign Rule Lesson 1736 — Direct Comparison Test: Proving Convergence Lesson 1769 — Operations on Power Series
smallest: common multiple.; Lesson 54 — Least Common Multiple (LCM)Lesson 309 — Applications: Minimization Problems
Smoothness: Because *u* at each point is an average of nearby values, harmonic functions cannot have sudden jumps or sharp corners—they're infinitely differentiable.; Lesson 2444 — Mean Value Property and Applications , is the sum of just the first n terms: Lesson 1713 — Definition of an Infinite Series
Social networks: An influencer who is the *only* link between two friend groups.; Lesson 2661 — Cut Vertices and Cut Edges
Socks in a drawer: You have 10 pairs of socks (5 black, 5 blue) mixed randomly.; Lesson 2570 — The Pigeonhole Principle: Statement and Intuition
SOH-CAH-TOA: Lesson 947 — Introduction to Trigonometric Ratios Lesson 949 — The Sine Ratio (SOH)Lesson 951 — The Tangent Ratio (TOA)
SOH, CAH, or TOA: based on what you know and what you're finding; Lesson 970 — Using SOH-CAH-TOA with Depression Angles
Solution: All numbers between -3 and 7 (not including -3 and 7).; Lesson 233 — Absolute Value Inequalities of the Form |x| < a Lesson 277 — Introduction to the Substitution Method Lesson 585 — Writing Transformed Functions from Descriptions Lesson 655 — Finding the First Term from Given Information Lesson 1878 — Solving Two-Variable Constrained Problems Lesson 2302 — Solutions and Solution Curves Lesson 2571 — Simple Applications of the Pigeonhole Principle Lesson 2582 — Permutations with Restrictions: Fixed Positions
Solution 1: y₁(x) = xʳ¹ Σ(n=0 to ∞) a xⁿ; Lesson 2403 — Frobenius Solutions with Distinct Roots
Solution 2: y₂(x) = xʳ² Σ(n=0 to ∞) b xⁿ; Lesson 2403 — Frobenius Solutions with Distinct Roots
Solution approach: Lesson 2812 — Applying the Law of Total Probability
Solution curves: follow the flow of the field, always staying tangent to the segments; Lesson 2305 — Direction Fields
solution set: is the collection of all values that work.; Lesson 197 — Solution Sets and Checking Solutions Lesson 227 — Solving Basic Absolute Value Equations
Solutions: The number of pivots tells you how many independent equations you have; Lesson 2117 — Leading Entries and Pivot Positions
Solutions when assumptions fail: combine categories (if sensible), collect more data, or use Fisher's exact test (for 2×2 tables) or other exact methods.; Lesson 3046 — Expected Frequencies and Assumptions
Solve: 8/12 + 9/12 = 17/12 = 1 5/12 miles; Lesson 82 — Word Problems Involving Fraction Operations Lesson 130 — Real-World Inverse Variation Problems Lesson 192 — Age Problems and Consecutive Integer Problems Lesson 193 — Perimeter and Area Word Problems Lesson 216 — Mixed Practice: Complex Linear Equations Lesson 233 — Absolute Value Inequalities of the Form |x| < a Lesson 235 — Applications and Mixed Absolute Value Problems Lesson 403 — Solving Quadratics by Completing the Square (a = 1) (+15 more)
Solve algebraically: Move all terms containing I to one side, then divide; Lesson 1597 — Cyclic Integration by Parts
Solve and Check: Lesson 982 — Mixed Application Problems and Problem-Solving Strategies
Solve by factoring: – get everything to one side (standard form), factor, and use the Zero Product Property; Lesson 400 — Applications and Word Problems
Solve division problems: Don't know 56 ÷ 8?; Lesson 24 — The Relationship Between Multiplication and Division
Solve each piece: x = -4 or x = 4; Lesson 398 — Solving Special Forms: Difference of Squares
Solve each sub-problem: Use separation of variables on each (now manageable) piece; Lesson 2440 — Superposition and Non-Homogeneous Boundaries
Solve each triangle separately: Lesson 1124 — Solving for Two Possible Triangles
Solve easily: Each component **y ᵢ** satisfies **dyᵢ/dt = λᵢyᵢ**, giving **yᵢ(t) = yᵢ(0)e^(λ ᵢt)**; Lesson 2278 — Applications to Differential Equations and Dynamics
Solve for ∂z/∂x: Lesson 1829 — Implicit Partial Differentiation Lesson 1856 — Implicit Differentiation in Multiple Variables
Solve for angle B: using the Law of Sines; Lesson 1099 — Two Solutions Case in SSA
Solve for C: algebraically; Lesson 2324 — Initial Value Problems for Linear ODEs
Solve for dy/dx: dy/dx = y · [result from step 4]; Lesson 1435 — Derivatives of Exponential Expressions with Variable Bases Lesson 1449 — The Basic Technique of Implicit Differentiation Lesson 1459 — Logarithmic Differentiation: Basic Technique Lesson 1460 — Differentiating Products with Many Factors Lesson 1462 — Variable Bases and Variable Exponents Lesson 1463 — Logarithmic Differentiation with Radicals
solve for n: .; Lesson 657 — Finding the Number of Terms in a Finite Sequence Lesson 1787 — Determining Required Degree for Given Accuracy
Solve for roots: The roots are your eigenvalues.; Lesson 2188 — Using the Characteristic Polynomial in Practice
Solve for that variable: directly; Lesson 1312 — Back Substitution from Echelon Form
Solve for the unknown: Use algebra to isolate and calculate the missing length; Lesson 757 — Finding Missing Sides Using Similarity
Solve for the variable: Lesson 477 — Finding Restricted Values and Domain Lesson 551 — Domain Restrictions from Rational Expressions Lesson 615 — Natural Exponential Equations Lesson 638 — Exponential and Logarithmic Equations with Quadratic Form
Solve for x: Lesson 405 — Solving General Quadratics by Completing the Square Lesson 409 — Complex Solutions via Completing the Square Lesson 1646 — Finding Intersection Points
Solve for y: isolate y on one side of the equation; Lesson 563 — Finding Inverse Functions Algebraically Lesson 2323 — Solving Non-Homogeneous Linear ODEs Lesson 2325 — Linear ODEs with Constant Coefficients
Solve for δ: Lesson 1364 — Proving Simple Linear Limits Using ε-δ
Solve for θ: this gives your MLE, θ̂; Lesson 3005 — Finding MLEs by Differentiation
Solve for λ: by factoring or using the cubic formula (though factoring is usually easier in practice problems); Lesson 2178 — Computing Eigenvalues for 3×3 Matrices
Solve in order: find simpler angles before complex ones; Lesson 712 — Solving Multi-Step Angle Problems
Solve problems: Sometimes you need to know where `f(x) = g(x)` or which function gives a larger value; Lesson 542 — Working with Multiple Functions
Solve problems faster: Know one fact, and you actually know four facts; Lesson 13 — The Relationship Between Addition and Subtraction
Solve recurrence relations: by turning them into algebraic equations; Lesson 2638 — What Is a Generating Function?
Solve simultaneously: Find all (x, y) pairs that satisfy both equations; Lesson 1862 — Critical Points in Two Variables
Solve step-by-step: , often finding one angle before you can find another; Lesson 712 — Solving Multi-Step Angle Problems
Solve the algebraic system: You now have a system of algebraic equations in the transformed variables (like X(s) and Y(s) instead of x(t) and y(t)).; Lesson 2397 — Solving Systems of ODEs with Laplace Transforms
Solve the equation: – Use inverse operations to find the unknown; Lesson 193 — Perimeter and Area Word Problems Lesson 804 — Finding Missing Lengths with Geometric Means Lesson 2643 — Solving Recurrence Relations with Generating Functions
Solve the homogeneous equation: first (find the complementary solution $y_c$); Lesson 2357 — Higher-Order ODEs and Undetermined Coefficients
Solve the inequality: find which x-values make it true; Lesson 550 — Domain Restrictions from Square Roots
Solve the leading variables: in terms of these parameters; Lesson 1315 — Systems with Infinitely Many Solutions
Solve the normal equations: $A^T A \mathbf{c} = A^T \mathbf{y}$ to get optimal coefficients; Lesson 2239 — Applications: Data Fitting and Approximation
Solve the ODE: using the integrating factor method to get the general solution *y(x) = [something] + C*; Lesson 2324 — Initial Value Problems for Linear ODEs
Solve the quadratic: Use factoring, the quadratic formula, or completing the square; Lesson 638 — Exponential and Logarithmic Equations with Quadratic Form
Solve the recurrence relation: to express later coefficients in terms of earlier ones (usually a₀ and a₁, your initial conditions); Lesson 2400 — Finding Power Series Solutions Near Ordinary Points
Solve the reduced system: Use only the independent constraints with their corresponding multipliers; Lesson 1889 — Dealing with Redundant Constraints
Solve the resulting equation: using familiar methods; Lesson 209 — Equations with Fractions: Clearing Denominators Lesson 277 — Introduction to the Substitution Method Lesson 444 — Finding Restricted Values Lesson 478 — Solving by Cross- Multiplication Lesson 524 — Solving Simple Radical Equations Lesson 526 — Equations with Radicals on Both Sides Lesson 528 — Radical Equations with Higher Index Roots Lesson 530 — Mixed Radical and Rational Equations (+2 more)
Solve the simplified equation: You get a direct answer for one coefficient; Lesson 1624 — Solving for Coefficients: Substitution Method
Solve the spatial problem: X'' + λX = 0 with boundary conditions yields eigenfunctions X_n(x) = sin(nπx/L); Lesson 2431 — Fourier Series Solutions for Bounded Domains
Solve the system: You have two equations and two unknowns (height and distance_A).; Lesson 980 — Two Right Triangle Problems and Indirect Measurement Lesson 1269 — Tension in Cables and Ropes Lesson 2064 — Applications: Expressing Solutions as Linear Combinations Lesson 3003 — Method of Moments for Multiple Parameters
Solve the temporal problem: T'' + λc²T = 0 gives T_n(t) = A_n cos(ω t) + B_n sin(ω t), where ω = nπc/L; Lesson 2431 — Fourier Series Solutions for Bounded Domains
Solve the trajectory: (solve the resulting IVP using methods you know); Lesson 2406 — Solving Boundary Value Problems: Shooting Method Concept
Solve using cross multiplication: Lesson 123 — Real-World Proportion Problems
Solve using one operation: Lesson 190 — Solving One-Step Word Problems
Solve using substitution: – Use the method you've learned to find both values; Lesson 282 — Application Problems Using Systems and Substitution
Solve: |x| = 7: Lesson 227 — Solving Basic Absolute Value Equations
Something cubed: (m + 2)³ → cube of a binomial; Lesson 341 — Recognizing Special Products in Context
Sort all edges: by weight, from lightest to heaviest; Lesson 2675 — Minimum Spanning Trees: Kruskal's Algorithm
Sound waves: Pitch corresponds to frequency (how fast the wave oscillates); loudness relates to amplitude.; Lesson 1022 — Applications: Modeling Periodic Phenomena
Source (s): A special vertex where flow originates (like a water pump); Lesson 2704 — Networks and Flows: Definitions and Notation
Space complexity: O(V + E), where V = vertices, E = edges.; Lesson 2656 — Graph Representations: Adjacency List and Edge List
span: of those vectors.; Lesson 2056 — Computing Linear Combinations in R² Lesson 2059 — The Span of a Set of Vectors Lesson 2061 — Geometric Span in R²: Lines and Planes Lesson 2064 — Applications: Expressing Solutions as Linear Combinations Lesson 2082 — Span as a Subspace
Span(v₁, v₂, ..., v ): Lesson 2059 — The Span of a Set of Vectors
Spanning: ensures completeness: every vector in the space can be written as a linear combination of basis vectors.; Lesson 2086 — What is a Basis?Lesson 2087 — Verifying a Set is a Basis
Spanning ℝⁿ: means the columns can build any vector in n-dimensional space; Lesson 2144 — The Invertible Matrix Theorem
spanning tree: does for a graph.; Lesson 2672 — Spanning Trees of a Graph Lesson 2674 — Finding Spanning Trees: Breadth-First Search
Spans all of Rⁿ: (can reach any point); Lesson 2088 — Standard Bases
Sparse: (many zero entries); Lesson 2257 — Finding Convenient Bases
Spatial equation: X''(x) + λX(x) = 0; Lesson 2420 — Eigenvalue Problems from Boundary Conditions
Spatial ODE: $X''(x) + \lambda X(x) = 0$; Lesson 2419 — The Method of Separation of Variables Lesson 2447 — Separating Variables in the Heat Equation
Spearman's rank correlation: solves this by converting each observation to its rank within its variable, then calculating how similarly the two sets of ranks behave.; Lesson 3109 — Spearman's Rank Correlation
special angles: are like the "landmark exits" on a highway—memorizing them makes navigation effortless.; Lesson 993 — Special Angles in Degrees and Radians Lesson 1061 — Sum Formulas with Special Angle Combinations
Special case: When multiplying conjugates like (a + √b)(a - √b), the radical terms cancel out, leaving you with just integers—very handy for simplifying!; Lesson 159 — Multiplying Radical Expressions Lesson 223 — Special Cases: All Reals and No Solution Lesson 1110 — The Law of Cosines and the Pythagorean Theorem Lesson 1773 — The Taylor Series Formula Lesson 2061 — Geometric Span in R²: Lines and Planes
Special Case — Resonance: If sin(ωx) or cos(ωx) is already part of your homogeneous solution (meaning ωi is a root of the characteristic equation), multiply your guess by x:; Lesson 2352 — Particular Solutions for Sine and Cosine Functions
special cases: of others—meaning they have all the properties of their "parent" shape, plus additional unique features.; Lesson 825 — The Quadrilateral Family Tree Lesson 1217 — Finding the Argument (Angle)
Special triangle ratios: (45-45-90 and 30-60-90 patterns); Lesson 799 — Comparing Special Triangles with Pythagorean Theorem
Special value: Γ(1) = 1; Lesson 2916 — Introduction to the Gamma Function
Specificity: (True Negative Rate): P(Negative Test | Disease Absent) — how good the test is at returning negative when you're healthy; Lesson 2818 — Medical Testing and Screening Problems
Specify everything: significance level α, sample size n, population standard deviation (or estimate), and the specific alternative parameter value μ₁ you want to detect; Lesson 3058 — Calculating Power for Simple Tests
Spectral Decomposition: takes this one step further by expanding this product into an explicit sum.; Lesson 2273 — Spectral Decomposition
Spectral Theorem: states that every symmetric matrix can be orthogonally diagonalized.; Lesson 2272 — The Spectral Theorem for Symmetric Matrices Lesson 2273 — Spectral Decomposition
Speed: 60 miles per hour (miles ÷ hours); Lesson 109 — Introduction to Rates
Speed of Convergence: When both converge, geometric series (with small |r|) typically converge much faster than p- series because exponential decay beats polynomial decay.; Lesson 1728 — Comparing Geometric and p-Series
sphere: is a perfectly round three-dimensional object, like a basketball or a soap bubble.; Lesson 882 — Volume of Spheres Lesson 1480 — Volume Related Rates: Spheres and Cones Lesson 2040 — Applying the Theorem to a Sphere
Spheres: The equation **ρ = c** (where c is constant) describes a sphere of radius c centered at the origin.; Lesson 1939 — Graphing Surfaces in Spherical Coordinates
spherical: (ρ, θ, φ).; Lesson 1943 — Choosing the Best Coordinate System Lesson 2454 — Separation in Other Coordinate Systems
Split: the integral: (1/2)∫1 dx - (1/2)∫cos(2x)dx; Lesson 1601 — Powers of Sine and Cosine: Even Powers
Split into two limits: Lesson 1419 — Derivative of sin(x) from First Principles
Split the integral: at those points; Lesson 1573 — Area and Signed Area Lesson 1643 — Area Between a Curve and the x-axis
Split the middle term: Rewrite 7x as 6x + 1x:; Lesson 364 — Factoring by Grouping After Splitting the Middle Term
Split the problem: Create multiple simpler problems, each with only *one* non-homogeneous boundary and all others zero; Lesson 2440 — Superposition and Non-Homogeneous Boundaries
Split the radical: into the product of two roots; Lesson 782 — Simplifying Radical Answers
Splitting: the polynomial into smaller groups (usually pairs of terms); Lesson 374 — Introduction to Factoring by Grouping
Spot inner products: – Does the numerator or denominator contain functions multiplied together?; Lesson 1406 — Combining Product and Quotient Rules
Spot the operation words: which math action is happening?; Lesson 172 — Writing Expressions from Word Problems
spread: (width); Lesson 2905 — Parameters: Mean and Standard Deviation Lesson 2992 — Histograms and Frequency Distributions Lesson 2994 — Comparing Distributions
Spread differences: Which group shows more variability (wider IQR)?; Lesson 2994 — Comparing Distributions
spring constant: (measures the spring's stiffness); Lesson 1694 — Hooke's Law and Spring Problems Lesson 2368 — Spring-Mass Systems: Setting Up the Differential Equation
square: this is always true for maximum area with fixed perimeter!; Lesson 435 — Area and Perimeter Optimization Problems Lesson 529 — Word Problems Involving Radical Equations Lesson 812 — Finding Individual Exterior Angles in Regular Polygons Lesson 820 — Squares: The Ultimate Special Quadrilateral Lesson 825 — The Quadrilateral Family Tree Lesson 857 — Area of Rectangles and Squares Lesson 862 — Area of Circles Lesson 894 — Similar Solids and Surface Area (+4 more)
Square all three numbers: Lesson 784 — Verifying Pythagorean Triples
Square both sides: to eliminate the square root; Lesson 524 — Solving Simple Radical Equations Lesson 526 — Equations with Radicals on Both Sides Lesson 527 — Equations Requiring Multiple Squaring Steps Lesson 530 — Mixed Radical and Rational Equations
Square both sides again: to eliminate it; Lesson 527 — Equations Requiring Multiple Squaring Steps
Square each leg: Take the length of each leg and multiply it by itself; Lesson 775 — Finding the Hypotenuse
Square it: to work with `D² = (x₂ - x₁)² + (y₂ - y₁)²` (easier derivatives!; Lesson 1512 — Distance Optimization Problems Lesson 1676 — Computing Arc Length: Polynomial Functions
square matrix: has the same number of rows and columns.; Lesson 1272 — Types of Matrices: Square, Row, Column, and Zero Lesson 2097 — Matrix Notation and Dimensions
Square root functions: like `f(x) = √x` only produce non-negative outputs, so the range is `y ≥ 0`.; Lesson 553 — Finding Range from Equations
Square roots: Require non-negative inputs (for real functions); Lesson 1803 — Domain and Range of Multivariable Functions
Square roots (n=2): Two points, 180° apart (opposite ends of a diameter); Lesson 1230 — Geometric Interpretation of nth Roots
square units: (cm², m², in², etc.; Lesson 829 — Area of a Circle Lesson 857 — Area of Rectangles and Squares
Squared terms opposite signs: → Hyperbolic paraboloid (saddle); Lesson 1807 — Common Surfaces: Planes, Paraboloids, and Saddles
Squares: (all sides equal + right angles); Lesson 826 — Coordinate Geometry of Quadrilaterals
Squaring both sides: (turns negative values positive); Lesson 1091 — Extraneous Solutions and Verification
Squaring something: (3y - 5)² → square of a binomial; Lesson 341 — Recognizing Special Products in Context
Squeeze it: Since *x² ≤ x² + y²* and *y² ≤ x² + y²*, we have:; Lesson 1816 — Squeeze Theorem in Several Variables
Squeeze Theorem: (which you just learned!; Lesson 1345 — Special Trigonometric Limit: sin(x)/x Lesson 1709 — The Squeeze Theorem for Sequences Lesson 1820 — Limits and Continuity in Three or More Variables
SS_Between: measures squared deviations of group means from the grand mean (weighted by group size); Lesson 3095 — Between-Group and Within-Group Variation
SS_Total: measures total squared deviations from the grand mean; Lesson 3095 — Between-Group and Within-Group Variation
SS_Within: measures squared deviations of individual observations from their own group mean; Lesson 3095 — Between-Group and Within-Group Variation
SSA: when you know two sides and an angle that's *not* between them—is special and trickier.; Lesson 1096 — The SSA Configuration Introduction Lesson 1102 — Introduction to the Law of Cosines Lesson 1106 — Choosing Between Law of Sines and Law of Cosines
SSA is ambiguous: meaning it might produce no triangle, one triangle, or even two completely different triangles!; Lesson 1121 — Introduction to the Ambiguous Case (SSA)
SSS: (all three sides but no angles)?; Lesson 1102 — Introduction to the Law of Cosines Lesson 1106 — Choosing Between Law of Sines and Law of Cosines
SSS Congruence Postulate: states that if three sides of one triangle are congruent (equal in length) to the three corresponding sides of another triangle, then the two triangles are congruent.; Lesson 741 — SSS (Side-Side-Side) Congruence Postulate
SSS Similarity Criterion: states that if the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar.; Lesson 755 — SSS Similarity Criterion
SSS triangles: (all three sides known).; Lesson 1116 — Applying Heron's Formula
Stability: in numerical algorithms; Lesson 2211 — Orthogonal Bases Lesson 2412 — Well-Posed Problems: Existence, Uniqueness, Stability
Stable node: Both eigenvalues negative → all trajectories spiral inward toward equilibrium; Lesson 2383 — Phase Portraits and Stability Analysis
Stable spiral: Real part α < 0 → trajectories spiral inward; Lesson 2383 — Phase Portraits and Stability Analysis
Stack the polynomials: one above the other, aligning like terms in columns (though this alignment is flexible); Lesson 330 — Multiplying Polynomials Vertically
Stadium seating: If the first row has 20 seats, and each row has 2 more seats than the one in front, with 30 rows total; Lesson 693 — Stacking and Construction Problems
Staircases: Blocks needed where each step adds a constant amount; Lesson 693 — Stacking and Construction Problems
Stan: Uses HMC for faster convergence; popular in research; Lesson 3121 — Computational Tools: Introduction to MCMC and Bayesian Software
Standard: `Ax + By = C`; Lesson 261 — Converting Between Forms of Linear Equations
standard basis: as the "factory settings" of a vector space—it's the simplest, most obvious way to describe any vector using coordinates.; Lesson 2088 — Standard Bases Lesson 2213 — Coordinates with Respect to Orthogonal Bases Lesson 2218 — Standard Orthonormal Bases in Euclidean Space
Standard continuous functions: like sin, cos, exponentials; Lesson 1813 — Computing Limits Using Direct Substitution
standard deviation: σ = √Var(X) for interpretation.; Lesson 2856 — Variance: Definition and Interpretation Lesson 2858 — Standard Deviation Lesson 2891 — Mean and Variance of the Poisson Distribution Lesson 2905 — Parameters: Mean and Standard Deviation Lesson 2989 — Measures of Spread: Range, Variance, and Standard Deviation
Standard Deviation of S: SD(S ) = σ√n (not σ/√n); Lesson 2984 — CLT for Sample Sums
standard error: tells you how much the sample mean typically varies from the true population mean.; Lesson 2978 — The Standard Error and Standardization Lesson 2979 — CLT for Independent Identically Distributed Variables Lesson 3015 — Confidence Interval for a Population Proportion Lesson 3016 — Confidence Interval for the Difference of Two Means
Standard form: is the normal way we write numbers, all digits together:; Lesson 5 — Expanded Form and Standard Form Lesson 264 — Choosing the Best Form for a Problem Lesson 314 — Standard Form of a Polynomial Lesson 317 — Leading Coefficient and Leading Term Lesson 332 — Simplifying Products by Combining Like Terms Lesson 395 — Standard Form and Setting Equal to Zero Lesson 406 — Converting to Vertex Form Lesson 412 — Understanding the Structure of the Quadratic Formula (+14 more)
Standard forms: Some tests (upcoming in your studies) work best when series start at a particular index; Lesson 1720 — Reindexing and Shifting Series
Standardize: the values using z-scores; Lesson 2909 — Finding Probabilities with Normal Distributions Lesson 2978 — The Standard Error and Standardization
Standardize and find probabilities: Convert to *Z*-scores and use the standard normal table; Lesson 2985 — Approximating Binomial Distributions
Standardized or studentized residuals: help identify outliers (typically |residual| > 2 or 3 suggests an outlier); Lesson 3089 — Outliers in Regression
Start: y = f(x)^g(x); Lesson 1462 — Variable Bases and Variable Exponents Lesson 2226 — Gram-Schmidt in Inner Product Spaces Lesson 2673 — Finding Spanning Trees: Depth-First Search Lesson 2674 — Finding Spanning Trees: Breadth-First Search Lesson 2702 — The Hungarian Algorithm for Maximum Bipartite Matching
Start at one: – Always begin with the first number; Lesson 1 — Counting Objects from 1 to 10
Start with: a linearly independent set of vectors in a vector space (say, in R³); Lesson 2090 — Extending to a Basis
Start with 1: Every number has 1 as a factor (1 × 24 = 24); Lesson 48 — Understanding Factors
Start with a universe: of all candidates; Lesson 2623 — The Sieve Method in Practice
Start with easier facts: (dividing by 1, 2, 5, 10) and work toward the trickier ones (7, 8, 9).; Lesson 25 — Division Facts and Basic Division Fluency
Start with givens: – Write down what you know is true; Lesson 768 — Direct Proof Techniques
Start with your basis: Put the basis vectors of your subspace as columns of matrix A; Lesson 2231 — The Projection Matrix
Start with your matrix: (any size); Lesson 2127 — Computing Rank via Row Reduction
Starting number: (how many you begin with); Lesson 11 — Understanding Subtraction as Taking Away
Starting point: Assume P(A ∩ B) = P(A) · P(B).; Lesson 2806 — Independence of Complements
Starting with: `y = ax² + bx + c`; Lesson 406 — Converting to Vertex Form
State hypotheses: about the population mean μ (null: μ = μ₀; alternative: μ ≠ μ₀, μ > μ₀, or μ < μ₀); Lesson 3034 — One-Sample t-Test: Testing a Population Mean Lesson 3043 — Chi-Squared Goodness-of- Fit Test
State restrictions: from the original denominator; Lesson 448 — Simplifying with Trinomial Factors Lesson 458 — Applications of Multiplying and Dividing Rationals
State the domain: write your answer in interval notation or set-builder notation; Lesson 550 — Domain Restrictions from Square Roots
State your assumption: (the hypothesis); Lesson 2493 — Direct Proof Examples and Practice
statement: is a fact or conclusion you make.; Lesson 762 — Introduction to Geometric Proof Lesson 766 — Common Reasons in Proofs Lesson 2496 — Contrapositive Proof Examples
Statement 1: AB ≅ CD → **Reason:** Given; Lesson 762 — Introduction to Geometric Proof
Statement 2: AB = CD → **Reason:** Definition of congruent segments; Lesson 762 — Introduction to Geometric Proof
statements: (what you know or what you're claiming), and the right column contains **reasons** (why each statement is true).; Lesson 748 — Writing Two-Column Congruence Proofs Lesson 763 — Two-Column Proof Format
Statements and Reasons: Lesson 762 — Introduction to Geometric Proof
Statistical significance: means your result landed in the critical region—it's unlikely enough under H₀ to warrant rejection; Lesson 3023 — Significance Levels and Critical Regions
stays constant: (steady state); Lesson 2179 — Applications: Steady States and Dynamics Lesson 2887 — Hypergeometric vs. Binomial: When to Use Each
stays the same: .; Lesson 218 — Solving One-Step Inequalities Lesson 1294 — Determinant Properties: Row Operations
Steady-state heat distribution: When a heated object reaches equilibrium (temperature no longer changing with time), the heat equation simplifies to Laplace's equation.; Lesson 2436 — Introduction to Laplace's Equation
Steady-state solution: b/a (what y approaches as x → ∞ if a > 0); Lesson 2325 — Linear ODEs with Constant Coefficients Lesson 2371 — Forced Harmonic Motion and Resonance
Step: Consider *(k+1)³ - (k+1)*.; Lesson 2504 — Proving Divisibility Properties Lesson 2514 — Proving Divisibility Results with Strong Induction
Step 0: Prior probability P(D) = 0.; Lesson 2820 — Sequential Updating with Bayes' Theorem
Step 1 (inner): Integrate with respect to y, treating x as constant:; Lesson 1899 — Double Integrals of Non-Separable Functions
Step 1: Base Case: Lesson 2501 — The Principle of Mathematical Induction
Step 1: Define Variables: Lesson 291 — Word Problems Solved by Elimination
Step 1: Differentiate u: Lesson 1581 — Computing du and Substituting
Step 1: Distribute: Lesson 180 — Using the Distributive Property to Simplify Lesson 208 — Equations with Parentheses and the Distributive Property
Step 1: Find Eigenvalues: Lesson 2275 — Computing Orthogonal Diagonalization
Step 1: Identify u: Lesson 1582 — Simple u-Substitution Examples
Step 1: Read carefully: and identify all the variables involved.; Lesson 131 — Mixed Variation Applications Lesson 191 — Multi-Step Word Problems with Combined Operations
Step 1: Substitute: Lesson 183 — Order of Operations in Evaluation Lesson 1086 — Quadratic-Type Trigonometric Equations
Step 2 (outer): Now integrate this result with respect to x:; Lesson 1899 — Double Integrals of Non-Separable Functions
Step 2: Apply PEMDAS: Lesson 183 — Order of Operations in Evaluation
Step 2: Check Spanning: Lesson 2087 — Verifying a Set is a Basis
Step 2: Compute du: Lesson 1582 — Simple u-Substitution Examples
Step 2: Inductive Step: Lesson 2501 — The Principle of Mathematical Induction
Step 3 (optional): Convert back to radical/fraction form if needed; Lesson 1396 — Rewriting Functions Before Differentiating
Step 3: Check cross-sections: Lesson 902 — Problem Solving with Cavalieri's Principle
Step 3: Substitute: Lesson 1582 — Simple u-Substitution Examples
Step 3: Substitute back: Lesson 1086 — Quadratic-Type Trigonometric Equations
Step 3: Substitute completely: Lesson 1581 — Computing du and Substituting
Step 3: Translate back: Lesson 936 — Dilations Centered at Points Other Than the Origin
Step 3: Use Elimination: Lesson 291 — Word Problems Solved by Elimination
Step 4: Add periodicity: Lesson 1081 — Linear Trigonometric Equations
Step 4: Apply inclusion-exclusion: Plug into the formula, then subtract from the universe if counting "good" objects.; Lesson 2626 — Complex Applications and Problem-Solving Strategies
Step 4: Integrate: Lesson 1582 — Simple u-Substitution Examples
Step 4: Use symmetry: to plot matching points on the opposite side of the axis.; Lesson 427 — Graphing Parabolas in Vertex Form
Step 5: Back-substitute: Lesson 1582 — Simple u-Substitution Examples
Step 5: Check: if your answer makes sense in context.; Lesson 191 — Multi-Step Word Problems with Combined Operations
Step 5: State restrictions: Lesson 453 — Multiplying Complex Rational Expressions
Step 6: Final answer: `(2x - 3)(3x² - 2)`; Lesson 381 — Applications and Mixed Practice Lesson 408 — Deriving the Quadratic Formula Lesson 411 — Deriving the Quadratic Formula by Completing the Square Lesson 1614 — Substitution for √(a²+x²): Using x = a tan θ Lesson 2757 — Solving Two-Congruence Systems
Step 6: Verify: Lesson 2275 — Computing Orthogonal Diagonalization
Step 7: Subtract `b/(2a)` from both sides:; Lesson 411 — Deriving the Quadratic Formula by Completing the Square
Step 8: Combine into one fraction:; Lesson 411 — Deriving the Quadratic Formula by Completing the Square
step-by-step: , applying one rule at a time until the expression is fully simplified.; Lesson 149 — Combining Multiple Exponent Rules Lesson 1766 — Interval of Convergence Lesson 2758 — Solving Systems with Three or More Congruences
Step-by-step approach: Lesson 938 — Combining Rotations and Dilations with Other Transformations
Step-by-step process: Lesson 155 — Simplifying Square Roots Lesson 452 — Simplifying Before Multiplying
Steps: Lesson 63 — Converting Between Improper and Mixed Lesson 214 — Literal Equations: Solving for a Specified Variable Lesson 459 — Adding Rationals with Like Denominators Lesson 658 — Arithmetic Means and Inserting Terms Lesson 1309 — Writing Systems as Augmented Matrices Lesson 1839 — Applications to Measurement and Error Propagation Lesson 2681 — Finding Eulerian Paths and Circuits: Algorithms
Steps to estimate error: Lesson 1838 — Estimating Error in Linear Approximation
Stokes' Theorem: , which works for curved surfaces in three dimensions.; Lesson 2003 — Applications and Extensions of Green's Theorem Lesson 2033 — Conservative Fields and Stokes' Theorem Lesson 2034 — Physical Interpretations: Circulation and Rotation
Stokes' Theorem still works: , but you must account for how pieces connect.; Lesson 2032 — Stokes' Theorem on Piecewise Smooth Surfaces
stop: .; Lesson 1314 — Identifying Inconsistent Systems Lesson 1340 — Indeterminate Forms: 0/0 and ∞/∞Lesson 2675 — Minimum Spanning Trees: Kruskal's Algorithm Lesson 2713 — The Euclidean Algorithm
Stop when numbers repeat: Once you reach pairs you've already found (like 6 × 4), you're done.; Lesson 48 — Understanding Factors
straight: no bends or curves; Lesson 696 — Lines: Infinite Straight Paths Lesson 704 — Classifying Angles by Measure
straight angle: measures 180° (a straight line); Lesson 703 — Measuring Angles and Angle Notation Lesson 704 — Classifying Angles by Measure
Straight diagonal line: = residuals are normally distributed; Lesson 3086 — Normal Probability Plots (Q-Q Plots)
Straightforward limits: When x approaches a value where the function is defined, substitute directly:; Lesson 1347 — Exponential and Logarithmic Limits
Strategic choice: Combine equations 1 and 3 first to eliminate z quickly, then tackle another pair.; Lesson 295 — Choosing Strategic Pairs for Elimination
Strategic ordering: Tackle the simplest constraint equations first; Lesson 1888 — Three or More Constraints
Strategic substitution: for linear factors (plug in values that zero out most terms); Lesson 1631 — Complex Mixed Denominator Cases
Strategy: Take only the prime factors that appear in BOTH numbers, using the smallest power of each.; Lesson 58 — Using Prime Factorization for GCF and LCM Lesson 499 — Simplifying Square Roots with Variables Lesson 561 — Decomposing Functions Lesson 1066 — Using Double Angle Formulas to Simplify Expressions Lesson 1264 — Equilibrium: Balancing Multiple Forces Lesson 1492 — The MVT and Constant Functions Lesson 1816 — Squeeze Theorem in Several Variables Lesson 1899 — Double Integrals of Non- Separable Functions (+2 more)
Strengths: Always works when f′ exists near the critical point.; Lesson 1502 — Comparing First and Second Derivative Tests
strictly increasing: on an interval if, whenever you pick two points where `x₁ < x₂`, you always have `f(x₁) < f(x₂)`.; Lesson 571 — Monotonicity: Increasing and Decreasing Functions Lesson 1490 — Applying the MVT to Prove Increasing/Decreasing Behavior
strong induction: prove statements for all integers n ≥ some base case.; Lesson 2512 — Review: Ordinary vs. Strong Induction Lesson 2513 — The Strong Induction Principle Lesson 2519 — Equivalence of Induction and Well-Ordering Lesson 2520 — Choosing the Right Proof Technique
Strong Induction Principle: To prove that a predicate `P(n)` is true for all natural numbers n ≥ n₀:; Lesson 2513 — The Strong Induction Principle
Strong induction structure: Lesson 2515 — Strong Induction for Sequences and Recurrences
Strong Law: The *entire sequence* of sample means approaches $\mu$, and this happens for almost every possible outcome; Lesson 2973 — Strong Law of Large Numbers: Statement
strongly connected: if there's a directed path from every vertex to every other vertex.; Lesson 2664 — Strongly Connected Digraphs Lesson 2665 — Weakly Connected Digraphs
Strongly connected components: in directed social graphs (like Twitter's follow relationships) reveal tight-knit communities where everyone can reach everyone else through the network.; Lesson 2667 — Applications of Connectivity
Structural Engineering: Predicting how beams and columns resist bending (larger *k* means better resistance to buckling); Lesson 1962 — Radius of Gyration and Applications
Structural induction: adapts the inductive principle to prove properties about these recursively defined objects.; Lesson 2516 — Structural Induction on Trees and Graphs
Structural Supports: Lesson 981 — Applications in Construction and Engineering
Structure: Lesson 765 — Flow Chart Proof Format Lesson 1728 — Comparing Geometric and p-Series Lesson 2117 — Leading Entries and Pivot Positions Lesson 2507 — Multiple Base Cases
Student loans: Plan repayment strategies; Lesson 689 — Loan Amortization
Student's t-distribution: (often just called the t-distribution) is a continuous probability distribution that looks similar to the standard normal distribution but with heavier tails.; Lesson 2923 — The Student's t-Distribution: Definition Lesson 3032 — Why t-Tests? When the Population Variance is Unknown
Sturm-Liouville problems: are a particular class of BVPs with the form:; Lesson 2405 — Eigenvalue Problems and Sturm-Liouville Theory
Subdivide intelligently: Break the region into 2 or more simpler subregions (usually rectangles, triangles, or simple curves); Lesson 1909 — Regions Requiring Multiple Integrals
subset: of B, written A ⊆ B.; Lesson 2523 — Subsets and Proper Subsets Lesson 2774 — Events as Subsets of the Sample Space
subspace: is a subset of a vector space that is itself a vector space using the same operations (addition and scalar multiplication).; Lesson 2078 — Subspaces: Definition and Subspace Test Lesson 2082 — Span as a Subspace Lesson 2128 — The Column Space
Substitute: the fractional value for each variable; Lesson 185 — Evaluating Expressions with Fractions Lesson 187 — Using Formulas and Literal Equations Lesson 278 — Substitution When One Equation is Already Solved Lesson 372 — Factoring Higher-Degree Trinomials Using Substitution Lesson 443 — Evaluating Rational Expressions Lesson 638 — Exponential and Logarithmic Equations with Quadratic Form Lesson 659 — Applications of Arithmetic Sequences Lesson 954 — Finding Side Lengths Using Tangent (+11 more)
Substitute and calculate: as you would with any function; Lesson 545 — Evaluating Piecewise-Defined Functions
Substitute and solve: for the unknown value; Lesson 129 — Real-World Direct Variation Problems
Substitute back: into one of your reduced two-variable equations to find the second variable (like y).; Lesson 296 — Back-Substitution in Three Variables Lesson 372 — Factoring Higher-Degree Trinomials Using Substitution Lesson 1415 — Chain Rule with Product and Quotient Rules Lesson 1462 — Variable Bases and Variable Exponents Lesson 1579 — What is u-Substitution?
Substitute both known values: back into one of the original three-variable equations to find the last variable (like x).; Lesson 296 — Back-Substitution in Three Variables
Substitute carefully: , especially with negative signs; Lesson 414 — Solving Quadratics When a ≠ 1
Substitute each integer: from start to stop into the expression; Lesson 670 — Evaluating Finite Sums with Sigma Notation
Substitute everything: Replace all instances of x with expressions in u; Lesson 1585 — u-Substitution with Exponential and Logarithmic Functions Lesson 2400 — Finding Power Series Solutions Near Ordinary Points
Substitute immediately: plug in the numbers right away, then calculate; Lesson 186 — Simplifying Before Evaluating
Substitute into the formula: Lesson 245 — Calculating Slope from Two Points
Substitute it back: into the original equation for *x*; Lesson 421 — Verifying Solutions by Substitution
Substitute known values: and solve for the unknown rate; Lesson 1478 — Related Rates with Rectangles and Squares Lesson 1479 — Related Rates with Triangles Lesson 1480 — Volume Related Rates: Spheres and Cones Lesson 1483 — Angle Related Rates
Substitute that simplified result: back into the next layer out; Lesson 474 — Nested Complex Fractions
Substitute the given values: into the equation; Lesson 128 — Finding the Constant of Variation
Substitute the initial condition: values into your general solution; Lesson 2311 — Initial Value Problems with Separable ODEs
Substitute the known values: Lesson 804 — Finding Missing Lengths with Geometric Means
Substitute u = tan(x): , so du = sec²(x) dx; Lesson 1603 — Powers of Tangent and Secant: Even Powers of Secant
substitution: plug your answer back into the original problem and see if it makes sense.; Lesson 195 — Checking Solutions and Problem-Solving Strategies Lesson 372 — Factoring Higher-Degree Trinomials Using Substitution Lesson 373 — Applications and Strategy Selection for Trinomial Factoring Lesson 1878 — Solving Two-Variable Constrained Problems Lesson 1885 — Solving Systems with Multiple Lagrange Multipliers Lesson 1888 — Three or More Constraints
substitution method: is a way to solve for both variables by converting the system into a single equation with just one variable.; Lesson 277 — Introduction to the Substitution Method Lesson 1624 — Solving for Coefficients: Substitution Method
Subtler case: After simplifying, you might get something like `|x - 1| - |x - 1| = 0`, which reduces to `0 = 0` — always true!; Lesson 229 — Absolute Value Equations with No Solution or All Solutions
Subtract: when things are being taken away, separated, or compared (finding the difference); Lesson 18 — Word Problems with Addition and Subtraction Lesson 29 — Long Division Algorithm Lesson 73 — Subtracting Fractions with Unlike Denominators Lesson 150 — Common Mistakes with Exponent Rules Lesson 199 — Solving One-Step Addition/Subtraction Equations Lesson 843 — Segment Area Lesson 865 — Area of Segments Lesson 869 — Finding Area Using Subtraction Method (+5 more)
Subtract 3 first: (undo the addition); Lesson 202 — Combining Addition/Subtraction with Multiplication/Division
Subtract projections: For each subsequent vector v , compute:; Lesson 2226 — Gram-Schmidt in Inner Product Spaces
Subtract tens: 4 - 2 = 2; Lesson 16 — Subtraction with Regrouping (Borrowing)
subtract the exponents: .; Lesson 143 — Quotient Rule for Exponents Lesson 166 — Dividing Numbers in Scientific Notation
Subtract the numerators: (subtract the top numbers); Lesson 70 — Subtracting Fractions with Like Denominators Lesson 464 — Subtracting Rationals with Unlike Denominators
subtracting: angles, you **add** the sine terms.; Lesson 1053 — Sum and Difference Formulas for Cosine Lesson 2618 — Inclusion-Exclusion for Three Sets
Subtracting inside shifts RIGHT: `f(x - 3)` moves the graph 3 units to the right; Lesson 579 — Horizontal Shifts of Functions
Subtracting like radicals: Lesson 158 — Adding and Subtracting Radicals
Subtraction: "There were 42 birds on a fence.; Lesson 18 — Word Problems with Addition and Subtraction Lesson 172 — Writing Expressions from Word Problems Lesson 285 — Eliminating a Variable by Subtracting Equations Lesson 424 — Finding the Vertex from Vertex Form Lesson 1197 — Adding and Subtracting Complex Numbers
Subtraction clues: left, remaining, fewer than, how many more, difference; Lesson 18 — Word Problems with Addition and Subtraction
Subtraction phrases: Lesson 188 — Translating Words into Mathematical Expressions
Subtraction Property of Equality: If you subtract the same number from both sides of an equation, the sides remain equal.; Lesson 198 — Addition and Subtraction Properties of Equality
Success: (often coded as 1); Lesson 2872 — The Bernoulli Trial and Bernoulli Distribution
Sufficiency ( ): (By induction on |X|) Assume Hall's condition holds.; Lesson 2700 — Hall's Marriage Theorem
sum: itself.; Lesson 677 — What is a Series?Lesson 1819 — Properties and Composition of Continuous Functions Lesson 2084 — Sum of Subspaces Lesson 2355 — Superposition for Multiple Terms Lesson 2912 — Linear Combinations of Normal Variables Lesson 2984 — CLT for Sample Sums
Sum all frustums: Total area ≈ Σ 2πr(x)Δs; Lesson 1684 — Deriving the Surface Area Formula
Sum all rectangle areas: Right Sum = Δx · [f(x₁) + f(x₂) + .; Lesson 1548 — Right Riemann Sums
Sum all three terms: Lesson 2150 — Determinant of a 3×3 Matrix: Cofactor Expansion
sum and difference formulas: let you rewrite these as combinations of known angles.; Lesson 1052 — Sum and Difference Formulas for Sine Lesson 1059 — Sum and Difference Formulas in Equation Solving
Sum Formula: Geometric series have a closed-form sum when convergent.; Lesson 1728 — Comparing Geometric and p-Series
Sum of constants: Σ(i=1 to n) c = cn; Lesson 1556 — Computing Simple Integrals Using Riemann Sum Limits
Sum of Cubes: Lesson 339 — Sum and Difference of Cubes Lesson 389 — Applying Cube Formulas with Coefficients
Sum of integers: Σ(i=1 to n) i = n(n+1)/2; Lesson 1556 — Computing Simple Integrals Using Riemann Sum Limits
sum of squares: a² + b² should factor too, but it doesn't—at least not with real numbers.; Lesson 386 — Sum of Squares (Why It Doesn't Factor)Lesson 1556 — Computing Simple Integrals Using Riemann Sum Limits
Sum ranks: for each group (call them R₁ and R₂); Lesson 3106 — Mann-Whitney U Test (Wilcoxon Rank-Sum)
Sum ranks per group: Calculate the sum of ranks for each group separately.; Lesson 3107 — Kruskal-Wallis Test
Sum Rule: says: the derivative of `f(x) + g(x)` is `f'(x) + g'(x)`; Lesson 1424 — Derivatives of Trig Functions with Coefficients and Sums Lesson 1708 — Limit Laws for Sequences
Sum the flux contributions: or use the triple integral of divergence instead!; Lesson 2042 — The Divergence Theorem for General Regions
Sum the results: Add all the integrals together; Lesson 1909 — Regions Requiring Multiple Integrals Lesson 1979 — Line Integrals Over Piecewise Smooth Curves
Sum to one: The sum over all possible pairs (x, y) equals 1; Lesson 2927 — Joint Probability Distributions: Discrete Case
Sum/Difference: If f(x) and g(x) are continuous at x = a, then f(x) + g(x) and f(x) - g(x) are also continuous at a.; Lesson 1355 — Continuity of Combinations of Functions
Sum/Difference Rule: Lesson 672 — Properties of Summation: Linearity Lesson 1559 — Properties of Definite Integrals: Linearity
Sum/Difference Rules: you learned earlier for polynomials.; Lesson 1424 — Derivatives of Trig Functions with Coefficients and Sums
summation: Lesson 2933 — Joint Cumulative Distribution Functions Lesson 2937 — Mixed Discrete-Continuous Joint Distributions
summation notation: (also called sigma notation) to express sums efficiently.; Lesson 669 — Introduction to Summation Notation Lesson 674 — Writing Series in Summation Notation
superposition principle: lets you combine *all* of these individual solutions into one general solution.; Lesson 2421 — The General Series Solution Lesson 2466 — Fourier Series and Partial Differential Equations
supplementary: , their measures sum to 180°.; Lesson 707 — Supplementary Angles Lesson 708 — Linear Pairs
support: ) of a discrete random variable is the complete set of all distinct numerical values it can possibly produce.; Lesson 2824 — The Range of a Discrete Random Variable Lesson 3042 — Chi-Squared Distribution Basics
surface area: of a pyramid is the total area of all its outer surfaces.; Lesson 889 — Surface Area of Pyramids Lesson 891 — Surface Area of Spheres Lesson 893 — Nets and Surface Area Lesson 1683 — Introduction to Surface Area of Revolution Lesson 1690 — Comparing Surface Area and Volume Calculations
Surface Area = 4πr²: Lesson 891 — Surface Area of Spheres
Surface element: dS = ρ₀² sin φ dφ dθ (this captures how area elements distort on a sphere).; Lesson 2025 — Surface Integrals in Different Coordinate Systems
surface of revolution: .; Lesson 1683 — Introduction to Surface Area of Revolution Lesson 1801 — Surfaces of Revolution
Surface parametrization: Lesson 2025 — Surface Integrals in Different Coordinate Systems
surjective: (or "onto") when its range equals the entire codomain:; Lesson 2262 — The Range (Image) of a Linear Transformation Lesson 2559 — Introduction to Function Types Lesson 2562 — Surjective Functions (Onto)Lesson 2563 — Testing for Surjectivity Lesson 2564 — Bijective Functions (One-to-One Correspondence)Lesson 2565 — Inverse Functions and Bijections Lesson 2568 — Examples with Finite Sets Lesson 2569 — Applications to Infinite Sets
Surjective (onto): Every element in W is reached; Lesson 2269 — Isomorphisms Between Vector Spaces
Surjectivity (onto): Every possible output actually appears.; Lesson 2565 — Inverse Functions and Bijections
Surprising result: Even with a positive test, you only have about a 16% chance of being sick, because the disease is rare!; Lesson 2815 — Basic Applications of Bayes' Theorem
Surveying: calculating land boundaries or distances across rivers and canyons; Lesson 1100 — Applications of the Law of Sines
Swap: the main diagonal elements (swap `a` and `d`); Lesson 1302 — Finding the Inverse of a 2×2 Matrix Lesson 1997 — Using Green's Theorem to Evaluate Line Integrals
swap two rows: , the determinant changes sign (gets multiplied by -1).; Lesson 1294 — Determinant Properties: Row Operations Lesson 2105 — Introduction to Gaussian Elimination
Swap x and y: this reverses the input-output relationship; Lesson 563 — Finding Inverse Functions Algebraically
Swapping two rows: Multiplies the determinant by **-1**; Lesson 2155 — Computing Determinants via Row Reduction
Switch to spherical coordinates: (ρ, φ, θ) where 0 ≤ ρ ≤ R; Lesson 2040 — Applying the Theorem to a Sphere
Switch tools mid-problem: After finding that gateway piece, reassess.; Lesson 1111 — Complex Problems Combining Triangle Laws
Sylvester's Criterion: gives us a shortcut: just check the determinants of certain submatrices.; Lesson 2294 — Principal Minors and Sylvester's Criterion Lesson 2298 — Applications to Optimization and Hessian Matrices
symmetric: if it equals its own transpose: **A** = **A ᵀ**.; Lesson 2270 — Symmetric Matrices and Their Properties Lesson 2543 — Symmetric Relations Lesson 2544 — Antisymmetric Relations Lesson 2549 — Inverse Relations Lesson 2550 — Definition of Equivalence Relations Lesson 2551 — Verifying Equivalence Relations Lesson 2556 — Partitions Induce Equivalence Relations Lesson 2557 — Congruence Modulo n (+3 more)
Symmetric arguments: If z has argument θ, then z̄ has argument −θ (or equivalently, 360° − θ).; Lesson 1213 — Complex Conjugates on the Plane
Symmetric matrices: No need to handle complex numbers when computing eigenvalues; Lesson 2176 — Eigenvalues of Special Matrices: Symmetric and Orthogonal Lesson 2276 — The Spectral Theorem for Normal Matrices
Symmetric matrices are special: For every eigenvalue λ, these two multiplicities are always equal.; Lesson 2274 — Eigenspaces and Multiplicity for Symmetric Matrices
symmetric matrix: .; Lesson 1864 — The Hessian Matrix Lesson 2176 — Eigenvalues of Special Matrices: Symmetric and Orthogonal Lesson 2291 — Quadratic Forms and Their Matrix Representation
Symmetry: Lesson 824 — Kites: Properties and Diagonal Relationships Lesson 1517 — Domain, Intercepts, and Symmetry Lesson 1525 — Comprehensive Curve Sketching Algorithm Lesson 1888 — Three or More Constraints Lesson 2200 — Definition of Inner Product Spaces Lesson 2202 — Inner Products on Function Spaces Lesson 2232 — Properties of Projection Matrices Lesson 2550 — Definition of Equivalence Relations (+3 more)
Synthetic division: is a faster, cleaner method that works specifically when dividing by a linear binomial like (x - 3) or (x + 2).; Lesson 349 — Synthetic Division Setup
System: Lesson 1877 — Setting Up the Lagrange System Lesson 2387 — Applications: Coupled Oscillators and Mixing Problems Lesson 2397 — Solving Systems of ODEs with Laplace Transforms
System form: Lesson 2379 — Converting Higher-Order ODEs to Systems
system of linear equations: is simply a collection of two or more linear equations that we consider simultaneously.; Lesson 273 — Introduction to Systems of Linear Equations Lesson 274 — Solving Systems by Graphing: Basic Technique
System solvability: A zero determinant signals either no solution or infinitely many solutions; Lesson 2165 — Determinant and Matrix Rank

T

T (t): are time-evolution factors (how each pattern changes over time); Lesson 2421 — The General Series Solution
T is one-to-one: (injective): different inputs give different outputs; Lesson 2268 — Invertible Linear Transformations
T is onto: (surjective): every output is reachable; Lesson 2268 — Invertible Linear Transformations
T_ambient: = surrounding temperature (the "goal" temperature); Lesson 644 — Newton's Law of Cooling
t-distribution: looks like a normal distribution but with thicker tails, meaning it places more probability on extreme values.; Lesson 2924 — Properties and Uses of the t-Distribution Lesson 3014 — Confidence Interval for a Population Mean (σ Unknown)Lesson 3036 — One-Sample t-Test: Confidence Intervals
T: V → W: is **invertible** if there exists another linear transformation **T ¹: W → V** such that:; Lesson 2268 — Invertible Linear Transformations
T(n) = Θ(f(n)): Lesson 2636 — Divide-and-Conquer Recurrences
T(n) = Θ(n^(log_b a)): Lesson 2636 — Divide-and-Conquer Recurrences
T(t): = temperature at time *t*; Lesson 644 — Newton's Law of Cooling
T(v) = 0: .; Lesson 2261 — The Kernel (Null Space) of a Linear Transformation
T⁻¹: rotates it back -45°.; Lesson 1308 — Applications of Inverse Matrices
T⁻¹: W → V: such that:; Lesson 2268 — Invertible Linear Transformations
T₀: = initial temperature of the object; Lesson 644 — Newton's Law of Cooling
Take absolute values: Form the series ∑|a | from your original series ∑a; Lesson 1757 — The Absolute Convergence Test
Take each solution: you found using the quadratic formula; Lesson 421 — Verifying Solutions by Substitution
Take the absolute value: of each piece; Lesson 1573 — Area and Signed Area
Take the dot product: **F**(**r**(t)) · **r**'(t); Lesson 1977 — Computing Vector Line Integrals via Parametrization Lesson 2021 — Computing Flux for Parametric Surfaces
Take the limit: lim[h → 0] m = m; Lesson 1381 — Computing Derivatives from the Definition: Linear Functions Lesson 1634 — Type 1 Improper Integrals: Infinite Upper Limit
Take the square root: of both sides, writing √(negative) as √(positive) · i.; Lesson 409 — Complex Solutions via Completing the Square Lesson 775 — Finding the Hypotenuse Lesson 1676 — Computing Arc Length: Polynomial Functions
Taking the limit: lim(n → ∞) 2(1 + 1/n) = 2(1 + 0) = **2**; Lesson 1556 — Computing Simple Integrals Using Riemann Sum Limits
Tangent: = opposite side ÷ adjacent side; Lesson 947 — Introduction to Trigonometric Ratios Lesson 951 — The Tangent Ratio (TOA)Lesson 977 — Finding Heights Using Angles of Elevation Lesson 996 — Third Quadrant Points Lesson 1032 — Comparing and Analyzing All Six Trig Functions Lesson 1604 — Reduction Formulas for Tangent and Secant
Tangent (TOA): Use when working with **opposite** and **adjacent**; Lesson 956 — Choosing the Appropriate Trig Ratio Lesson 969 — Using SOH-CAH-TOA with Elevation Angles
Tangent & Cotangent: Unbounded; shoot to ±∞ at asymptotes; Lesson 1032 — Comparing and Analyzing All Six Trig Functions
Tangent and Cotangent: Both have a period of **π** (180°).; Lesson 1032 — Comparing and Analyzing All Six Trig Functions
tangent line: to a circle is a straight line that touches the circle at exactly one point—no more, no less.; Lesson 847 — Tangent Lines to Circles Lesson 1143 — Tangent Lines to Circles Lesson 1373 — From Secant Lines to Tangent Lines Lesson 1388 — The Derivative of a Constant Function
tangent plane: does the same thing for a surface in 3D—it's the flat plane that "just touches" the surface at a point and best approximates the surface locally.; Lesson 1832 — Tangent Planes: Geometric Intuition Lesson 1835 — Normal Lines to Surfaces Lesson 1836 — Linear Approximation in Two Variables
Taylor series: of a function *f(x)* centered at *x = a* is:; Lesson 1773 — The Taylor Series Formula
Taylor's Theorem: makes the relationship between a function and its Taylor polynomial precise.; Lesson 1784 — Taylor's Theorem Statement and Motivation Lesson 1785 — The Lagrange Form of the Remainder
Technology: Computer algebra systems become essential for k ≥ 3; Lesson 1888 — Three or More Constraints
telescoping series: works the same way mathematically—when you write out the terms, consecutive terms cancel each other, and only the first few and last few terms remain.; Lesson 684 — Telescoping Series Lesson 1717 — Telescoping Series Lesson 1729 — Telescoping Series Lesson 1750 — Strategy: Choosing the Right Convergence Test
Telescoping series clues: Lesson 1730 — Mixed Examples: Known Series
Temperature: A city averages 70°F with 15° variation over a year (period = 365 days), warmest on day 200.; Lesson 1022 — Applications: Modeling Periodic Phenomena
Temperature change: How fast is temperature rising right now?; Lesson 1376 — Applications: Velocity and Other Rates
Temperature function: If `T(h) = 68 - 2h` represents temperature in degrees Fahrenheit as a function of hours after noon, then:; Lesson 546 — Function Evaluation in Context Lesson 577 — Zeros and Intercepts
Temporal ODE: $T'(t) + \alpha^2 \lambda T(t) = 0$; Lesson 2419 — The Method of Separation of Variables Lesson 2447 — Separating Variables in the Heat Equation
Tens column: 3 + 2 = 5; Lesson 14 — Adding and Subtracting with Larger Numbers
Tens place: (the left digit) — tells you how many groups of 10; Lesson 3 — Understanding Place Value: Ones and Tens Lesson 4 — Place Value: Hundreds Lesson 7 — Rounding to the Nearest Ten and Hundred
Tenths place: (1st position right of the point) = 1/10; Lesson 83 — Understanding Decimal Place Value
term: is a single part of an expression separated by `+` or `−` signs.; Lesson 171 — Terms, Coefficients, and Constants Lesson 312 — Terms, Coefficients, and Constants Lesson 318 — What is a Polynomial? Terms and Like Terms Lesson 1703 — Definition of a Sequence and Notation
Term 0: C(4,0)x⁴(2)⁰ = 1·x⁴·1 = x⁴; Lesson 2601 — Expanding Binomials with Numerical Examples
Term 1: C(4,1)x³(2)¹ = 4·x³·2 = 8x³; Lesson 2601 — Expanding Binomials with Numerical Examples
Term 2: C(4,2)x²(2)² = 6·x²·4 = 24x²; Lesson 2601 — Expanding Binomials with Numerical Examples
Term 3: C(4,3)x¹(2)³ = 4·x·8 = 32x; Lesson 2601 — Expanding Binomials with Numerical Examples
Term 4: C(4,4)x⁰(2)⁴ = 1·1·16 = 16; Lesson 2601 — Expanding Binomials with Numerical Examples
Term 5: C(5,5)(2x)⁰(-y)⁵ = 1·1·(-y⁵) = -y⁵; Lesson 2601 — Expanding Binomials with Numerical Examples
Terms: `5x`, `2y`, and `−8` (three terms); Lesson 171 — Terms, Coefficients, and Constants Lesson 310 — What is a Polynomial?Lesson 318 — What is a Polynomial? Terms and Like Terms
Terms below were extracted from bolded phrases in lesson content. Click a lesson reference to jump
Test: Compute ∂Q/∂x and ∂P/∂y; Lesson 1988 — Testing for Conservative Fields: Cross-Partial Test in ℝ² Lesson 2336 — Strategy Guide: Choosing the Right Solution Method Lesson 2939 — Independence via Joint and Marginal Distributions
Test 1: f(g(x)); Lesson 564 — Verifying Inverse Functions
Test 2: g(f(x)); Lesson 564 — Verifying Inverse Functions
Test combinations: systematically:; Lesson 365 — Trial and Error Method for Factoring Trinomials
Test critical points: Use the First or Second Derivative Test to confirm it's a maximum; Lesson 1513 — Cost and Revenue Optimization
Test for exactness: The equation $M\,dx + N\,dy = 0$ is exact if and only if:; Lesson 2328 — Exact Differential Equations: Definition and Test
Test for linear independence: using the matrix test you learned earlier; Lesson 2089 — Finding a Basis from a Spanning Set
Test intervals: These critical values divide the number line into intervals.; Lesson 1499 — Finding Intervals of Concavity
Test Nearby Points: Lesson 1868 — Inconclusive Cases and Further Analysis
Test specific points: Pick a value of *x* in your interval and evaluate both functions.; Lesson 1645 — Identifying Top and Bottom Functions
test statistic: is a single number computed from your sample data that measures how far your data deviates from what H₀ predicts.; Lesson 3020 — Introduction to Hypothesis Testing Lesson 3022 — Test Statistics and P-values Lesson 3026 — Two-Tailed z-Tests Lesson 3028 — z-Test for Proportions Lesson 3029 — Two-Sample z-Tests Lesson 3037 — Two-Sample t-Test: Independent Samples with Equal Variances
Test the absolute series: Use any convergence test that works with non-negative terms:; Lesson 1757 — The Absolute Convergence Test
Test the sign: of f''(x) on intervals around these candidate points; Lesson 1522 — Inflection Points
Test type: one-tailed or two-tailed; Lesson 3035 — Calculating the t-Statistic and Finding Critical Values
Testing invertibility: If `det(A) = 0`, you immediately know A is not invertible (singular); Lesson 2165 — Determinant and Matrix Rank
Their combinations become manageable: Complex-looking terms unravel into limits you can actually evaluate; Lesson 1748 — Series with Factorials and Exponentials
Then: `F'(x) = f(x)` for every `x` in `(a, b)`.; Lesson 1567 — The Fundamental Theorem of Calculus: Part 1 Lesson 1737 — Direct Comparison Test: Proving Divergence Lesson 1742 — The Ratio Test: Statement and Logic Lesson 2394 — Second Shifting Theorem and Unit Step Function Lesson 2491 — Direct Proof: Basic Structure
Then divide by 2: (undo the multiplication); Lesson 202 — Combining Addition/Subtraction with Multiplication/Division
Then multiply: 2 × 9 = 18; Lesson 33 — Exponents in Order of Operations
Then separately: examine the boundary circle x² + y² = 1 (often using substitution or Lagrange multipliers, which you'll learn later); Lesson 1869 — Boundary Considerations for Optimization
Theorem: A bipartite graph G = (X ∪ Y, E) has a matching that covers all of X (a perfect matching from X to Y) **if and only if** Hall's condition holds.; Lesson 2700 — Hall's Marriage Theorem
Theorems: Lesson 766 — Common Reasons in Proofs
Theoretical insights: This connects matrix products to geometric scaling in a clean way; Lesson 2159 — Determinant of a Product
there.
Therefore: Lesson 2185 — The Trace and Determinant via Characteristic Polynomial Lesson 2491 — Direct Proof: Basic Structure
thermal diffusivity: .; Lesson 2416 — Introduction to the Heat Equation Lesson 2425 — Physical Interpretation and Decay Rates
Thermodynamics: Minimize energy subject to entropy and volume constraints; Lesson 1890 — Application: Optimization in Physics and Engineering
They match: the value equals the limit: *f(a)* equals the limit; Lesson 576 — Continuity of Functions Lesson 1817 — Continuity at a Point
Thickness: An infinitesimally small width (we'll use dx or dy); Lesson 1653 — Volume of Revolution: Introduction and Disk Method Concept Lesson 1654 — Disk Method: Rotation About the x-axis Lesson 1659 — Washer Method: Rotation About the x-axis Lesson 1664 — Introduction to the Shell Method
Think binomial: If the true median equals M₀, then each observation has probability 0.; Lesson 3104 — Sign Test for Median
Think in layers: , like peeling an onion:; Lesson 1406 — Combining Product and Quotient Rules
Think of it as: For each possible outcome pair \((x, y)\), you measure how far each is from its mean, multiply those deviations together, weight by the joint probability, and sum everything up.; Lesson 2948 — Computing Covariance from Joint Distributions
Third: f'''(x), f ³ (x), d³f/dx³; Lesson 1397 — Higher-Order Derivatives
Third column: Get a 1 in position (3,3) and zeros above it; Lesson 1305 — Row Reduction to Find Inverses of 3×3 Matrices
Third Law (Harmonic Law): The square of a planet's orbital period is proportional to the cube of the semi-major axis length (the value *a* in your ellipse equation).; Lesson 1184 — Planetary Orbits and Kepler's Laws
third position: , you have **(n-1) choices** (any remaining book); Lesson 2579 — Counting Permutations of n Distinct Objects Lesson 2580 — Permutations of n Objects Taken r at a Time
Third quartile: Q₃ (75th percentile): Solve F(x) = 0.; Lesson 2849 — Percentiles and Quantiles from the CDF
Third term: ∫7 dx = 7x; Lesson 1542 — Antiderivatives of Polynomial Functions
Third term times everything: Multiply *3* by each of the three terms: *3·x*, *3·1*, *3·4*; Lesson 329 — Multiplying Two Trinomials
Thirteen: = ten + three; Lesson 2 — Counting from 11 to 100
This is Bayes' Theorem: Lesson 2814 — Deriving Bayes' Theorem
This is constrained optimization: finding extreme values of a function *f(x, y)* subject to a constraint equation *g(x, y) = c*.; Lesson 1873 — Optimization Problems with Constraints
Thousandths place: (3rd position) = 1/1000; Lesson 83 — Understanding Decimal Place Value
Three angles: the right angle (always 90°) and the two acute angles; Lesson 957 — What Does 'Solving a Right Triangle' Mean?
Three common approaches: Lesson 1546 — Approximating Area Under a Curve with Rectangles
three conditions: on the interval [1, ∞):; Lesson 1731 — The Integral Test: Statement and Conditions Lesson 2078 — Subspaces: Definition and Subspace Test
three dimensions: .; Lesson 1808 — Level Surfaces for Functions of Three Variables Lesson 2437 — Basic Solutions in Cartesian Coordinates
three or more: categories?; Lesson 2609 — From Binomial to Multinomial Coefficients Lesson 3093 — Introduction to Analysis of Variance (ANOVA)
Three sides: the two legs and the hypotenuse; Lesson 957 — What Does 'Solving a Right Triangle' Mean?
three terms: `7y`, `−2`, and `4z`; Lesson 171 — Terms, Coefficients, and Constants Lesson 391 — Mixed Practice: Identifying Special Forms Lesson 2608 — Multinomial Theorem Preview via Trinomial Expansion
three variables: (x, y, and z), everything moves into **three-dimensional space**.; Lesson 293 — Visualizing Three-Variable Systems Lesson 1840 — Higher-Dimensional Linear Approximation
Three vectors: (not all on the same plane) span **all of R³**; Lesson 2062 — Geometric Span in R³: Lines, Planes, and All of R³
three-dimensional wave equation: is:; Lesson 2434 — The Three-Dimensional Wave Equation Lesson 2435 — Applications: Vibrating Strings, Membranes, and Sound
Three-term cubes or sums/differences: Possibly sum or difference of cubes; Lesson 341 — Recognizing Special Products in Context
Three+ independent groups: Kruskal-Wallis Test; Lesson 3111 — Choosing and Interpreting Nonparametric Tests
Three+ paired samples: Friedman Test; Lesson 3111 — Choosing and Interpreting Nonparametric Tests
Tides: If high tide is 12 feet at noon, low tide is 4 feet, and the cycle repeats every 12 hours, you'd use:; Lesson 1022 — Applications: Modeling Periodic Phenomena
Time: usually starts at `t ≥ 0`; Lesson 557 — Real-World Context and Restricted Domains
time constant: it tells you how fast the circuit decays.; Lesson 2327 — Applications: RC Circuits and Decay Lesson 2425 — Physical Interpretation and Decay Rates
Time on a clock: Hours modulo 12 creates a quotient set with 12 equivalence classes; Lesson 2558 — Applications and Quotient Sets
Time-consuming: Drawing accurate graphs takes effort; Lesson 281 — Comparing Graphing and Substitution Methods
Time-dependent boundary conditions: Lesson 2455 — Limitations and Extensions of the Method
Timing attacks: Measuring how long decryption takes can leak information about `d`.; Lesson 2770 — Security and Practical Considerations
To compute z^n: Lesson 1226 — Using De Moivre's Theorem to Compute Powers
To condense (combine): Lesson 621 — The Product Rule for Logarithms
To expand (break apart): Lesson 621 — The Product Rule for Logarithms
To find ∂f/∂x: Lesson 1824 — Computing First-Order Partial Derivatives
To find ∂f/∂y: Lesson 1824 — Computing First-Order Partial Derivatives
To find earlier terms: Lesson 652 — Finding Terms Using the Common Difference
To find f(1): Since 1 < 2, use the first formula: f(1) = 1 + 3 = 4; Lesson 545 — Evaluating Piecewise-Defined Functions
To find f(2): Since 2 ≥ 2 (the boundary belongs to the second piece), use the second formula: f(2) = 2² - 1 = 4 - 1 = 3; Lesson 545 — Evaluating Piecewise-Defined Functions
To find f(5): Since 5 ≥ 2, use the second formula: f(5) = 5² - 1 = 25 - 1 = 24; Lesson 545 — Evaluating Piecewise-Defined Functions
To prove: "If *n* is even, then *n²* is even.; Lesson 2492 — Direct Proof: Working with Definitions
TOA: **T**angent = **O**pposite / **A**djacent.; Lesson 954 — Finding Side Lengths Using Tangent Lesson 970 — Using SOH-CAH-TOA with Depression Angles
Together: they fill it in 2 hours.; Lesson 439 — Work and Mixture Rate Problems
Tossing two coins: Lesson 2772 — Sample Spaces: The Set of All Possible Outcomes
Total: 24 + 14.; Lesson 871 — Composite Figures with Circles and Semicircles Lesson 3044 — Chi-Squared Test for Independence
Total approximation: = 0 + 1 + 4 = 5 square units; Lesson 1546 — Approximating Area Under a Curve with Rectangles
Total Area: = |∫ₐᶜ f(x) dx| + |∫ᶜᵇ f(x) dx|; Lesson 1573 — Area and Signed Area Lesson 1650 — Area Enclosed by Intersecting Curves Lesson 1651 — Applications: Net vs Total Area Lesson 2839 — Graphical Interpretation of PDFs
Total arrangements: 1 × 1 × 6 = **6 ways**; Lesson 2582 — Permutations with Restrictions: Fixed Positions
Total budget: g + a + e = 500; Lesson 299 — Applications of Three-Variable Systems
total energy: at time $t$ as:; Lesson 2432 — Energy Conservation in Wave Motion Lesson 2465 — Parseval's Theorem
Total interest paid: = \(n \cdot P - L\) (total paid minus the principal).; Lesson 689 — Loan Amortization
total mass: M of the rod.; Lesson 1700 — Center of Mass for One-Dimensional Systems Lesson 1932 — Applications: Density and Total Mass Lesson 1954 — Mass and Density Functions Lesson 1958 — Center of Mass in Two Dimensions Lesson 1959 — Center of Mass in Three Dimensions
Total probability: `12,600 × (1/6)^10 ≈ 0.; Lesson 2616 — Applications in Probability and Polynomials
Total probability equals 1: The sum of *p*(*x*) over all possible values equals 1; Lesson 2832 — Introduction to Probability Mass Functions (PMFs)
Total probability is 1: ∫∫ f(x,y) dx dy = 1 (integrating over all possible values); Lesson 2928 — Joint Probability Distributions: Continuous Case
total surface area: combines all three pieces:; Lesson 888 — Surface Area of Cylinders Lesson 889 — Surface Area of Pyramids Lesson 890 — Surface Area of Cones
Total work: = 1 complete job (usually); Lesson 468 — Applications of Adding and Subtracting Rationals Lesson 1695 — Pumping Liquids from Tanks
Touch or point: – Make sure you count each object exactly once; Lesson 1 — Counting Objects from 1 to 10
trace: is the curve you get when you slice a surface with a plane parallel to a coordinate plane.; Lesson 1799 — Cylindrical Surfaces and Traces Lesson 2188 — Using the Characteristic Polynomial in Practice
Trace from the left: Follow the curve as x approaches a from values less than a.; Lesson 1322 — Estimating Limits from Graphs
Trace from the right: Follow the curve as x approaches a from values greater than a.; Lesson 1322 — Estimating Limits from Graphs
Trace the outer boundary: Imagine walking around the outside of the figure with your finger.; Lesson 870 — Perimeter of Composite Figures
Traces: work the same way for mathematical surfaces: they're the curves you get when you hold one variable constant and let the others vary.; Lesson 1809 — Traces and Cross-Sections
Trade-off: Reducing Type I errors often increases Type II errors (failing to detect real effects).; Lesson 3052 — Type I Error: False Positives
Trail: A walk in which *no edge is repeated* (but vertices may repeat).; Lesson 2657 — Walks, Trails, and Paths
Transcendental functions: (exponentials, logarithms, trig): anticipate their characteristic growth/decay patterns and periodic behavior; Lesson 1526 — Sketching Complex Functions
transform: the difference into a quotient using algebra:; Lesson 1532 — Indeterminate Differences: ∞ - ∞Lesson 1600 — Powers of Sine and Cosine: Odd Powers
Transform back: **x(t) = Py(t)** gives the solution in original coordinates; Lesson 2278 — Applications to Differential Equations and Dynamics
Transform each equation: Apply the Laplace transform to every equation in your system, using the transforms of derivatives you already know (from lesson 2391).; Lesson 2397 — Solving Systems of ODEs with Laplace Transforms
Transform the matrix: D = P ¹AP is A represented in the eigenvector basis; Lesson 2258 — Applications: Diagonalization via Change of Basis
transformation: is a rule or mapping that takes every point in a geometric figure and moves it to a new location in the plane.; Lesson 921 — Introduction to Geometric Transformations Lesson 1945 — Transformations and the Jacobian Matrix
Transient solution: Ce^(-ax) (decays exponentially if a > 0); Lesson 2325 — Linear ODEs with Constant Coefficients Lesson 2371 — Forced Harmonic Motion and Resonance
Transition matrices compose naturally: If you change from basis B to C, then from C to D, the combined effect is the same as a single transition from B to D.; Lesson 2253 — Properties of Transition Matrices
transitive: if whenever one element relates to a second, and that second relates to a third, then the first must also relate directly to the third.; Lesson 2545 — Transitive Relations Lesson 2546 — Combining Properties: Partial Orders Lesson 2549 — Inverse Relations Lesson 2550 — Definition of Equivalence Relations Lesson 2551 — Verifying Equivalence Relations Lesson 2556 — Partitions Induce Equivalence Relations Lesson 2557 — Congruence Modulo n Lesson 2731 — Equivalence Classes in Modular Arithmetic
Transitivity: Lesson 2550 — Definition of Equivalence Relations Lesson 2709 — Divisibility: Definition and Basic Properties
Translate: words into an equation with radicals; Lesson 529 — Word Problems Involving Radical Equations Lesson 936 — Dilations Centered at Points Other Than the Origin Lesson 938 — Combining Rotations and Dilations with Other Transformations Lesson 974 — Setting Up Right Triangle Problems from Word Descriptions
Translate back: Reverse the first translation by adding h back to x-coordinates and k back to y-coordinates.; Lesson 932 — Rotations About Points Other Than the Origin Lesson 936 — Dilations Centered at Points Other Than the Origin
Translate the hypothesis: *n* is even means *n = 2k* for some integer *k*.; Lesson 2492 — Direct Proof: Working with Definitions
Translate to the origin: Move the center of rotation to (0, 0) by shifting the entire figure.; Lesson 932 — Rotations About Points Other Than the Origin
Translation: Sliding a figure in any direction without rotating or flipping it.; Lesson 724 — Congruence Transformations Lesson 922 — Translations in the Coordinate Plane Lesson 924 — Properties of Translations Lesson 1733 — Estimating Remainders with the Integral Test Lesson 2415 — Physical Interpretation and Modeling Lesson 2864 — Moment Generating Functions (MGFs)
Translation (adding a constant): If you add any constant *b* to every outcome, the variance stays exactly the same: **Var(X + b) = Var(X)**; Lesson 2859 — Properties of Variance: Scaling and Translation
Translation doesn't affect spread: Think of test scores: if everyone gets 5 bonus points, the average shifts up by 5, but the *spread* between scores stays identical.; Lesson 2859 — Properties of Variance: Scaling and Translation
Translations: – Sliding every point the same distance in the same direction; Lesson 939 — Introduction to Rigid Motions (Isometries)Lesson 941 — Congruence Through Rigid Motions
Transportation: A bridge (cut edge) connects two landmasses; destroying it isolates them.; Lesson 2661 — Cut Vertices and Cut Edges
transpose: of a matrix is a new matrix created by exchanging the rows and columns of the original matrix.; Lesson 1279 — The Transpose of a Matrix Lesson 2102 — The Transpose of a Matrix
transversal: is simply a line that intersects (crosses) two or more other lines at distinct points.; Lesson 713 — Introduction to Transversals Lesson 714 — Corresponding Angles
transverse axis: (the axis along which the hyperbola opens).; Lesson 1168 — Components of a Hyperbola: Center, Vertices, and Foci Lesson 1169 — The Transverse and Conjugate Axes Lesson 1172 — Hyperbolas with Horizontal vs Vertical Transverse Axes
Trapezoidal Rule: Lesson 1681 — Numerical Approximation of Arc Length
Trapezoids: sit on a separate branch.; Lesson 825 — The Quadrilateral Family Tree
Traveling in opposite directions: (combined rates add); Lesson 484 — Distance-Rate-Time and Mixture Problems
Traveling Salesman Problem: finding the shortest Hamiltonian cycle in a weighted graph where vertices represent cities and edge weights represent distances.; Lesson 2688 — Applications: Traveling Salesman Problem and DNA Sequencing
Treat each transversal separately: Apply the angle relationships you know (corresponding angles are equal, alternate interior angles are equal, same-side interior angles are supplementary, etc.; Lesson 720 — Parallel Lines Cut by Multiple Transversals
Treat y as constant: (since it's a partial derivative); Lesson 1829 — Implicit Partial Differentiation
Treating f(x) as Multiplication: Lesson 547 — Common Function Notation Mistakes
tree: is a connected graph that contains no cycles.; Lesson 2668 — Definition and Properties of Trees Lesson 2696 — Coloring Special Graphs
tree diagram: approach still applies: draw branches from `w` to each intermediate variable (`x`, `y`, `z`), then from each of those to the independent variables (`s`, `t`).; Lesson 1857 — Chain Rule with Three or More Variables Lesson 2796 — Sequential Experiments and Tree Diagrams
Trends: Residuals gradually increasing or decreasing over time; Lesson 3090 — Checking for Independence: Residual Sequence Plots
Trial and Error: Test different factor combinations of both a and c until the middle term checks out; Lesson 368 — Factoring Trinomials with Leading Coefficient Greater Than 1 Lesson 373 — Applications and Strategy Selection for Trinomial Factoring
Triangle: If a triangle has sides of 3 cm, 4 cm, and 5 cm:; Lesson 856 — Perimeter of Polygons
Triangle 1: From point A, you measure an angle of elevation to the cliff top.; Lesson 980 — Two Right Triangle Problems and Indirect Measurement Lesson 1124 — Solving for Two Possible Triangles
Triangle 2: Move to point B (closer or farther).; Lesson 980 — Two Right Triangle Problems and Indirect Measurement Lesson 1124 — Solving for Two Possible Triangles
Triangle ACD: (left smaller triangle); Lesson 800 — The Altitude to the Hypotenuse
Triangle Angle Sum Theorem: .; Lesson 732 — Triangle Angle Sum Theorem Lesson 733 — Exterior Angle Theorem Lesson 807 — Interior Angles of a Triangle
Triangle area: = ½(10)(10)sin(60°) = 43.; Lesson 865 — Area of Segments
Triangle CBD: (right smaller triangle); Lesson 800 — The Altitude to the Hypotenuse
triangle inequality: for norms; Lesson 2204 — The Cauchy-Schwarz Inequality Lesson 2205 — The Triangle Inequality
Triangle Inequality Theorem: states that *the sum of the lengths of any two sides of a triangle must be greater than the length of the third side*.; Lesson 737 — Triangle Inequality Theorem
Triangular: (upper or lower, when diagonalization isn't possible); Lesson 2257 — Finding Convenient Bases
triangular matrix: (upper triangular or lower triangular), the determinant equals simply the **product of the diagonal entries**.; Lesson 2154 — Determinant of Triangular Matrices Lesson 2174 — Eigenvalues of Triangular Matrices
Trigonometric Functions: sin(x), cos(x), and tan(x) are continuous on their domains.; Lesson 1354 — Continuity of Elementary Functions Lesson 1646 — Finding Intersection Points
Trigonometric Identities: If your derivative involves trig functions, use identities like `1 + tan²θ = sec²θ` to simplify under the radical.; Lesson 1680 — Simplifying Arc Length Integrals
Trigonometric substitution: solves this problem by exploiting Pythagorean identities.; Lesson 1611 — Introduction to Trigonometric Substitution Lesson 1612 — The Three Standard Forms Lesson 1620 — Applications and Strategy Selection Lesson 1677 — Arc Length with Radical and Rational Functions
Trinomial squared: (a + b + c)² → extended square formula; Lesson 341 — Recognizing Special Products in Context
triple integral: ∭ f(x,y,z)dV accumulates values throughout a solid region in space (3D); Lesson 1922 — Introduction to Triple Integrals Lesson 1932 — Applications: Density and Total Mass
Triple integrals: take this one step further: they integrate functions over three-dimensional regions in space.; Lesson 1922 — Introduction to Triple Integrals Lesson 1923 — Setting Up Triple Integrals over Rectangular Boxes
true: , shade the side containing that test point.; Lesson 300 — Graphing Linear Inequalities in Two Variables Lesson 525 — Checking for Extraneous Solutions Lesson 2468 — Propositions and Truth Values Lesson 2469 — Logical Negation Lesson 2470 — Conjunction (AND)Lesson 2471 — Disjunction (OR)Lesson 2473 — Biconditional Statements (If and Only If)Lesson 2479 — Predicates and Propositional Functions (+1 more)
True bearings: Measured clockwise from north (0° to 360°); Lesson 979 — Navigation and Bearing Problems
Try the Integral Test: if the function is continuous, positive, and decreasing; Lesson 1744 — When the Ratio Test is Inconclusive
twice: to reduce the system down to something you can solve.; Lesson 295 — Choosing Strategic Pairs for Elimination Lesson 432 — Using the Discriminant to Predict Intercepts Lesson 1593 — Polynomial Times Trigonometric Functions Lesson 2650 — Degree of a Vertex and the Handshaking Lemma
Twin Prime Conjecture: states there are infinitely many such pairs.; Lesson 2729 — Special Primes: Mersenne, Fermat, and Twin Primes
Twin primes: are pairs of primes that differ by 2: (3,5), (5,7), (11,13), (17,19), (29,31).; Lesson 2729 — Special Primes: Mersenne, Fermat, and Twin Primes
two: inputs: an `x` value and a `y` value; Lesson 543 — Function Notation with Multiple Variables Lesson 1884 — The Method with Two Constraints Lesson 2433 — The Two-Dimensional Wave Equation Lesson 2449 — Solving the Temporal ODE Lesson 2609 — From Binomial to Multinomial Coefficients
Two circular bases: (top and bottom); Lesson 888 — Surface Area of Cylinders
Two columns: Lesson 763 — Two-Column Proof Format
two complex solutions: (involving imaginary numbers) because you cannot take the square root of a negative number in the real number system.; Lesson 416 — The Discriminant: b² - 4ac Lesson 417 — Interpreting Discriminant Values
Two independent groups: Mann-Whitney U Test; Lesson 3111 — Choosing and Interpreting Nonparametric Tests
Two intersecting lines: Lesson 1134 — Degenerate Cases
Two main problem types: Lesson 1447 — Applications and Implicit Differentiation Problems
two numbers: (or infinity symbols) separated by a comma, surrounded by brackets or parentheses:; Lesson 222 — Interval Notation for Solutions Lesson 227 — Solving Basic Absolute Value Equations Lesson 2905 — Parameters: Mean and Standard Deviation
two or more: other variables simultaneously.; Lesson 126 — Joint Variation Lesson 2775 — Simple Events vs Compound Events
Two paired samples: Wilcoxon Signed-Rank Test; Lesson 3111 — Choosing and Interpreting Nonparametric Tests
Two parallel lines: Lesson 1134 — Degenerate Cases
Two right triangles: (each with a 90° angle); Lesson 745 — HL (Hypotenuse-Leg) Congruence Theorem
Two secants: Each secant intercepts two points, creating two arcs.; Lesson 841 — Angles Formed by Secants and Tangents Outside
two separate equations: Lesson 227 — Solving Basic Absolute Value Equations Lesson 228 — Absolute Value Equations with Expressions
Two sides: (leg-leg, leg-hypotenuse, etc.; Lesson 964 — Choosing the Right Strategy
two solutions: x = 5 and x = −5.; Lesson 393 — Solving x² = c by Square Roots Lesson 414 — Solving Quadratics When a ≠ 1 Lesson 920 — Intersection of Lines and Circles
Two tangents: Each tangent intercepts one point.; Lesson 841 — Angles Formed by Secants and Tangents Outside
two terms: `3x` and `5`; Lesson 171 — Terms, Coefficients, and Constants Lesson 391 — Mixed Practice: Identifying Special Forms
Two triangles: The opposite side is long enough to "overshoot" and create two valid configurations; Lesson 1096 — The SSA Configuration Introduction Lesson 1122 — The Ambiguous Case with Acute Angles Lesson 1123 — Using the Height to Determine Triangle Existence
Two vectors: (not on the same line) span a **plane** through the origin; Lesson 2062 — Geometric Span in R³: Lines, Planes, and All of R³
Two x-intercepts: The parabola crosses the x-axis at two different points (the discriminant b² - 4ac > 0); Lesson 431 — x-Intercepts and Zeros of Quadratic Functions
Two-tailed probabilities: Use symmetry around zero; Lesson 2908 — Using the Standard Normal Table
Two-way ANOVA without interaction: lets you test both factors at once, under the assumption that each factor's effect is *additive* and independent.; Lesson 3099 — Two-Way ANOVA Without Interaction
Type 1 region: in three dimensions is like a sandwich where the "bread slices" are surfaces that depend on x and y.; Lesson 1924 — Type 1 Regions: z Between Functions of x and y Lesson 1930 — Volume as a Triple Integral
Type I: (vertically simple): bounded by x = a to x = b, and y = g₁(x) to y = g₂(x); Lesson 1911 — Volume Under Surfaces over General Regions
Type I (vertically simple): regions where vertical slices were easiest.; Lesson 1903 — Type II Regions: Horizontally Simple Regions Lesson 1904 — Setting Up Integrals for Triangular Regions Lesson 1905 — Changing the Order of Integration Lesson 1908 — Evaluating Double Integrals over General Regions
Type I error: occurs when we reject the null hypothesis even though it's actually true.; Lesson 3052 — Type I Error: False Positives
Type I Error (α): Rejecting a true null hypothesis (false positive); Lesson 3054 — The Relationship Between Type I and Type II Errors
Type I error rate: .; Lesson 3055 — Significance Level and Error Control
Type I region: (or **vertically simple region**) is one where:; Lesson 1902 — Type I Regions: Vertically Simple Regions Lesson 1906 — Regions Bounded by Curves Lesson 1907 — Finding Limits from Graphs and Inequalities Lesson 1910 — Applications: Area of General Regions Lesson 1911 — Volume Under Surfaces over General Regions
Type II: (horizontally simple): bounded by y = c to y = d, and x = h₁(y) to x = h₂(y); Lesson 1911 — Volume Under Surfaces over General Regions
Type II (Horizontally Simple): For each fixed y-value, x ranges from a left boundary to a right boundary.; Lesson 1904 — Setting Up Integrals for Triangular Regions Lesson 1905 — Changing the Order of Integration Lesson 1908 — Evaluating Double Integrals over General Regions
Type II error: occurs when we **fail to reject the null hypothesis when it is actually false**.; Lesson 3053 — Type II Error: False Negatives
Type II Error (β): Failing to reject a false null hypothesis (false negative); Lesson 3054 — The Relationship Between Type I and Type II Errors Lesson 3056 — Introduction to Statistical Power
Type II region: is called **horizontally simple** when you can describe it by:; Lesson 1903 — Type II Regions: Horizontally Simple Regions Lesson 1906 — Regions Bounded by Curves Lesson 1907 — Finding Limits from Graphs and Inequalities Lesson 1910 — Applications: Area of General Regions
Typical scenario: You want the height of a cliff, but there's a lake in the way.; Lesson 980 — Two Right Triangle Problems and Indirect Measurement
Typing: 40 words per minute (words ÷ minutes); Lesson 109 — Introduction to Rates
Typos: per page in a manuscript; Lesson 2893 — Applications of the Poisson Distribution

U

u: } by normalizing each vector:; Lesson 2223 — Normalizing to Obtain an Orthonormal Basis Lesson 2225 — Why Gram-Schmidt Works: The Proof Lesson 2230 — Orthogonal Projection onto a Subspace
u: } is an orthonormal basis and you want to express vector **v** in this basis, the coordinates are simply:; Lesson 2212 — Orthonormal Bases Lesson 2219 — Motivation for Orthonormal Bases Lesson 2273 — Spectral Decomposition
U W: instead of U + W, and we say "U and W form a direct sum.; Lesson 2085 — Direct Sums
U_k: and **V_k** contain only the first *k* columns.; Lesson 2290 — Applications of SVD: Image Compression and Noise Reduction
U_r: is m×r (only the first r columns of U); Lesson 2285 — Reduced and Full SVD
u₁: = (3, 0); Lesson 2221 — Gram-Schmidt in ℝ²: First Example Lesson 2222 — Gram-Schmidt in ℝ³: Complete Example Lesson 2223 — Normalizing to Obtain an Orthonormal Basis Lesson 2230 — Orthogonal Projection onto a Subspace Lesson 2273 — Spectral Decomposition
u₂: = (2, 4); Lesson 2221 — Gram-Schmidt in ℝ²: First Example Lesson 2222 — Gram-Schmidt in ℝ³: Complete Example Lesson 2223 — Normalizing to Obtain an Orthonormal Basis Lesson 2230 — Orthogonal Projection onto a Subspace Lesson 2273 — Spectral Decomposition
uᵢ: is simply:; Lesson 2214 — Coordinates with Respect to Orthonormal Bases Lesson 2223 — Normalizing to Obtain an Orthonormal Basis Lesson 2273 — Spectral Decomposition
unbiased: if its expected value equals the true parameter.; Lesson 2999 — Properties of Good Estimators Lesson 3000 — Sample Mean and Variance as Estimators
Uncountable values: Between any two values of a continuous random variable, there are infinitely many other possible values.; Lesson 2825 — Continuous Random Variables: Definition and Examples
Undefined: Line is vertical |; Lesson 246 — Positive, Negative, Zero, and Undefined Slopes Lesson 250 — Slope and Direction of Lines Lesson 443 — Evaluating Rational Expressions Lesson 1009 — Reciprocal Trigonometric Functions: Secant, Cosecant, Cotangent Lesson 1456 — Horizontal and Vertical Tangents Lesson 1543 — The Special Case: Antiderivative of 1/x
Undefined slope: Vertical lines are infinitely steep—steeper than any line with a defined numerical slope; Lesson 249 — Comparing Steepness Using Slope Lesson 263 — Horizontal and Vertical Lines Lesson 270 — Special Cases: Horizontal and Vertical Lines
Underdamped: (Δ < 0, complex conjugate roots); Lesson 2347 — Applications: Free Oscillations and Damped Motion
Understand behavior differences: One function might grow faster than another; Lesson 542 — Working with Multiple Functions
Understanding behavior: The eigenvector basis reveals the "natural directions" along which A acts independently; Lesson 2258 — Applications: Diagonalization via Change of Basis
Understanding restrictions: If a function isn't one-to-one, its reflection won't pass the vertical line test, confirming it has no inverse; Lesson 565 — Graphs of Inverse Functions
Undetermined coefficients: is your power drill: fast, efficient, but limited.; Lesson 2367 — Comparing Methods: When to Use Variation of Parameters
Undirected: The connection goes both ways (like mutual friendship); Lesson 2647 — What is a Graph? Vertices and Edges
Undo addition/subtraction first: (get rid of the constant term); Lesson 220 — Solving Multi-Step Linear Inequalities
Undo multiplication/division next: Lesson 220 — Solving Multi-Step Linear Inequalities
Uniform convergence: is stronger: the series converges to f(x) *equally well* across the entire interval, with no "bad spots" where convergence is slow or fails.; Lesson 2460 — Convergence of Fourier Series
Uniform(a, b): The "flat" distribution:; Lesson 2962 — MGFs of Common Distributions
Unilaterally connected: For every pair of vertices u and v, there's a directed path from u to v *or* from v to u (or both), but not necessarily both ways.; Lesson 2665 — Weakly Connected Digraphs
union: (combination) of both sets—any value that satisfies *at least one* of the inequalities.; Lesson 232 — Solving 'Or' Compound Inequalities Lesson 2524 — Union of Sets Lesson 2529 — Laws of Set Operations Lesson 2620 — Counting with At Least One Property
Union ( ): The event "A or B" occurs when *at least one* of the events happens; Lesson 2777 — Operations on Events: Union and Intersection
Union over intersection: *A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)*; Lesson 2529 — Laws of Set Operations
unique: .; Lesson 2244 — Uniqueness and Existence of QR Decomposition Lesson 2295 — The Cholesky Decomposition Lesson 2306 — Existence and Uniqueness Theorems Lesson 2346 — Existence and Uniqueness for Second-Order ODEs Lesson 2710 — Division Algorithm Lesson 2724 — Unique Factorization: Statement of the Theorem
Unique divisibility: If a relatively prime pair both divide some number n, then their product must also divide n.; Lesson 2718 — Relatively Prime Integers
Unique factors: 2, 3, and x; Lesson 461 — Finding the Least Common Denominator (LCD)
Unique path property: – There's exactly one path between any two vertices; Lesson 2668 — Definition and Properties of Trees
unique solution: by calculating ratios of determinants.; Lesson 1317 — Cramer's Rule Lesson 2074 — Applications to Solutions of Linear Systems Lesson 2124 — Solving Systems Using RREF Lesson 2125 — Recognizing Inconsistent and Dependent Systems Lesson 2137 — Applications to System Solvability Lesson 2346 — Existence and Uniqueness for Second-Order ODEs
Unique solutions: means the transformation **A** is reversible—you can always work backward; Lesson 2144 — The Invertible Matrix Theorem
Uniquely determines: the distribution (like MGFs); Lesson 2967 — Characteristic Functions (Optional)
Uniqueness: Suppose v = w₁ + w₁ ⊥ = w₂ + w₂ ⊥.; Lesson 2217 — Decomposition into Orthogonal Subspaces Lesson 2412 — Well-Posed Problems: Existence, Uniqueness, Stability Lesson 2488 — Uniqueness Quantifier and Applications Lesson 2724 — Unique Factorization: Statement of the Theorem Lesson 2753 — The Chinese Remainder Theorem Statement Lesson 2864 — Moment Generating Functions (MGFs)
unit circle: is simply a circle with a radius of exactly 1 unit, centered at the origin (0, 0) of the coordinate plane.; Lesson 991 — Defining the Unit Circle Lesson 1011 — Applications: Angles of Any Measure in Problems
unit rate: is a special kind of rate where the second quantity is exactly 1.; Lesson 110 — Unit Rates Lesson 111 — Comparing Rates to Make Decisions Lesson 252 — Slope in Real- World Applications
unit vector: is simply a vector with magnitude (length) of exactly 1.; Lesson 1239 — Unit Vector Notation: i, j, and k Lesson 1249 — Unit Vectors and Normalization Lesson 1259 — Vector Projection Formula Lesson 1843 — Directional Derivatives: Motivation and Definition Lesson 1849 — Directional Derivatives in Non-Unit Directions Lesson 2054 — Unit Vectors and Normalization
Unitary matrices: (A* = A ¹) are normal; Lesson 2276 — The Spectral Theorem for Normal Matrices
Units: Density has units like kg/m² (2D) or kg/m³ (3D), so integrating gives total mass in kg.; Lesson 1954 — Mass and Density Functions Lesson 2738 — Complete and Reduced Residue Systems Lesson 3064 — Interpreting Slope and Intercept
Universal form: "For all x in domain D, if P(x), then Q(x)"; Lesson 2489 — Structure of Mathematical Statements
universe of discourse: .; Lesson 2482 — Domains of Discourse Lesson 2522 — Universal Set and Empty Set
Unknown: `w = ?; Lesson 193 — Perimeter and Area Word Problems Lesson 3016 — Confidence Interval for the Difference of Two Means Lesson 3034 — One-Sample t-Test: Testing a Population Mean
Unlike radicals: Lesson 506 — Identifying Like and Unlike Radicals
Unlike terms: (cannot be combined):; Lesson 318 — What is a Polynomial? Terms and Like Terms
Unpack definitions: into mathematical expressions; Lesson 2493 — Direct Proof Examples and Practice
Unpredictability: – You cannot predict the exact outcome in advance; Lesson 2771 — What is a Random Experiment?
Unreliable interpretation: Can't trust which variables truly matter; Lesson 3080 — Multicollinearity: Detection and Consequences
unstable: Lesson 2278 — Applications to Differential Equations and Dynamics Lesson 2383 — Phase Portraits and Stability Analysis
Unstable coefficients: Small changes in data can produce wildly different estimates; Lesson 3080 — Multicollinearity: Detection and Consequences
Unstable node: Both eigenvalues positive → all trajectories move away from equilibrium; Lesson 2383 — Phase Portraits and Stability Analysis
Unstable spiral: Real part α > 0 → trajectories spiral outward; Lesson 2383 — Phase Portraits and Stability Analysis
up: , you multiply all parts of a ratio by the same number to make them bigger.; Lesson 118 — Scaling Up and Down Lesson 239 — Plotting Points on the Coordinate Plane Lesson 578 — Vertical Shifts of Functions Lesson 1025 — Transformations of Tangent: Phase Shift and Vertical Shift
Update: the residual network; Lesson 2706 — The Ford-Fulkerson Algorithm
Updated: P(Disease | Positive) = ?; Lesson 2801 — Updating Probabilities with New Information
Upper fence: Q3 + 1.; Lesson 2991 — Box Plots and Outlier Detection
Upper Sum: U = Σ M_i · Δx, where M_i is the maximum value of *f* on the *i*-th subinterval; Lesson 1554 — Upper and Lower Sums
Upper sums: use the **maximum** function value on each subinterval as the rectangle height, guaranteeing an overestimate (assuming a positive function).; Lesson 1554 — Upper and Lower Sums
Upper triangular: matrices have zeros below the diagonal; **lower triangular** matrices have zeros above it.; Lesson 1295 — Determinants of Special Matrices Lesson 2154 — Determinant of Triangular Matrices
upward: (focus above vertex); Lesson 1145 — Standard Form: Vertex at Origin, Vertical Axis Lesson 1148 — Parabolas with Vertex at (h, k): Vertical Axis Lesson 1501 — The Second Derivative Test for Local Extrema
Urban Planning: Analyzing pedestrian flow and designing efficient inspection routes for infrastructure (like checking every bridge or road segment) relies on Eulerian path concepts.; Lesson 2682 — Applications of Eulerian Paths: The Königsberg Bridge Problem
Use addition or subtraction: to move variable terms to one side; Lesson 203 — Equations with Variables on One Side Lesson 221 — Inequalities with Variables on Both Sides
Use combinations: C(10,3); Lesson 2594 — Distinguishing When to Use Combinations vs Permutations
Use congruence when: Lesson 729 — Distinguishing Congruence from Similarity
Use constraints: (given conditions) to reduce everything to one variable; Lesson 1511 — Box and Container Optimization
Use context: If you're minimizing cost, the smallest function value wins.; Lesson 1891 — Verifying and Interpreting Solutions
Use Contradiction when: Lesson 2500 — Choosing the Right Proof Method
Use Contrapositive when: Lesson 2500 — Choosing the Right Proof Method
Use correct postulates: Match your given information to SSS, SAS, ASA, AAS, or HL; Lesson 748 — Writing Two-Column Congruence Proofs
Use Direct Comparison: if you can match it to a known series; Lesson 1744 — When the Ratio Test is Inconclusive
Use direct proof techniques: to show ¬P must follow; Lesson 2495 — Proof by Contrapositive: Method and Strategy
Use Direct Proof when: Lesson 2500 — Choosing the Right Proof Method
Use fact families: 3 × 9 = 27, 9 × 3 = 27, 27 ÷ 3 = 9, 27 ÷ 9 = 3 all belong together.; Lesson 25 — Division Facts and Basic Division Fluency
Use Form 1: when you only know cos θ and need to work with squares.; Lesson 1070 — Half Angle Formula for Tangent
Use Form 2: when you know both sin θ and cos θ.; Lesson 1070 — Half Angle Formula for Tangent
Use Green's Theorem when: Lesson 1997 — Using Green's Theorem to Evaluate Line Integrals
Use Higher-Order Derivatives: Lesson 1868 — Inconclusive Cases and Further Analysis
Use horizontal slices: at height *y*; Lesson 1660 — Washer Method: Rotation About the y-axis
Use integration by parts: for products like x·eˣ or x·ln(x) without radicals.; Lesson 1620 — Applications and Strategy Selection
Use invariant subspaces: If certain subspaces map to themselves, choose basis vectors adapted to that structure.; Lesson 2257 — Finding Convenient Bases
Use known values: (√2/2)(√3/2) − (√2/2)(1/2); Lesson 1055 — Evaluating Non-Standard Angles Using Sum Formulas
Use logarithm properties: to break down products, quotients, and powers; Lesson 1463 — Logarithmic Differentiation with Radicals
Use logical steps: to follow where that assumption leads; Lesson 767 — Proof by Contradiction
Use multinomial coefficients: when:; Lesson 2615 — Relationship to Combinations and Stars-and-Bars
Use multiplication or division: to isolate the variable; Lesson 203 — Equations with Variables on One Side
Use normal parameters: Approximate *X* with a normal distribution having:; Lesson 2985 — Approximating Binomial Distributions
Use other proof techniques: Combine with previously proven theorems (like vertical angles, linear pairs, or angle addition); Lesson 771 — Proving Properties of Parallel Lines
Use periodicity: to identify if that solution repeats within your interval; Lesson 1082 — Finding All Solutions in an Interval
Use permutations: P(10,3); Lesson 2594 — Distinguishing When to Use Combinations vs Permutations
Use point-slope form: with the new slope and the given point to write your equation; Lesson 268 — Writing Equations of Perpendicular Lines Lesson 1468 — Normal Lines to Curves
Use polar coordinates when: Lesson 1920 — Choosing Between Rectangular and Polar
Use rectangular coordinates when: Lesson 1920 — Choosing Between Rectangular and Polar
Use results: from early steps to unlock later steps; Lesson 712 — Solving Multi-Step Angle Problems
Use row operations: to create zeros in all positions directly below it; Lesson 2108 — Forward Elimination Process Lesson 2121 — Converting REF to RREF
Use similarity: The sun's rays create similar triangles (AA criterion—both have a right angle and share the sun's angle); Lesson 761 — Applications and Proofs with Similar Triangles
Use similarity when: Lesson 729 — Distinguishing Congruence from Similarity
Use special triangle values: Calculate cos and sin of the reference angle using your knowledge from the first quadrant (30-60- 90 and 45-45-90 triangles).; Lesson 997 — Fourth Quadrant Points
Use standard tables: All your probability calculations can reference a single Z-table instead of infinite possible normal distributions; Lesson 2907 — Standardization and Z-Scores
Use stars-and-bars: when:; Lesson 2615 — Relationship to Combinations and Stars-and-Bars
Use subtraction strategically: Sometimes it's faster to find the area of a large simple shape and subtract cut-out regions rather than add many small positive areas together.; Lesson 874 — Optimizing Composite Figure Problems
Use sum/difference rules: to combine the results; Lesson 1393 — Combining Basic Rules for Polynomials
Use symmetry: if the integrand has rotational symmetry in θ, that integral becomes just 2π; Lesson 1942 — Evaluating Triple Integrals Using Spherical Coordinates
Use that piece's formula: to calculate the output; Lesson 591 — Evaluating Piecewise Functions
Use the corresponding formula: for that interval; Lesson 545 — Evaluating Piecewise-Defined Functions
Use the distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]; Lesson 1482 — Distance and Position Related Rates
Use the linear approximation: to predict values near your known point; Lesson 1474 — Tangent Line Approximation in Applied Problems
Use the point-normal form: of the plane:; Lesson 1834 — Finding Tangent Planes to Level Surfaces
Use the power rule: to bring down the exponent:; Lesson 631 — Using Logarithms to Solve Exponential Equations
Use the quotient property: if you have a fraction under the radical; Lesson 504 — Combining Simplification Techniques
Use the reference angle: to find the "base" coordinates as if you were in quadrant I; Lesson 995 — Second Quadrant Points
Use the smaller U: to find your p-value from tables or approximations; Lesson 3106 — Mann-Whitney U Test (Wilcoxon Rank-Sum)
Use this angle: with your right triangle and SOH-CAH-TOA; Lesson 976 — Angles of Depression: Definition and Setup
Use trigonometric identities: for integrals like sin³(x)cos²(x) that don't involve algebraic radicals.; Lesson 1620 — Applications and Strategy Selection
Use trigonometry: The angle θ satisfies `tan(θ) = b/a`; Lesson 1210 — Argument of a Complex Number
Use u-substitution: if you spot an inner function whose derivative appears nearby (it's faster and simpler).; Lesson 1620 — Applications and Strategy Selection
Uses the fewest edges: necessary to maintain connectivity; Lesson 2672 — Spanning Trees of a Graph
Using a colon: 3:5; Lesson 105 — What is a Ratio?
Using ASA or AAS: Surveyors use angle measurements from known distances to map terrain.; Lesson 751 — Applications: Constructing Congruent Figures
Using colons: Lesson 114 — What is a Proportion?
Using derivatives: Treat b as a function f(x) and compute f'(x).; Lesson 1753 — Verifying Conditions for the Alternating Series Test
Using HL: For right-angled structures (like corner braces), measuring just the hypotenuse and one leg ensures you can build a congruent duplicate.; Lesson 751 — Applications: Constructing Congruent Figures
Using identities: that expand the domain or change restrictions; Lesson 1091 — Extraneous Solutions and Verification
Using Method 1: The LCD is `x`.; Lesson 472 — Complex Fractions with Single-Variable Expressions
Using Method 2: The denominator becomes `(x + 2)/x`, so:; Lesson 472 — Complex Fractions with Single-Variable Expressions
Using SAS: To duplicate a triangular roof truss, measure two beams and the angle between them.; Lesson 751 — Applications: Constructing Congruent Figures
Using SSS: If you know all three side lengths of a triangular garden bed, you can recreate an identical bed elsewhere by cutting three boards to the exact same lengths and assembling them.; Lesson 751 — Applications: Constructing Congruent Figures
Using the 360° Property: If you have multiple arcs making up the entire circle:; Lesson 844 — Arc Addition and Subtraction
Using the word "to": 3 to 5; Lesson 105 — What is a Ratio?Lesson 114 — What is a Proportion?

V

v: vᵢ**⟩ = ⟨**0**, **v ᵢ**⟩ = 0; Lesson 2210 — Orthogonal Sets of Vectors Lesson 2223 — Normalizing to Obtain an Orthonormal Basis Lesson 2225 — Why Gram-Schmidt Works: The Proof
v: and scalars c₁, c₂, .; Lesson 2055 — What is a Linear Combination?Lesson 2059 — The Span of a Set of Vectors Lesson 2060 — Determining if a Vector is in a Span
V = (1/3)Bh: Lesson 880 — Volume of Pyramids
V = (1/3)πr²h: Lesson 881 — Volume of Cones
V = (4/3)πr³: .; Lesson 882 — Volume of Spheres
V = ⅓Bh: .; Lesson 901 — Deriving Volume Formulas Using Cavalieri's Principle
V = ⅓πr²h: .; Lesson 901 — Deriving Volume Formulas Using Cavalieri's Principle
V = 3: vertices; Lesson 2691 — Euler's Formula for Planar Graphs
V = 4: vertices; Lesson 2691 — Euler's Formula for Planar Graphs
V = ⁴⁄₃πr³: .; Lesson 901 — Deriving Volume Formulas Using Cavalieri's Principle
V = Bh: .; Lesson 898 — Applying Cavalieri's Principle to Prisms
V = πr²h: Lesson 879 — Volume of Cylinders Lesson 899 — Cavalieri's Principle with Cylinders
V_k: contain only the first *k* columns.; Lesson 2290 — Applications of SVD: Image Compression and Noise Reduction
V_r^T: is r×n (only the first r rows of V^T); Lesson 2285 — Reduced and Full SVD
V-shape: Lesson 594 — The Absolute Value Function
V^T: is the transpose of an n×n orthogonal matrix (columns of V are orthonormal eigenvectors of A^TA); Lesson 2280 — What is the Singular Value Decomposition?Lesson 2281 — Geometric Interpretation of SVD Lesson 2285 — Reduced and Full SVD
v₁: , **v₂**, .; Lesson 2055 — What is a Linear Combination?Lesson 2057 — Computing Linear Combinations in R³ and Beyond Lesson 2059 — The Span of a Set of Vectors Lesson 2060 — Determining if a Vector is in a Span Lesson 2062 — Geometric Span in R³: Lines, Planes, and All of R³ Lesson 2068 — Testing Linear Independence in R³ Lesson 2089 — Finding a Basis from a Spanning Set Lesson 2191 — Constructing P and D from Eigenvectors (+4 more)
v₁, v₂, ..., v: are linearly independent if the only solution to the equation; Lesson 2065 — Definition of Linear Independence Lesson 2069 — The Matrix Test for Linear Independence Lesson 2081 — Column Space and Null Space Lesson 2163 — Determinants and Linear Independence
v₂: , .; Lesson 2055 — What is a Linear Combination?Lesson 2057 — Computing Linear Combinations in R³ and Beyond Lesson 2059 — The Span of a Set of Vectors Lesson 2060 — Determining if a Vector is in a Span Lesson 2062 — Geometric Span in R³: Lines, Planes, and All of R³ Lesson 2068 — Testing Linear Independence in R³ Lesson 2089 — Finding a Basis from a Spanning Set Lesson 2191 — Constructing P and D from Eigenvectors (+4 more)
v₃: = ⟨0, 1, -1⟩.; Lesson 2057 — Computing Linear Combinations in R³ and Beyond Lesson 2062 — Geometric Span in R³: Lines, Planes, and All of R³ Lesson 2068 — Testing Linear Independence in R³ Lesson 2089 — Finding a Basis from a Spanning Set Lesson 2222 — Gram-Schmidt in ℝ³: Complete Example
Valid partition: `{{1, 2}, {3}, {4}}`; Lesson 2554 — Partitions of a Set
Validation: Check assumptions through residual diagnostics before trusting conclusions; Lesson 3102 — ANOVA Applications and Experimental Design
Value from reference: sin(30°) = 1/2; Lesson 1005 — Evaluating Trig Functions for Angles Greater than 90°
Values between: Indicate the strength and direction of linear association; Lesson 2952 — Properties of Correlation
Var(aX) = a²Var(X): Lesson 2859 — Properties of Variance: Scaling and Translation
Var(X) = np(1-p): , derived by viewing X as a sum of independent Bernoulli trials.; Lesson 2877 — Expectation and Variance of Binomial Distribution
Var(X) = α/β²: Lesson 2918 — Properties of the Gamma Distribution
Var(X) = λ: Lesson 2862 — Expectation and Variance in Common Distributions
Variability in the data: More noise makes real signals harder to detect; Lesson 3053 — Type II Error: False Negatives
variable: is a symbol (usually a letter) that stands in for a number we don't know yet or a number that can change.; Lesson 169 — What is a Variable?Lesson 170 — Writing Expressions with Variables Lesson 171 — Terms, Coefficients, and Constants Lesson 310 — What is a Polynomial?
Variable Bases and Exponents: Lesson 1458 — When to Use Logarithmic Differentiation
Variable coefficients: At least one of `P(x)` or `Q(x)` depends on `x` (like `y'' + xy' + y = 0`); Lesson 2337 — Form and Classification of Second-Order Linear ODEs Lesson 2455 — Limitations and Extensions of the Method
Variable density: The density function captures how mass is distributed—higher values mean more mass concentrated there.; Lesson 1954 — Mass and Density Functions
Variables: `x` and `y`; Lesson 171 — Terms, Coefficients, and Constants Lesson 176 — Exponents in Variable Expressions Lesson 2479 — Predicates and Propositional Functions
Variables in denominators: – Dividing by a variable is forbidden; Lesson 311 — Identifying Polynomials vs Non-Polynomials
Variance: measures this spread.; Lesson 2856 — Variance: Definition and Interpretation Lesson 2858 — Standard Deviation Lesson 2861 — Higher Moments: Skewness and Kurtosis Lesson 2886 — Mean and Variance of the Hypergeometric Distribution Lesson 2891 — Mean and Variance of the Poisson Distribution Lesson 2897 — Applications of the Uniform Distribution Lesson 2900 — Mean and Variance of Exponential Distributions Lesson 2920 — Properties and Applications of the Beta Distribution (+6 more)
Variance Inflation Factor: quantifies how much the variance of a regression coefficient is inflated due to multicollinearity.; Lesson 3080 — Multicollinearity: Detection and Consequences
Variance of S: Var(S ) = nσ² (not σ²/n); Lesson 2984 — CLT for Sample Sums
Variance testing: Tests hypotheses about population variance, since (n-1)s²/σ² ~ χ²(n-1); Lesson 2922 — Properties and Applications of Chi-Squared
Variances scale with squares: The new variance is *a²* times the first variance plus *b²* times the second variance (because variance scales with the square of constants); Lesson 2912 — Linear Combinations of Normal Variables
variation of parameters: , a universal method that works for *any* continuous function on the right-hand side and any homogeneous solution you can find—no guessing required.; Lesson 2359 — Motivation and Limitations of Undetermined Coefficients Lesson 2360 — The Variation of Parameters Formula for Second-Order ODEs Lesson 2365 — Handling Logarithmic and Rational Forcing Functions Lesson 2367 — Comparing Methods: When to Use Variation of Parameters
vector: that:; Lesson 1212 — Geometric Interpretation of Subtraction Lesson 1235 — Introduction to Vectors: Magnitude and Direction Lesson 1841 — The Gradient Vector: Definition and Notation Lesson 2007 — The Curl of a Vector Field Lesson 2046 — Vectors as Ordered Lists of Numbers
Vector Addition: Add component-wise:; Lesson 1794 — Vectors in 3D Space Lesson 2048 — Vector Addition
Vector decomposition: lets you separate any vector into these two components relative to another direction.; Lesson 1260 — Decomposing Vectors into Components
vector field: at every point (x, y), there's a corresponding gradient vector.; Lesson 1842 — Computing Gradient Vectors Lesson 1963 — Introduction to Vector Fields Lesson 1964 — Component Functions of Vector Fields
vector projection: of vector **u** onto vector **v** tells us how much of **u** lies in the direction of **v**.; Lesson 1259 — Vector Projection Formula Lesson 1260 — Decomposing Vectors into Components Lesson 1268 — Component of Force Along a Direction
vector space: .; Lesson 2050 — Algebraic Properties of Vector Operations Lesson 2075 — Definition of a Vector Space
Velocity: 60 mph northbound on a highway (not just speed); Lesson 1235 — Introduction to Vectors: Magnitude and Direction Lesson 1265 — Velocity and Displacement Vectors Lesson 1376 — Applications: Velocity and Other Rates Lesson 1428 — Applications: Rates of Change in Oscillatory Motion Lesson 1545 — Applications: Position from Velocity, Velocity from Acceleration
Velocity vector: Shows both *speed and direction* of motion.; Lesson 1265 — Velocity and Displacement Vectors
Velocity with resistance: An object falls or moves through air/water where resistance is proportional to velocity.; Lesson 2317 — Applications: Mixing and Velocity Problems
Venn diagram: is a visual tool that uses overlapping circles (or other shapes) to show how sets relate to each other.; Lesson 2527 — Venn Diagrams: Two Sets
Verification: If you substitute `y = y_h + y_p` into the original ODE:; Lesson 2348 — Non-Homogeneous ODEs: Structure of Solutions Lesson 2669 — Counting Vertices and Edges in Trees Lesson 2757 — Solving Two-Congruence Systems
Verification and alternative paths: Sometimes computing flux directly is hard, but converting to a volume integral via divergence is easy—or vice versa with Stokes' converting circulation to surface curl.; Lesson 2045 — Combining with Stokes' Theorem and Applications
Verification Problems: Given three measurements (like fence posts, property boundaries, or garden edges), you can use the converse: calculate whether a² + b² = c².; Lesson 790 — Applications of Triples and the Converse
Verified: Lesson 2742 — Verifying Fermat's Little Theorem with Examples
Verify: by checking that the middle term matches.; Lesson 365 — Trial and Error Method for Factoring Trinomials Lesson 941 — Congruence Through Rigid Motions Lesson 1307 — Using Inverses to Solve Matrix Equations Lesson 1511 — Box and Container Optimization Lesson 1753 — Verifying Conditions for the Alternating Series Test Lesson 2742 — Verifying Fermat's Little Theorem with Examples Lesson 2930 — Conditional Distributions: Discrete Case
Verify continuity: Confirm that f is continuous on [a, b]; Lesson 1357 — Using the IVT to Show Roots Exist
Verify each one: Plug *y = c* back into the original ODE.; Lesson 2314 — Losing Solutions Through Division
Verify equality: Check that all three pairs of corresponding sides are congruent; Lesson 741 — SSS (Side-Side-Side) Congruence Postulate
Verify it's a maximum: (optional): check the second derivative is negative; Lesson 3005 — Finding MLEs by Differentiation
Verify it's a minimum: Use the first or second derivative test; Lesson 1510 — Geometric Optimization: Minimizing Perimeter
Verify the minimum: using the first or second derivative test; Lesson 1512 — Distance Optimization Problems
Verify when possible: check that your answer makes sense (e.; Lesson 2158 — Practice: Mixed Determinant Computation
Verify x = 2: Lesson 421 — Verifying Solutions by Substitution
Verify x = 3: Lesson 421 — Verifying Solutions by Substitution
Verify Your Answer: Lesson 1516 — Applied Optimization Strategy
Verifying: the theorem means computing both sides separately and checking they give the same answer.; Lesson 2028 — Verifying Stokes' Theorem on Simple Surfaces
Vertex: `(50, 5000)`.; Lesson 410 — Applications and Problem Solving Lesson 422 — Parabola Basics: Shape and Orientation Lesson 423 — Vertex Form of a Quadratic Function Lesson 433 — Key Features Summary and Complete Graphing Lesson 434 — Projectile Motion: Height and Time Lesson 436 — Revenue and Profit Maximization Lesson 594 — The Absolute Value Function Lesson 703 — Measuring Angles and Angle Notation (+6 more)
Vertex and directrix: Another way to describe the same information; Lesson 1152 — Finding Equations from Geometric Information
Vertex and focus: This tells you where the parabola sits and which way it opens; Lesson 1152 — Finding Equations from Geometric Information
vertex connectivity: (denoted κ(G)) of a connected graph G is the minimum number of vertices that must be removed to either:; Lesson 2662 — Vertex Connectivity Lesson 2663 — Edge Connectivity Lesson 2666 — Menger's Theorem Lesson 2667 — Applications of Connectivity
Vertex form: `y = a(x - h)² + k` reveals this instantly—the vertex sits at the point `(h, k)`.; Lesson 406 — Converting to Vertex Form Lesson 407 — Finding the Vertex by Completing the Square Lesson 410 — Applications and Problem Solving Lesson 423 — Vertex Form of a Quadratic Function Lesson 424 — Finding the Vertex from Vertex Form Lesson 425 — The Axis of Symmetry Lesson 427 — Graphing Parabolas in Vertex Form
Vertex Version: Lesson 2666 — Menger's Theorem
Vertex: (4, 6): Lesson 302 — Vertices of the Feasible Region
Vertical: Clearer for complex polynomials with many terms; reduces alignment errors; Lesson 323 — Subtracting Polynomials Horizontally and Vertically Lesson 491 — Graphing Rational Functions with Asymptotes Lesson 1171 — Graphing Hyperbolas Centered at the Origin Lesson 1172 — Hyperbolas with Horizontal vs Vertical Transverse Axes Lesson 1456 — Horizontal and Vertical Tangents Lesson 1468 — Normal Lines to Curves
Vertical and horizontal lines: are perpendicular to each other.; Lesson 268 — Writing Equations of Perpendicular Lines
Vertical angles: are the pairs of angles that sit directly opposite each other—across the intersection point, not next to each other.; Lesson 709 — Vertical Angles Lesson 718 — Using Angle Relationships to Find Measures
vertical asymptote: is an invisible vertical line that a rational function approaches but never touches or crosses.; Lesson 486 — Vertical Asymptotes Lesson 576 — Continuity of Functions Lesson 1327 — Infinite Limits and Vertical Asymptotes
Vertical asymptotes: create breaks—x-values where the function doesn't exist; Lesson 493 — Domain and Range from Graphs Lesson 494 — Applications of Rational Functions Lesson 552 — Domain from Graphs Lesson 1023 — Graphing the Tangent Function Lesson 1025 — Transformations of Tangent: Phase Shift and Vertical Shift Lesson 1026 — Graphing the Cotangent Function Lesson 1028 — Graphing the Secant Function Lesson 1030 — Graphing the Cosecant Function (+4 more)
vertical axis: (y-axis) represents the **imaginary part**; Lesson 1207 — Complex Numbers as Points in the Plane Lesson 1208 — The Argand Diagram Lesson 3084 — Residual Plots: Fitted Values vs. Residuals
Vertical axis (imaginary axis): represents the imaginary part of the complex number; Lesson 1192 — Plotting Complex Numbers on the Complex Plane
Vertical change: (difference in output values): `f(b) - f(a)`; Lesson 574 — Average Rate of Change
Vertical component: F_y = ||**F**|| sin(θ); Lesson 1263 — Force as a Vector: Magnitude and Direction
Vertical component (y-component): `v_y = |v| sin(θ)`; Lesson 1252 — Resolving Vectors into Components
Vertical hyperbola: (opens up/down): vertices at (h, k ± a); Lesson 1168 — Components of a Hyperbola: Center, Vertices, and Foci Lesson 1169 — The Transverse and Conjugate Axes Lesson 1170 — Asymptotes of a Hyperbola
vertical leg: (running parallel to the y-axis); Lesson 779 — Right Triangles on the Coordinate Plane Lesson 903 — Distance Formula in the Coordinate Plane Lesson 968 — Drawing Diagrams for Angle Problems Lesson 975 — Angles of Elevation: Definition and Setup Lesson 978 — Finding Distances Using Angles of Depression
vertical line: has undefined slope because there's no run—you can't move left or right.; Lesson 246 — Positive, Negative, Zero, and Undefined Slopes Lesson 425 — The Axis of Symmetry Lesson 1662 — Rotation About Vertical Lines x = h
Vertical lines: run straight up and down.; Lesson 263 — Horizontal and Vertical Lines
Vertical major axis: `x²/b² + y²/a² = 1` (where a > b); Lesson 1158 — Graphing Ellipses Centered at the Origin Lesson 1162 — Finding the Equation from Given Information
Vertical parabolas: Lesson 1151 — Graphing Parabolas from Standard Form
vertical shells: .; Lesson 1666 — Revolving Around the y-axis Lesson 1669 — Revolving Around Horizontal Lines (y = k)
vertical shift: (up or down) changes the y-coordinate; Lesson 922 — Translations in the Coordinate Plane Lesson 1019 — Combined Transformations: y = A·sin(B(x - C)) + D Lesson 1031 — Transformations of Cosecant Lesson 1039 — Vertical Shift: D Parameter
Vertical Shift (D): The midline or average value (e.; Lesson 1022 — Applications: Modeling Periodic Phenomena Lesson 1031 — Transformations of Cosecant Lesson 1042 — Applications: Modeling Periodic Phenomena
Vertical Shift = D: The midline moves to y = D instead of y = 0; Lesson 1019 — Combined Transformations: y = A·sin(B(x - C)) + D
Vertical shifts: (these happen "outside" the function); Lesson 584 — Combining Multiple Transformations Lesson 587 — Even and Odd Functions Under Transformation Lesson 589 — Inverse Relationships in Transformations
vertical stretch: (amplitude-like factor) and reflection; Lesson 1031 — Transformations of Cosecant Lesson 1033 — Amplitude: Vertical Stretch and Compression
Vertical Stretch (A): Multiplying by A stretches the branches of cosecant vertically away from the sine curve it wraps around.; Lesson 1031 — Transformations of Cosecant
Vertical stretches/compressions: Lesson 584 — Combining Multiple Transformations Lesson 587 — Even and Odd Functions Under Transformation Lesson 589 — Inverse Relationships in Transformations
vertical stretching: (if |A| > 1) or **vertical compression** (if 0 < |A| < 1).; Lesson 1015 — Amplitude: Vertical Stretching with y = A·sin(x)Lesson 1024 — Transformations of Tangent: Period and Vertical Stretch
Vertical tangent lines: These occur where the curve goes straight up or down for an instant — like a cliff face.; Lesson 1467 — Horizontal and Vertical Tangent Lines
Vertical tangents: occur when the slope is **undefined**: the denominator of dy/dx = 0 (but the numerator ≠ 0); Lesson 1456 — Horizontal and Vertical Tangents Lesson 1467 — Horizontal and Vertical Tangent Lines
Vertical traces: (fix y = 0): \(z = x^2\) is a parabola opening upward; Lesson 1809 — Traces and Cross-Sections
Vertical transverse axis: Lesson 1173 — Writing Equations from Given Information Lesson 1174 — Hyperbolas Centered at (h, k)
vertically: .; Lesson 578 — Vertical Shifts of Functions Lesson 1148 — Parabolas with Vertex at (h, k): Vertical Axis Lesson 1172 — Hyperbolas with Horizontal vs Vertical Transverse Axes Lesson 1649 — Choosing Vertical vs Horizontal Slicing Lesson 1655 — Disk Method: Rotation About the y-axis
vertices: (corner points) of this polygon are critically important because, in optimization problems, the best solutions typically occur at these corners.; Lesson 302 — Vertices of the Feasible Region Lesson 306 — The Corner Point Theorem Lesson 725 — Corresponding Parts of Congruent Figures Lesson 1156 — Parts of an Ellipse: Vertices, Co-vertices, and Foci Lesson 1168 — Components of a Hyperbola: Center, Vertices, and Foci Lesson 1174 — Hyperbolas Centered at (h, k)Lesson 2647 — What is a Graph? Vertices and Edges
Vertices (nodes): for each element in the set(s); Lesson 2541 — Representing Relations: Matrices and Digraphs Lesson 2704 — Networks and Flows: Definitions and Notation
vᵢ: , **vⱼ**⟩ = 0 whenever i ≠ j; Lesson 2210 — Orthogonal Sets of Vectors Lesson 2211 — Orthogonal Bases Lesson 2223 — Normalizing to Obtain an Orthonormal Basis Lesson 2225 — Why Gram-Schmidt Works: The Proof Lesson 2380 — Eigenvalue Method for 2×2 Systems: Real Distinct Eigenvalues
VIF: measures how much a predictor's variance is "inflated" due to correlation with other predictors.; Lesson 3091 — Multicollinearity Diagnostics
Visual analogy: Think of a bowl that can hold water.; Lesson 1497 — Introduction to Concavity
Visual clarity: You can literally *see* where lines intersect; Lesson 281 — Comparing Graphing and Substitution Methods
Visual inspection: Graph both functions.; Lesson 1645 — Identifying Top and Bottom Functions
Visual verification: You can quickly check if two complex numbers are conjugates by seeing if they're mirror images across the real axis.; Lesson 1213 — Complex Conjugates on the Plane
Visualize: the clockwise rotation from the positive x-axis; Lesson 1007 — Negative Angles and Trigonometric Values Lesson 2302 — Solutions and Solution Curves
Voltage across the capacitor: *V_C = Q/C* (stores charge, like a compressed spring); Lesson 2372 — RLC Circuits: Kirchhoff's Laws and the Governing ODE
Voltage across the inductor: *V_L = L·(dI/dt)* (resists changes in current, like mass resisting acceleration); Lesson 2372 — RLC Circuits: Kirchhoff's Laws and the Governing ODE
Voltage across the resistor: *V_R = R·I* (proportional to current, like friction); Lesson 2372 — RLC Circuits: Kirchhoff's Laws and the Governing ODE
Volume: is the amount of three-dimensional space that an object occupies or contains.; Lesson 875 — Volume Concept and Cubic Units Lesson 1290 — What is a Determinant?Lesson 1659 — Washer Method: Rotation About the x-axis Lesson 1690 — Comparing Surface Area and Volume Calculations Lesson 1901 — Applications: Volume Under a Surface Lesson 1922 — Introduction to Triple Integrals Lesson 1930 — Volume as a Triple Integral
Volume (Disk Method): Lesson 1690 — Comparing Surface Area and Volume Calculations
Volume element: dS adapts naturally: for a cylinder, dS = a dθ dz (the "a" comes from the radius stretching the arc length).; Lesson 2025 — Surface Integrals in Different Coordinate Systems
Volume of one disk: π × (radius)² × thickness; Lesson 1653 — Volume of Revolution: Introduction and Disk Method Concept Lesson 1654 — Disk Method: Rotation About the x-axis
volume scaling: instead.; Lesson 1946 — The Jacobian Determinant Lesson 2148 — What is a Determinant?
Vorticity: Lesson 2009 — Interpreting Curl Physically

W

W ⁺: , the sum of ranks for positive differences.; Lesson 3105 — Wilcoxon Signed-Rank Test (read "W perp"), is the set of *all* vectors in **V** that are orthogonal to *every* vector in **W**. Lesson 2215 — Orthogonal Complements Lesson 2216 — Properties of Orthogonal Complements Lesson 2233 — Orthogonal Complement and Direct Sum Decomposition
W = ||F|| ||d||: .; Lesson 1267 — Work Done by a Constant Force
W = W(y₁, y₂): is the Wronskian.; Lesson 2360 — The Variation of Parameters Formula for Second-Order ODEs
Waiting time problems: If you've flipped a coin 10 times without heads, the expected number of *additional* flips until heads is the same as if you just started: 1/p flips.; Lesson 2883 — Memoryless Property and Applications
Walk: Any sequence of vertices where each consecutive pair is connected by an edge.; Lesson 2657 — Walks, Trails, and Paths
washer: is simply a disk with a hole punched out of the center.; Lesson 1657 — Washer Method: Introduction and Setup Lesson 1663 — Choosing Between Disk and Washer Methods
washer method: comes in.; Lesson 1657 — Washer Method: Introduction and Setup Lesson 1659 — Washer Method: Rotation About the x-axis Lesson 1661 — Rotation About Horizontal Lines y = k Lesson 1663 — Choosing Between Disk and Washer Methods
Watch for constant bounds: that make integration easier (common with θ from 0 to 2π); Lesson 1937 — Evaluating Triple Integrals Using Cylindrical Coordinates
Watch for opposites: Sometimes you'll see `(a - b)` and `(b - a)` — these are opposites and need special handling (later!; Lesson 377 — Identifying the Common Binomial Factor
Watch for the cycle: Notice when your original integral reappears; Lesson 1597 — Cyclic Integration by Parts
Watch the pattern: as h shrinks: 0.; Lesson 1374 — Estimating Instantaneous Rates Numerically
Watch the signs carefully: The equation shows **(x - h)** and **(y - k)**, so:; Lesson 917 — Finding Center and Radius from Circle Equations
Wave propagation: in strings, membranes, and electromagnetic fields; Lesson 2408 — What is a Partial Differential Equation?
Weak Law: For any given large $n$, the sample mean is *probably* close to $\mu$; Lesson 2973 — Strong Law of Large Numbers: Statement
weighted average: the point "leans" toward whichever endpoint has the larger ratio number.; Lesson 908 — Partitioning Segments with Ratios Lesson 2852 — Definition of Expectation (Mean) for Discrete Random Variables Lesson 3115 — Conjugate Priors: Normal-Normal Model
Weighted average temperature: Integrate temperature weighted by area; Lesson 2018 — Surface Integrals of Scalar Functions
Welch-Satterthwaite approximation: Lesson 3038 — Two-Sample t-Test: Welch's Test for Unequal Variances
Welch's ANOVA: or transformations instead of standard ANOVA.; Lesson 3101 — Checking ANOVA Assumptions
Welch's t-test: solves this by *not pooling* the sample variances.; Lesson 3038 — Two-Sample t-Test: Welch's Test for Unequal Variances
Well-Ordering: when:; Lesson 2520 — Choosing the Right Proof Technique
Well-Ordering → Induction: Suppose P(0) holds and P(k) → P(k+1) for all k, but P doesn't hold for all n.; Lesson 2519 — Equivalence of Induction and Well-Ordering
Well-Ordering Principle: states: *Every nonempty set of positive integers has a least element.; Lesson 2517 — The Well-Ordering Principle Lesson 2518 — Proofs Using Well-Ordering
What: that maximum or minimum value is (it's `k`); Lesson 410 — Applications and Problem Solving Lesson 929 — Composition of Translations and Reflections
What function: f(x) tells you which function to examine; Lesson 1320 — Limit Notation and Reading Limits
What Gram-Schmidt does: Lesson 2225 — Why Gram-Schmidt Works: The Proof
What happens: Lesson 580 — Vertical Stretches and Compressions Lesson 759 — Parallel Lines and Triangle Similarity Lesson 1320 — Limit Notation and Reading Limits
What input value: x→ a tells you which point to approach; Lesson 1320 — Limit Notation and Reading Limits
What to do: Don't automatically delete influential points—investigate *why* they're unusual.; Lesson 3088 — Cook's Distance and Influence Measures
What to look for: Lesson 3086 — Normal Probability Plots (Q-Q Plots)
What you know immediately: Lesson 2274 — Eigenspaces and Multiplicity for Symmetric Matrices
What's missing: The current denominator is x; we need 5x.; Lesson 462 — Building Equivalent Rationals
What's the probability: of any one specific arrangement?; Lesson 2875 — Binomial Probability Mass Function
When: the maximum or minimum occurs (at `x = h`); Lesson 410 — Applications and Problem Solving
When |B| > 1: The coefficient is larger, making the function cycle *faster*.; Lesson 1035 — Period: Horizontal Stretch and Compression
When decimals suffice: Lesson 521 — Applications and When Not to Rationalize
When gcd = 1: If gcd(*a*, *b*) = 1, we say *a* and *b* are **relatively prime** or **coprime**; Lesson 2711 — Greatest Common Divisor (GCD): Definition
When it works: You need $np_0 \geq 10$ and $n(1-p_0) \geq 10$ for the normal approximation to be valid.; Lesson 3028 — z-Test for Proportions
When n is composite: φ(n) gives us the right exponent to make a^φ(n) ≡ 1 (mod n); Lesson 2748 — Statement of Euler's Theorem
When n is prime: φ(n) = n - 1, so Euler's Theorem reduces exactly to Fermat's Little Theorem; Lesson 2748 — Statement of Euler's Theorem
When rationalization is preferred: Lesson 521 — Applications and When Not to Rationalize
When to reconsider: If your substitution leads to an integral messier than the original, backtrack and try a different *u*.; Lesson 1589 — Practice Problems and Strategy
When to use: You have reliable prior knowledge, want to incorporate expert opinion, or are working with limited new data.; Lesson 3118 — Prior Elicitation and Informative vs. Noninformative Priors
Where are the asymptotes: Lesson 1030 — Graphing the Cosecant Function
Where arrows converge: Often indicates local minima (arrows point inward) or maxima (arrows point outward); Lesson 1850 — Gradient Fields and Visualization
Where it ends: (the **terminal point** or head); Lesson 1236 — Geometric Representation of Vectors
Where it starts: (the **initial point** or tail); Lesson 1236 — Geometric Representation of Vectors
Where your model struggles: Which observations are hardest to predict; Lesson 3083 — What Are Residuals?
Whether assumptions hold: Patterns in residuals can reveal problems (non-linearity, unequal variance, outliers); Lesson 3083 — What Are Residuals?
Which to choose: Whichever makes differentiation easier!; Lesson 1531 — Indeterminate Products: 0·∞
Whispering galleries: Elliptical domes in buildings where whispers carry across large distances; Lesson 1165 — Reflective Property of Ellipses Lesson 1187 — Real-World Conic Applications Review
Why: Think of integration as moving *from* `a` *to* `b`.; Lesson 1560 — Properties of Definite Integrals: Intervals Lesson 1815 — Using Polar Coordinates to Evaluate Limits Lesson 2145 — Inverses of Products and Transposes Lesson 2157 — Determinant of Special Matrices Lesson 2227 — Numerical Stability and Modified Gram-Schmidt Lesson 2577 — Advanced Pigeonhole Arguments and Ramsey Theory Lesson 2580 — Permutations of n Objects Taken r at a Time Lesson 2663 — Edge Connectivity (+1 more)
Why does this matter: At an ordinary point, we're *guaranteed* that a power series solution exists and converges in some neighborhood around $x_0$.; Lesson 2399 — Power Series Solutions: Introduction and Ordinary Points
Why does this work: Recall that a triangular matrix already has zeros either above or below the diagonal.; Lesson 2174 — Eigenvalues of Triangular Matrices
Why it works: Each row corresponds to one round of integration by parts.; Lesson 1596 — Tabular Integration Lesson 2067 — Testing Linear Independence in R² Lesson 2142 — The Gauss-Jordan Method for Finding Inverses Lesson 2225 — Why Gram-Schmidt Works: The Proof Lesson 2857 — Computing Variance Using the Shortcut Formula
Why the wider interval: Because 2x runs *twice as fast* as x, you must solve over [0, 4π) for the multiple angle to capture all solutions in [0, 2π) for x.; Lesson 1084 — Equations with Multiple Angles
Why this formula: The PDF must integrate to 1.; Lesson 2895 — PDF and CDF of the Uniform Distribution
Why this works: Recall that the derivative of ln(u) is 1/u · u'.; Lesson 1627 — Integrating Partial Fractions with Linear Terms Lesson 1833 — Equation of a Tangent Plane
Why uniqueness matters: Suppose two different matrices **B** and **C** both acted as inverses to **A**.; Lesson 2139 — Properties of Invertible Matrices
Widely spaced curves: indicate gentle slopes; Lesson 1800 — Level Curves and Contour Maps
Width: Larger |a| makes the parabola narrower; smaller |a| makes it wider.; Lesson 423 — Vertex Form of a Quadratic Function Lesson 1552 — Sigma Notation for Riemann Sums
Wilcoxon Signed-Rank Test: builds on the sign test by incorporating not just the *direction* of differences, but also their *magnitude*.; Lesson 3105 — Wilcoxon Signed-Rank Test
Wind/Current vector: The push from external forces (wind for planes, water current for boats); Lesson 1266 — Navigation Problems: Heading and Course
With binomials in numerators: Lesson 459 — Adding Rationals with Like Denominators
With D: For `y = sin(x) + 3`, the midline shifts up to `y = 3`; Lesson 1039 — Vertical Shift: D Parameter
With grouping: Lesson 32 — Parentheses and Grouping Symbols
With multiple terms: Lesson 174 — The Distributive Property with Variables
With negative terms: Lesson 1197 — Adding and Subtracting Complex Numbers
With preprocessing: Lesson 1464 — Combining Logarithm Properties Before Differentiating
With the Product Rule: Lesson 1415 — Chain Rule with Product and Quotient Rules
With the Quotient Rule: Lesson 1415 — Chain Rule with Product and Quotient Rules
With variables: ∛(x⁷) = ∛(x⁶ · x) = ∛(x⁶) · ∛x = x²∛x; Lesson 501 — Higher-Order Radicals: Cube Roots and Beyond
Within-group variation: How much do individual observations vary within each group?; Lesson 3049 — One-Way ANOVA F Test Introduction Lesson 3093 — Introduction to Analysis of Variance (ANOVA)
Within-group variation (SSW): How much do individual scores vary *within* each method?; Lesson 3095 — Between-Group and Within-Group Variation
Without D: For `y = sin(x)`, the midline is at `y = 0`; Lesson 1039 — Vertical Shift: D Parameter
Without grouping: Lesson 32 — Parentheses and Grouping Symbols
Without preprocessing: , you'd need the product rule, quotient rule, and chain rule all tangled together.; Lesson 1464 — Combining Logarithm Properties Before Differentiating
without replacement: , where each item can be classified as a "success" or "failure.; Lesson 2885 — Hypergeometric Probability Formula Lesson 2887 — Hypergeometric vs. Binomial: When to Use Each
Word form: "3 to 4"; Lesson 106 — Writing Ratios in Different Forms
Work: is the line integral of a force field **F** along a curve *C*:; Lesson 1981 — Applications: Work and Circulation
Work = 0: around closed paths in conservative fields (like gravity); Lesson 1981 — Applications: Work and Circulation
Work backwards: from what you need to what you have; Lesson 866 — Perimeter and Area Problem Solving
Work forward: Substitute: *n² = (2k)² = 4k² = 2(2k²)*.; Lesson 2492 — Direct Proof: Working with Definitions
Work from outside in: Lesson 1338 — Combining Multiple Limit Laws
Work rate: = 1/(time to complete alone); Lesson 468 — Applications of Adding and Subtracting Rationals
Work rate problems: describe how fast tasks get completed when people or machines work together.; Lesson 439 — Work and Mixture Rate Problems
Work systematically: – Apply the outer rule first, then handle the derivatives of the composite pieces.; Lesson 1406 — Combining Product and Quotient Rules
Working backward: If angle XYZ = 50° and YW bisects it, you know angle XYW and angle WYZ are each 25°.; Lesson 710 — Angle Bisectors
Working Backwards: Start from the answer you want and trace steps backward to understand what you need.; Lesson 195 — Checking Solutions and Problem-Solving Strategies Lesson 585 — Writing Transformed Functions from Descriptions
Wrapping a gift: You need enough paper to cover all six faces of a rectangular box (a prism).; Lesson 895 — Word Problems Involving Surface Area
Write: the resulting fraction; Lesson 65 — Simplifying Fractions Lesson 113 — Solving Rate Problems Lesson 598 — Writing Functions as Piecewise Definitions Lesson 1965 — Evaluating Vector Fields at Points
Write a formula: for that quantity using variables for dimensions; Lesson 1511 — Box and Container Optimization
Write an equation: translate the relationships in the problem into math using that variable; Lesson 189 — Writing Equations from Word Problems Lesson 1479 — Related Rates with Triangles
Write constraint inequalities: These represent your limits—flour, sugar, time, plus *x* ≥ 0 and *y* ≥ 0 (you can't make negative treats!; Lesson 308 — Applications: Maximization Problems
Write down the expression: Lesson 182 — Evaluating Expressions by Substitution
Write equations: from problem constraints; Lesson 1318 — Applications: Real-World Matrix Systems Lesson 1516 — Applied Optimization Strategy
Write expressions: for related quantities using your variable; Lesson 192 — Age Problems and Consecutive Integer Problems Lesson 343 — Special Products in Word Problems
Write in polar form: *z = r(cos θ + i sin θ)*; Lesson 1215 — Converting from Rectangular to Polar Form
Write in standard form: Combine them as a + bi; Lesson 1219 — Converting from Polar to Rectangular Form
Write new limits: accordingly; Lesson 1905 — Changing the Order of Integration
Write out all terms: explicitly; Lesson 670 — Evaluating Finite Sums with Sigma Notation
Write relationships as expressions: – use the problem's words to build algebraic expressions; Lesson 400 — Applications and Word Problems
Write the constraint equation: (usually the volume you must maintain); Lesson 896 — Optimization Problems with Surface Area Lesson 1481 — Ladder and Shadow Problems
Write the equation: using `y = kx`; Lesson 129 — Real-World Direct Variation Problems Lesson 1478 — Related Rates with Rectangles and Squares
Write the expression: combine the parts logically; Lesson 172 — Writing Expressions from Word Problems
Write the factors: (x + 3)(x + 4); Lesson 362 — Factoring Trinomials with Leading Coefficient 1
Write the formula: C(7,3) = 7!; Lesson 2590 — Computing Basic Combinations
Write the general solution: as a linear combination of basis vectors; Lesson 2170 — Finding Eigenvectors from Eigenvalues
Write the integral: `V = 2π∫[radius][height]dx` (or `dy`); Lesson 1665 — Setting Up a Shell Integral
Write the log-likelihood function: ℓ(θ) = log L(θ); Lesson 3005 — Finding MLEs by Differentiation
Write the objective function: This is what you want to maximize.; Lesson 308 — Applications: Maximization Problems
Write the parametric equation: The normal line is:; Lesson 1835 — Normal Lines to Surfaces
Write the particular solution: by plugging this specific *C* value back in; Lesson 2311 — Initial Value Problems with Separable ODEs Lesson 2324 — Initial Value Problems for Linear ODEs
Write the profit function: P(x) = R(x) - C(x); Lesson 1513 — Cost and Revenue Optimization
Write the proportion: Set up an equation showing that one pair of corresponding sides has the same ratio as another pair; Lesson 757 — Finding Missing Sides Using Similarity
Write the range: using interval notation or set notation; Lesson 554 — Range from Graphs
Write the ratio: Lesson 727 — Scale Factor
Write the solution: as an ordered pair (x, y); Lesson 274 — Solving Systems by Graphing: Basic Technique Lesson 2130 — The Null Space
Write the solution set: as expressions involving the parameters; Lesson 1315 — Systems with Infinitely Many Solutions
Write the tangent equation: tan(angle) = opposite/adjacent; Lesson 954 — Finding Side Lengths Using Tangent
Write the vertex: (2, -5); Lesson 429 — Finding the Vertex Using x = -b/(2a)
Write the volume formula: for the shape.; Lesson 1480 — Volume Related Rates: Spheres and Cones Lesson 1484 — Water Tank and Fluid Flow Problems
Write theoretical moments: as functions of the parameters: E[X], E[X²], E[X³], etc.; Lesson 3003 — Method of Moments for Multiple Parameters
Write two equations: – Translate each fact into mathematical form; Lesson 282 — Application Problems Using Systems and Substitution
Write what remains: of each term inside the parentheses; Lesson 360 — GCF with Four or More Terms Lesson 445 — Simplifying by Canceling Common Factors
Write what you know: (equations for each relationship); Lesson 712 — Solving Multi-Step Angle Problems
Write your answer: Lesson 14 — Adding and Subtracting with Larger Numbers
Write your constraints: – these are your minimum requirements ("at least 10 grams of protein," "no fewer than 3 workers," etc.; Lesson 309 — Applications: Minimization Problems
WRONG: 2³ + 2⁴ = 2⁷; Lesson 150 — Common Mistakes with Exponent Rules Lesson 547 — Common Function Notation Mistakes Lesson 1398 — Why We Need the Product Rule Lesson 1534 — Common Mistakes and When Not to Use L'Hôpital's Rule Lesson 2809 — Common Misconceptions About Independence Lesson 3019 — Interpreting Confidence Intervals Correctly
Wrong coefficient: Volume uses `π`, surface area uses `2π` (the circumference factor).; Lesson 1690 — Comparing Surface Area and Volume Calculations
Wronskian: of two functions $y_1$ and $y_2$ is defined as:; Lesson 2342 — Linear Independence and the Wronskian Lesson 2361 — Computing the Wronskian Lesson 2366 — Higher-Order ODEs and Variation of Parameters

X

X̃: Lesson 2289 — Principal Component Analysis (PCA) and SVD
x̄: .; Lesson 3011 — Confidence Interval for a Population Mean (σ Known)Lesson 3035 — Calculating the t- Statistic and Finding Critical Values Lesson 3036 — One-Sample t-Test: Confidence Intervals
X (x): are your eigenfunctions (spatial patterns); Lesson 2421 — The General Series Solution
x (x² ≥ 0): means "For all real numbers x, x squared is non-negative.; Lesson 2480 — The Universal Quantifier ( ∀)Lesson 2484 — Negating Quantified Statements
x → ∞: (x approaches positive infinity), we observe whether:; Lesson 575 — End Behavior and Limits at Infinity Lesson 1515 — Optimization on Open Intervals
x < 0: and positive for **x > 0**, confirming a concavity change at **(0, 0)**.; Lesson 1500 — Inflection Points Lesson 1543 — The Special Case: Antiderivative of 1/x
x = -2: Lesson 394 — Solving Quadratics in Factored Form Lesson 396 — Solving by Factoring: Simple Trinomials
x = -5: Lesson 392 — Zero Product Property Lesson 394 — Solving Quadratics in Factored Form
x = −3: the equals sign stays put.; Lesson 219 — The Multiplication/Division Sign Rule Lesson 399 — Solving Special Forms: Perfect Square Trinomials
x = 0: .; Lesson 1500 — Inflection Points Lesson 1774 — Maclaurin Series: Taylor Series at Zero Lesson 2069 — The Matrix Test for Linear Independence
x = 2: Lesson 609 — Solving Simple Exponential Equations Lesson 1327 — Infinite Limits and Vertical Asymptotes Lesson 1527 — Indeterminate Forms 0/0 and ∞/∞Lesson 1668 — Revolving Around Vertical Lines (x = k)
x = 3: , OR; Lesson 392 — Zero Product Property Lesson 394 — Solving Quadratics in Factored Form Lesson 425 — The Axis of Symmetry Lesson 487 — Holes in Rational Functions Lesson 609 — Solving Simple Exponential Equations Lesson 1312 — Back Substitution from Echelon Form
x = 4: Lesson 202 — Combining Addition/Subtraction with Multiplication/Division Lesson 1316 — Solving with Matrix Inverses
x = 5: cm.; Lesson 437 — Geometry: Pythagorean Applications Lesson 1668 — Revolving Around Vertical Lines (x = k)
x = a: , we:; Lesson 1372 — The Concept of Instantaneous Rate of Change Lesson 1693 — Work Done by a Variable Force Lesson 1700 — Center of Mass for One-Dimensional Systems
X = A ⁻¹B: Lesson 1307 — Using Inverses to Solve Matrix Equations Lesson 1308 — Applications of Inverse Matrices Lesson 1316 — Solving with Matrix Inverses Lesson 2146 — Solving Systems Using Matrix Inverses
x = b: , the work done is:; Lesson 1693 — Work Done by a Variable Force Lesson 1700 — Center of Mass for One-Dimensional Systems
X̃ = UΣV ᵀ: Lesson 2289 — Principal Component Analysis (PCA) and SVD
x > 0: , confirming a concavity change at **(0, 0)**.; Lesson 1500 — Inflection Points Lesson 1543 — The Special Case: Antiderivative of 1/x
X ~ Binomial(n, p): (number of successes in n independent trials with success probability p):; Lesson 2862 — Expectation and Variance in Common Distributions
X ~ Geometric(p): (number of trials until first success):; Lesson 2862 — Expectation and Variance in Common Distributions
X ~ N(μ, σ²): Lesson 2905 — Parameters: Mean and Standard Deviation
X̄ ~ N(μ, σ²/n): Lesson 2977 — Introduction to the Central Limit Theorem Lesson 2979 — CLT for Independent Identically Distributed Variables
X ~ N(μ₁, σ₁²): and **Y ~ N(μ₂, σ₂²)** are independent, then any linear combination **aX + bY** (where *a* and *b* are constants) is also normal:; Lesson 2912 — Linear Combinations of Normal Variables
X ~ Poisson(λ): (number of events in a fixed interval, where λ is the rate):; Lesson 2862 — Expectation and Variance in Common Distributions
x P(x): Lesson 2480 — The Universal Quantifier ( ∀)Lesson 2481 — The Existential Quantifier (∃)
x_{n+1} = Ax_n: , where **A** is a fixed matrix and **n** represents the time step.; Lesson 2179 — Applications: Steady States and Dynamics Lesson 2199 — Applications of Diagonalization
x_p: is a particular solution.; Lesson 2385 — Non-Homogeneous Systems: Variation of Parameters Lesson 2386 — Non-Homogeneous Systems: Undetermined Coefficients
x-axis: is a horizontal number line that runs left and right across the page.; Lesson 237 — The x-axis and y-axis Lesson 238 — The Four Quadrants Lesson 583 — Reflections Across the y-axis Lesson 584 — Combining Multiple Transformations Lesson 925 — Reflections Across the Axes Lesson 1667 — Shell Method with Horizontal Shells Lesson 1671 — Shell Method with Functions of y Lesson 1688 — Parametric Curves and Surface Area (+1 more)
x-axis reflection: (x, y) → (x, -y); Lesson 242 — Reflections and Symmetry in the Coordinate Plane Lesson 925 — Reflections Across the Axes
x-coordinate: – it shows horizontal position; Lesson 236 — Understanding Ordered Pairs and Coordinates Lesson 1000 — Using the Unit Circle to Find Exact Values Lesson 1020 — Comparing Sine and Cosine: The Phase Relationship Lesson 1062 — Proof and Geometric Interpretation of Sum Formulas Lesson 1687 — Surface Area for Revolution Around y-axis
x-intercepts: of a parabola are the points where the graph touches or crosses the x-axis.; Lesson 431 — x-Intercepts and Zeros of Quadratic Functions Lesson 491 — Graphing Rational Functions with Asymptotes
x': (t) = A**x**(t); Lesson 2378 — Introduction to Systems of Linear ODEs Lesson 2380 — Eigenvalue Method for 2×2 Systems: Real Distinct Eigenvalues Lesson 2382 — Repeated Eigenvalues and Generalized Eigenvectors Lesson 2383 — Phase Portraits and Stability Analysis Lesson 2385 — Non-Homogeneous Systems: Variation of Parameters Lesson 2386 — Non-Homogeneous Systems: Undetermined Coefficients
X'X: captures how your predictors relate to each other, while **X'Y** captures how they relate to the response.; Lesson 3074 — Least Squares Estimation in Multiple Regression
X'Xβ̂ = X'Y: Lesson 3074 — Least Squares Estimation in Multiple Regression
X'Y: captures how they relate to the response.; Lesson 3074 — Least Squares Estimation in Multiple Regression
X(0) = 0: and **X(L) = 0**:; Lesson 2448 — Solving the Spatial ODE: Eigenvalue Problems
X(L) = 0: Lesson 2448 — Solving the Spatial ODE: Eigenvalue Problems
x₁: (t) = e^(λt)**v**; Lesson 2382 — Repeated Eigenvalues and Generalized Eigenvectors Lesson 2384 — Fundamental Matrices and the Matrix Exponential
x₂: (t) = e^(λt)(t**v** + **w**); Lesson 2382 — Repeated Eigenvalues and Generalized Eigenvectors Lesson 2384 — Fundamental Matrices and the Matrix Exponential
X̃V: and keep only the top *k* columns for dimensionality reduction; Lesson 2289 — Principal Component Analysis (PCA) and SVD

Y

Y = g(X): .; Lesson 2870 — The Jacobian Method for Transformations
y = x: .; Lesson 565 — Graphs of Inverse Functions Lesson 926 — Reflections Across y = x and y = -x
Y ~ N(μ₂, σ₂²): are independent, then any linear combination **aX + bY** (where *a* and *b* are constants) is also normal:; Lesson 2912 — Linear Combinations of Normal Variables
y-axis: is a vertical number line that runs up and down.; Lesson 237 — The x-axis and y-axis Lesson 238 — The Four Quadrants Lesson 583 — Reflections Across the y-axis Lesson 925 — Reflections Across the Axes Lesson 1655 — Disk Method: Rotation About the y-axis Lesson 1660 — Washer Method: Rotation About the y-axis Lesson 1664 — Introduction to the Shell Method Lesson 1666 — Revolving Around the y-axis (+3 more)
y-axis reflection: (x, y) → (-x, y); Lesson 242 — Reflections and Symmetry in the Coordinate Plane Lesson 925 — Reflections Across the Axes
y-axis symmetry: the left side mirrors the right side perfectly.; Lesson 569 — Even and Odd Functions Lesson 1001 — Symmetry Properties of the Unit Circle
y-coordinate: – it shows vertical position; Lesson 236 — Understanding Ordered Pairs and Coordinates Lesson 1000 — Using the Unit Circle to Find Exact Values Lesson 1020 — Comparing Sine and Cosine: The Phase Relationship Lesson 1062 — Proof and Geometric Interpretation of Sum Formulas
y-intercept: (where the line crosses the y-axis); Lesson 254 — Slope-Intercept Form: y = mx + b Lesson 255 — Graphing Lines from Slope-Intercept Form Lesson 433 — Key Features Summary and Complete Graphing Lesson 491 — Graphing Rational Functions with Asymptotes Lesson 612 — Properties of e^x: Growth and Behavior Lesson 1517 — Domain, Intercepts, and Symmetry Lesson 3064 — Interpreting Slope and Intercept
y-values: Lesson 1655 — Disk Method: Rotation About the y-axis Lesson 1660 — Washer Method: Rotation About the y-axis Lesson 1671 — Shell Method with Functions of y
y₁(t): and **y₂(t)** are both solutions to the equation, then:; Lesson 2343 — The Principle of Superposition Lesson 2360 — The Variation of Parameters Formula for Second-Order ODEs
y₂(t): are both solutions to the equation, then:; Lesson 2343 — The Principle of Superposition Lesson 2360 — The Variation of Parameters Formula for Second-Order ODEs
Yes: Use the first piece: 2x + 1; Lesson 591 — Evaluating Piecewise Functions Lesson 784 — Verifying Pythagorean Triples Lesson 1193 — Equality of Complex Numbers Lesson 2060 — Determining if a Vector is in a Span Lesson 2091 — Dimension of a Vector Space
yᵢ(t) = y ᵢ(0)e^(λ ᵢt): , where λᵢ are eigenvalues.; Lesson 2199 — Applications of Diagonalization Lesson 2278 — Applications to Differential Equations and Dynamics
Your data is ordinal: (rankings, Likert scales) rather than truly continuous; Lesson 3103 — Introduction to Nonparametric Methods Lesson 3111 — Choosing and Interpreting Nonparametric Tests
Your position: (where the angle of elevation is measured from); Lesson 975 — Angles of Elevation: Definition and Setup
Your system: Three equations in three unknowns (x, y, λ).; Lesson 1877 — Setting Up the Lagrange System
Your toolkit: Lesson 1051 — Verifying Trigonometric Identities

Z

Zero: is still the center point; Lesson 40 — Representing Negative Numbers on the Number Line Lesson 241 — Points on the Axes Lesson 246 — Positive, Negative, Zero, and Undefined Slopes Lesson 250 — Slope and Direction of Lines Lesson 392 — Zero Product Property Lesson 394 — Solving Quadratics in Factored Form Lesson 395 — Standard Form and Setting Equal to Zero Lesson 557 — Real-World Context and Restricted Domains (+4 more)
Zero always wins: any nonzero number to the power of zero is 1; Lesson 141 — Mixed Practice with Integer Exponents
Zero covariance: No linear relationship between X and Y; Lesson 2947 — Covariance: Definition and Interpretation
Zero Covariance → Independence: (NOT always true); Lesson 2950 — Covariance and Independence
Zero divergence: is a region where people just pass through without accumulating or dispersing; Lesson 2006 — Interpreting Divergence Physically
Zero divergence (div: F** = 0):** The amount of fluid entering equals the amount leaving.; Lesson 2006 — Interpreting Divergence Physically
Zero exponent: = equals 1; Lesson 135 — Negative Integer Exponents Lesson 141 — Mixed Practice with Integer Exponents Lesson 147 — Zero Exponent Rule Lesson 601 — Evaluating Exponential Expressions
Zero flux: field flows parallel to the surface, or inflow equals outflow; Lesson 2019 — Vector Fields and Surface Integrals
zero matrix: is any matrix where every entry is zero.; Lesson 1272 — Types of Matrices: Square, Row, Column, and Zero Lesson 2103 — Special Matrices: Identity and Zero Lesson 2157 — Determinant of Special Matrices
Zero Product Property: states: **If a × b = 0, then a = 0 or b = 0 (or both).; Lesson 392 — Zero Product Property Lesson 394 — Solving Quadratics in Factored Form Lesson 395 — Standard Form and Setting Equal to Zero Lesson 399 — Solving Special Forms: Perfect Square Trinomials
Zero real parts: neutrally stable or needs further analysis; Lesson 2278 — Applications to Differential Equations and Dynamics
Zero residual: Perfect prediction (the point lies exactly on the regression line); Lesson 3083 — What Are Residuals?
Zero Rule: Every nonzero integer divides 0, but 0 only divides 0; Lesson 2709 — Divisibility: Definition and Basic Properties
zero vector: , denoted **0** or **0**, is the vector with all components equal to zero:; Lesson 2047 — Vector Equality and Zero Vector Lesson 2084 — Sum of Subspaces
Zero-length vectors: Mark critical points where ∇f = **0**; Lesson 1850 — Gradient Fields and Visualization
zeros: or **roots** of the quadratic function.; Lesson 431 — x-Intercepts and Zeros of Quadratic Functions Lesson 577 — Zeros and Intercepts Lesson 578 — Vertical Shifts of Functions