AP Calculus BC

85 core topics + 85 prerequisite topics taught as needed · approximately 46 hours of instruction including spaced review

How the course runs

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.

Core curriculum

Limits & Continuity · 10 topics

Limits: Graphical & Numerical [E]	What value a function approaches — which need not be the value it takes.
Evaluating Limits Algebraically [M]	Direct substitution — and the factor-and-cancel fix for 0/0.
One-Sided Limits [M]	Approaching from the left or right — and when the two disagree.
Limits by Rationalization [H]	Clearing a 0/0 form by multiplying by the conjugate.
Infinite Limits & Vertical Asymptotes [M]	Where a function blows up: reading the sign of an infinite limit.
Limits at Infinity [M]	End behavior of rational functions: compare the degrees.
Trig Limits [M]	The two special limits sin(x)/x → 1 and (1 − cos x)/x → 0.
The Squeeze Theorem [M]	Trapping a wild function between two tame ones with the same limit.
Continuity [H]	No jumps, holes, or blow-ups: the limit equals the value.
Intermediate Value Theorem [M]	A continuous function can't skip values: sign changes force roots.

Differentiation · 16 topics

The Limit Definition of the Derivative [M]	The derivative is the limit of average rates of change.
Derivatives Graphically [M]	Reading slopes off a graph.
The Power Rule [M]	d/dx xⁿ = n·xⁿ⁻¹ for any real n.
Sum & Constant-Multiple Rules [M]	Differentiate term by term.
The Product Rule [M]	(fg)′ = f′g + fg′.
The Quotient Rule [M]	(f/g)′ = (f′g − fg′)/g².
Derivatives of Trig Functions [M]	d/dx sin x = cos x, d/dx cos x = −sin x, d/dx tan x = sec²x.
Derivatives of Exponentials & Logs [M]	eˣ is its own derivative; (ln x)′ = 1/x.
The Chain Rule [H]	d/dx f(g(x)) = f′(g(x)) · g′(x).
Combining Differentiation Rules [H]	Product, quotient and chain rules together.
Implicit Differentiation [H]	Differentiating equations that mix x and y.
Derivatives of Inverse Trig [M]	(arcsin x)′ = 1/√(1−x²), (arctan x)′ = 1/(1+x²).
Derivatives of Inverse Functions [M]	(f⁻¹)′(b) = 1 / f′(f⁻¹(b)).
Higher-Order Derivatives [M]	Differentiating again: f″, f‴, …
Differentiability & Continuity [M]	Differentiable ⇒ continuous, but not conversely.
Tangent Lines & Linear Approximation [M]	The tangent line is the best local linear stand-in for f.

Applications of Differentiation · 10 topics

Motion: Position, Velocity, Acceleration [M]	v = s′, a = v′; at rest when v = 0.
Related Rates [H]	Differentiating a geometric relationship with respect to time.
Critical Points & Extrema [M]	Where f′ = 0 or is undefined — the candidates for extrema.
The Mean Value Theorem [M]	Somewhere, instantaneous rate equals average rate.
Increasing & Decreasing Intervals [M]	Sign of f′ decides the direction of f.
Concavity & Inflection Points [M]	f″ > 0 bends up, f″ < 0 bends down.
Curve Sketching & f, f', f'' [M]	Reading the shape of f from its derivatives.
Optimization [H]	Maximizing or minimizing with calculus.
L'Hôpital's Rule [M]	For 0/0 or ∞/∞, differentiate top and bottom.
Indeterminate Forms [M]	0·∞ and repeated applications.

Integration · 12 topics

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.

Applications of Integration · 8 topics

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 · 6 topics

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, Polar & Vector Calculus · 8 topics

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.

Infinite Series · 15 topics

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)!.

Prerequisite material — taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 8 topics

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.

Fractions · 6 topics

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.

Decimals, Percents & Ratios · 4 topics

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.

Expressions & Equations · 7 topics

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 <, >, ≤, ≥.

Linear Functions · 4 topics

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.

Quadratics & Polynomials · 10 topics

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.

Radicals & Exponentials · 7 topics

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.
Operations with Radicals	Adding like radicals and multiplying square roots.
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.

Geometry · 10 topics

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.
Volume: Cones, Pyramids & Spheres	Pointed solids hold one third of the matching prism; spheres use 4/3 πr³.
Special Right Triangles	The 45-45-90 and 30-60-90 side ratios.

Functions & Algebra II · 16 topics

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.

Trigonometry · 5 topics

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.

Precalculus · 8 topics

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).

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