AP Calculus AB Notes — Limits, Derivatives, and Integrals

AP Calculus AB covers the fundamentals of differential and integral calculus across 8 units. These notes summarize the key theorems, rules, and problem types at the level the FRQ section actually tests — proofs aren't required, but conceptual understanding and correct notation are.

Unit 1: Limits and Continuity

Limit notation and evaluation (substitution, factoring, rationalizing, L'Hôpital's rule). One-sided limits. Continuity: a function is continuous at x = c if the limit exists, f(c) is defined, and they're equal. Intermediate Value Theorem (IVT): if f is continuous on [a,b] and k is between f(a) and f(b), then there exists c such that f(c) = k.

Units 2–3: Differentiation

Definition of derivative as limit of difference quotient. Basic rules: power rule, constant multiple, sum/difference. Product rule: (uv)' = u'v + uv'. Quotient rule: (u/v)' = (u'v − uv')/v². Chain rule: [f(g(x))]' = f'(g(x))·g'(x). Implicit differentiation (differentiate both sides with respect to x, solve for dy/dx). Derivatives of trig, exponential (eˣ), and logarithmic functions. Mean Value Theorem (MVT): if f is differentiable on (a,b) and continuous on [a,b], then there exists c where f'(c) = [f(b)−f(a)]/(b−a).

Unit 4: Contextual Applications of Derivatives

Position, velocity (v = s'), acceleration (a = v' = s''). Related rates: write an equation relating variables, differentiate implicitly with respect to time, substitute known values. Local linearization (tangent line approximation).

Unit 5: Analytical Applications of Derivatives

Critical points (f'(c) = 0 or undefined). First Derivative Test: f increases where f' > 0, decreases where f' < 0. Second Derivative Test: f''(c) > 0 → local min, f''(c) < 0 → local max. Concavity and inflection points. Optimization: find the critical point of an objective function subject to a constraint equation.

Units 6–8: Integration and Differential Equations

Riemann sums (left, right, midpoint, trapezoid). Definite integral as signed area. Fundamental Theorem of Calculus: Part 1: d/dx[∫ₐˣ f(t)dt] = f(x). Part 2: ∫ₐᵇ f(x)dx = F(b) − F(a). U-substitution. Area between curves: ∫[top − bottom]dx. Volume by disk/washer: π∫[R²−r²]dx. Separable differential equations: separate variables, integrate both sides, solve for y. Slope fields and exponential growth/decay (y = Ce^(kt)).

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