49 core topics + 105 prerequisite topics taught as needed · approximately 41 hours of instruction including spaced review
An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph — topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. A cumulative quiz follows every 6 lessons. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.
| Antiderivatives [M] | Reversing differentiation: the power rule backwards. |
| Antiderivatives: Trig & Exponential [M] | ∫cos = sin, ∫sin = −cos, ∫eˣ = eˣ, ∫1/x = ln|x|. |
| Riemann Sums [M] | Approximating area with rectangles. |
| The Trapezoidal Rule [M] | Averaging left and right sums. |
| Properties of Definite Integrals [M] | Linearity, additivity, and orientation. |
| The Fundamental Theorem: Evaluating Integrals [M] | ∫ₐᵇ f = F(b) − F(a). |
| Accumulation Functions & FTC Part 1 [H] | d/dx ∫ₐˣ f(t) dt = f(x). |
| u-Substitution [H] | The chain rule in reverse. |
| Integration by Parts (BC) [H] | ∫u dv = uv − ∫v du. |
| Partial Fractions (BC) [H] | Splitting rational functions to integrate them. |
| Improper Integrals (BC) [H] | Integrals to infinity, defined by limits. |
| Integrals Yielding Inverse Trig [H] | 1/(1+x²) → arctan, 1/√(1−x²) → arcsin. |
| Average Value of a Function [M] | f_avg = (1/(b−a)) ∫ₐᵇ f. |
| Motion: Displacement & Distance [H] | Displacement is ∫v; distance is ∫|v|. |
| Accumulation & Net Change [H] | Final amount = initial + ∫(rate). |
| Analyzing Accumulation Functions [H] | Reading g(x) = ∫f from the graph of f. |
| Area Between Curves [H] | ∫(top − bottom) between the intersections. |
| Volumes: Disc & Washer [H] | V = π∫R² dx (discs), π∫(R² − r²) dx (washers). |
| Volumes by Cross-Section [H] | V = ∫A(x) dx for known cross-sectional areas. |
| Arc Length (BC) [H] | L = ∫√(1 + (y′)²) dx. |
| Differential Equations: Verifying Solutions [M] | A solution is a function that satisfies the equation. |
| Slope Fields [M] | A picture of dy/dx at every point. |
| Euler's Method (BC) [H] | Stepping along tangent lines. |
| Separation of Variables [H] | Move all y's left, all x's right, integrate both sides. |
| Exponential Growth & Decay Models [M] | dy/dt = ky means y = y₀e^{kt}. |
| Logistic Growth (BC) [H] | Growth limited by a carrying capacity L. |
| Parametric Derivatives (BC) [H] | dy/dx = (dy/dt)/(dx/dt). |
| Parametric Second Derivatives (BC) [H] | Differentiate dy/dx with respect to t, divide by dx/dt again. |
| Parametric Arc Length (BC) [H] | L = ∫√((dx/dt)² + (dy/dt)²) dt. |
| Vector-Valued Functions (BC) [H] | Differentiate component by component. |
| Motion in the Plane (BC) [H] | Speed is the magnitude of velocity. |
| Slopes of Polar Curves (BC) [H] | Convert to parametric: x = r cos θ, y = r sin θ. |
| Area in Polar Coordinates (BC) [H] | A = ½∫r² dθ. |
| Area Between Polar Curves (BC) [H] | Subtract the inner sweep from the outer sweep. |
| Convergence of Sequences (BC) [M] | A sequence converges if aₙ approaches a limit. |
| Geometric Series (BC) [M] | Σarⁿ = a/(1−r) when |r| < 1. |
| The nth-Term Test (BC) [M] | If terms don't go to 0, the series diverges — but 0 proves nothing. |
| Integral Test & p-Series (BC) [M] | Σ1/nᵖ converges iff p > 1. |
| Comparison Tests (BC) [M] | Compare with a series you already understand. |
| Alternating Series (BC) [M] | Alternating + decreasing to 0 ⇒ converges. |
| Alternating Series Error Bound (BC) [H] | |error| ≤ first omitted term. |
| The Ratio Test (BC) [H] | L = lim|aₙ₊₁/aₙ|: L<1 converges, L>1 diverges, L=1 says nothing. |
| Absolute vs Conditional Convergence (BC) [M] | Does it still converge with all terms made positive? |
| Radius of Convergence (BC) [H] | The ratio test gives |x − c| < R. |
| Interval of Convergence (BC) [H] | Check both endpoints separately. |
| Taylor Polynomials (BC) [H] | Matching derivatives at a point: Pₙ(x) = Σ f⁽ᵏ⁾(a)(x−a)ᵏ/k!. |
| Taylor & Maclaurin Series (BC) [H] | The big four: eˣ, sin x, cos x, 1/(1−x). |
| Manipulating Known Series (BC) [H] | Substitute, multiply, differentiate, integrate known series. |
| Lagrange Error Bound (BC) [H] | |Rₙ| ≤ M|x−a|ⁿ⁺¹/(n+1)!. |
| Adding & Subtracting Whole Numbers | Multi-digit addition and subtraction. |
| Multiplication | Multiplying whole numbers. |
| Division | Dividing whole numbers. |
| Order of Operations | Parentheses first, then multiplication/division, then addition/subtraction. |
| Negative Numbers: Adding & Subtracting | Working with numbers below zero on the number line. |
| Negative Numbers: Multiplying & Dividing | Sign rules for products and quotients. |
| Exponents | Repeated multiplication in shorthand. |
| Square Roots | Undoing a square. |
| Equivalent Fractions | Different fractions can name the same amount. |
| Simplifying Fractions | Reducing a fraction to lowest terms. |
| Adding Fractions (Like Denominators) | Same-denominator addition. |
| Adding Fractions (Unlike Denominators) | Rewrite over a common denominator first. |
| Multiplying Fractions | Multiply straight across. |
| Dividing Fractions | Multiply by the reciprocal. |
| Fractions ↔ Decimals | Converting between the two notations. |
| Percent of a Number | Percent means per hundred. |
| Percent Increase & Decrease | Applying a percent change to a quantity. |
| Ratios & Proportions | Two quantities that scale together. |
| Evaluating Expressions | Substituting a value for a variable. |
| Combining Like Terms | Adding the coefficients of matching variable parts. |
| The Distributive Property | Multiplying across a sum. |
| One-Step Equations | Undoing a single operation. |
| Two-Step Equations | Undo addition/subtraction first, then multiplication. |
| Multi-Step Equations | Equations needing distribution or variables on both sides. |
| Linear Inequalities | Solving with <, >, ≤, ≥. |
| The Coordinate Plane | Locating points with (x, y) pairs. |
| Slope of a Line | Rise over run between two points. |
| Slope-Intercept Form | y = mx + b describes a whole line. |
| Systems of Equations (Substitution) | Two equations, two unknowns. |
| Adding & Subtracting Polynomials | Combining polynomials by collecting like terms. |
| Multiplying Binomials (FOIL) | Expanding products of binomials. |
| Factoring Out the GCF | Undoing the distributive property. |
| Factoring Trinomials | Reversing FOIL: finding two numbers that multiply to c and add to b. |
| Special Factoring Patterns | Difference of squares and perfect-square trinomials. |
| Solving Quadratics by Factoring | Zero-product property: if a·b = 0 then a = 0 or b = 0. |
| Solving x² = k | Taking square roots of both sides — remembering ±. |
| Completing the Square | Turning any quadratic into a perfect square plus a constant. |
| The Quadratic Formula | x = (−b ± √(b² − 4ac)) / 2a solves any quadratic. |
| Quadratic Models | Projectile motion and other parabolic models. |
| Product Rule for Exponents | Multiplying powers of the same base adds the exponents. |
| Quotient & Power Rules | Dividing powers subtracts exponents; a power of a power multiplies them. |
| Zero & Negative Exponents | Anything (nonzero) to the 0 power is 1; a negative exponent flips to a reciprocal. |
| Simplifying Radicals | Pulling perfect-square factors out of a square root. |
| Rational Exponents | Fractional exponents are roots: x^(p/q) is the q-th root of x, raised to the p. |
| Exponential Growth & Decay | Quantities that multiply by the same factor each time step: y = a·bᵗ. |
| Angle Relationships | Vertical, complementary, and supplementary angle pairs. |
| Triangle Angle Sum | The three angles of a triangle always add to 180°. |
| The Pythagorean Theorem | In a right triangle, a² + b² = c². |
| Distance & Midpoint | Measuring segments in the coordinate plane. |
| Similar Triangles | Same shape, different size: corresponding sides are proportional. |
| Perimeter & Area | Measuring around and inside basic shapes. |
| Circles: Area & Circumference | C = 2πr and A = πr². |
| Volume: Prisms & Cylinders | Volume = base area × height. |
| Special Right Triangles | The 45-45-90 and 30-60-90 side ratios. |
| Function Notation & Evaluation | Reading f(x) notation and plugging in inputs. |
| Domain & Range | Which inputs a function accepts, and which outputs it can produce. |
| Function Composition | Feeding one function's output into another: f(g(x)). |
| Inverse Functions | The function that undoes f: f⁻¹(b) is the input that f sends to b. |
| Piecewise Functions | Functions defined by different rules on different intervals. |
| Nonlinear Systems | Where a line meets a parabola: set the two formulas equal. |
| Polynomial Division | Dividing a polynomial by (x − a) with long or synthetic division. |
| Remainder & Factor Theorems | The remainder when p(x) is divided by (x − a) is simply p(a). |
| Zeros of Polynomials | Finding all the roots of a cubic by factoring it down. |
| End Behavior of Polynomials | Far from the origin, only the leading term matters. |
| Simplifying Rational Expressions | Factor top and bottom, then cancel the common factor. |
| Operations on Rational Expressions | Multiplying and dividing algebraic fractions. |
| Logarithms | log_b(x) asks: to what power must b be raised to get x? |
| Properties of Logarithms | Logs turn products into sums, quotients into differences, powers into multiples. |
| Arithmetic Sequences | Sequences that grow by a constant difference each step. |
| Geometric Sequences | Sequences that grow by a constant ratio each step. |
| Right-Triangle Trigonometry | SOH-CAH-TOA: the three trig ratios of an acute angle in a right triangle. |
| Degrees & Radians | Two ways to measure the same angle: 180° equals π radians. |
| The Unit Circle | Exact sine, cosine, and tangent values at the special angles. |
| Trig of Any Angle | Reference angles plus quadrant signs extend trig beyond 90°. |
| Inverse Trig Functions | arcsin, arccos, and arctan undo the trig functions on restricted ranges. |
| Asymptotes of Rational Functions | Where rational functions blow up and where they level off. |
| Graphs of Rational Functions | Holes, asymptotes, and intercepts tell the whole story of the graph. |
| Vectors: Components & Magnitude | A vector is a displacement: components ⟨Δx, Δy⟩ and a length. |
| Vector Operations | Scaling, adding, and dotting vectors — all component by component. |
| Parametric Equations | Describing a moving point by giving x and y as functions of time. |
| Polar Coordinates | Locating points by distance from the origin and angle from the x-axis. |
| Sigma Notation & Series | Σ compresses a sum: read the limits, add up the terms. |
| Average Rate of Change | The slope of the secant line: (f(b) − f(a)) / (b − a). |
| Limits: Graphical & Numerical | What value a function approaches — which need not be the value it takes. |
| Evaluating Limits Algebraically | Direct substitution — and the factor-and-cancel fix for 0/0. |
| One-Sided Limits | Approaching from the left or right — and when the two disagree. |
| Infinite Limits & Vertical Asymptotes | Where a function blows up: reading the sign of an infinite limit. |
| Limits at Infinity | End behavior of rational functions: compare the degrees. |
| The Limit Definition of the Derivative | The derivative is the limit of average rates of change. |
| Derivatives Graphically | Reading slopes off a graph. |
| The Power Rule | d/dx xⁿ = n·xⁿ⁻¹ for any real n. |
| Sum & Constant-Multiple Rules | Differentiate term by term. |
| The Product Rule | (fg)′ = f′g + fg′. |
| Derivatives of Trig Functions | d/dx sin x = cos x, d/dx cos x = −sin x, d/dx tan x = sec²x. |
| Derivatives of Exponentials & Logs | eˣ is its own derivative; (ln x)′ = 1/x. |
| The Chain Rule | d/dx f(g(x)) = f′(g(x)) · g′(x). |
| Implicit Differentiation | Differentiating equations that mix x and y. |
| Derivatives of Inverse Trig | (arcsin x)′ = 1/√(1−x²), (arctan x)′ = 1/(1+x²). |
| Higher-Order Derivatives | Differentiating again: f″, f‴, … |
| Tangent Lines & Linear Approximation | The tangent line is the best local linear stand-in for f. |
| Motion: Position, Velocity, Acceleration | v = s′, a = v′; at rest when v = 0. |
| Critical Points & Extrema | Where f′ = 0 or is undefined — the candidates for extrema. |
| Increasing & Decreasing Intervals | Sign of f′ decides the direction of f. |
| Concavity & Inflection Points | f″ > 0 bends up, f″ < 0 bends down. |
| Curve Sketching & f, f', f'' | Reading the shape of f from its derivatives. |