AP Calculus Review — Complete Study Guide

AP Calculus AB Topics

Limits and continuity (ε-δ not required, but conceptual understanding is tested). Derivatives — all rules, implicit differentiation, related rates, curve analysis. Integrals — antiderivatives, FTC Parts 1 and 2, substitution, area, accumulation. Differential equations — slope fields, separable DEs, exponential growth/decay.

AP Calculus BC Additional Topics

Parametric equations and polar coordinates. Vector-valued functions. Sequences and series — convergence tests, power series, Taylor and Maclaurin series. Integration techniques — integration by parts, partial fractions. Improper integrals.

Exam Format

• Multiple Choice: 45 questions total (30 no-calculator, 15 with calculator). 1 hour 45 minutes.
• Free Response: 6 questions (2 with calculator, 4 without). 1 hour 30 minutes.
• Score: 1–5. Most universities accept 4 or 5 for credit.

High-Yield Formulas to Memorise

Common AP Exam Mistakes

• Not applying FTC Part 1 correctly when the upper limit is a function of x — requires Chain Rule.
• Forgetting to check both endpoints AND critical points on closed-interval optimisation.
• Missing +C on indefinite integrals in free response.
• Using L'Hôpital's Rule without verifying the indeterminate form first.
• Treating slope fields as direction fields for autonomous DEs only.

Free Response Strategy

Show all work — partial credit is awarded at each step. Define variables clearly. Include units in context problems. State theorems by name when you use them (MVT, IVT, FTC). Check your answer makes sense — plug a value back in, verify sign of derivative.

What the AP Exam Actually Tests

The AP Calculus exam tests conceptual understanding as much as procedural skill. In the free response section especially, you are expected to communicate mathematical reasoning clearly — not just get the right number. Graders award partial credit for correct reasoning even when the final answer is wrong, and withhold credit for correct answers with incorrect justification. This means: show your work, state theorems by name, define your variables, and include units in applied problems.

The Four Big Conceptual Areas

Limits and Continuity form the logical foundation. You need to evaluate limits algebraically, understand the formal definition conceptually (not necessarily prove with ε-δ), apply the Intermediate Value Theorem, and identify types of discontinuities. The FRQ section regularly asks you to justify continuity or differentiability at a point — a conceptual question, not a computation.

Derivatives are tested in three modes: computational (apply rules correctly), applied (related rates, optimisation), and analytical (use sign of f' and f'' to analyse functions). The latter is the most commonly mishandled — students know how to compute f'' but forget to state what it means and use the Candidates Test correctly on closed intervals.

Integrals appear in definite integral computation (FTC Part 2), accumulation problems (FTC Part 1 in context), Riemann sum approximations, and area/volume calculations. The integral as accumulation — understanding ∫ₐˣ f(t)dt as a running total — is tested almost every year in an FRQ.

Differential Equations on AB: slope fields and separable equations only. BC adds Euler's method and logistic differential equations. Slope field interpretation questions appear frequently — you must be able to read the slope field and match it to a DE.

BC-Only Topics — Prioritisation

If you are taking BC, these additional topics require focused study: series convergence tests (ratio test, comparison, alternating series, integral test — know when each applies), Taylor polynomials and error bounds (Lagrange remainder formula), parametric derivatives dy/dx and d²y/dx², polar area (∫½r²dθ), and L'Hôpital's Rule for indeterminate forms including 0·∞ and 1^∞ (requires rewriting).

Commonly Tested Theorems — Know By Name

• Intermediate Value Theorem (IVT): f continuous on [a,b], y₀ between f(a) and f(b) → ∃c: f(c) = y₀. Used to prove existence of a root or a specific value.
• Mean Value Theorem (MVT): f continuous on [a,b], differentiable on (a,b) → ∃c: f'(c) = (f(b)−f(a))/(b−a).
• Extreme Value Theorem (EVT): f continuous on [a,b] → f attains its absolute max and min on [a,b].
• FTC Part 1: d/dx[∫ₐˣ f(t)dt] = f(x). When the upper limit is g(x) not x: multiply by g'(x) (Chain Rule).
• FTC Part 2: ∫ₐᵇ f(x)dx = F(b) − F(a).

Calculator Section Strategy

On the calculator sections, four things are expected: finding zeros of functions (use solve/intersect), computing numerical derivatives at a point, computing definite integrals numerically, and plotting functions to inform analysis. The calculator does not replace mathematical reasoning — it handles arithmetic so you can focus on setting up problems correctly. Common mistake: using the calculator to evaluate a limit or derivative without setting up the mathematics correctly first.

Free Response Scoring — How to Get Points

AP FRQ scoring is analytic — each part of each question has specific scoring criteria. Typically 9 points per question (54 points total for 6 questions). Points are awarded for: setting up the correct integral or derivative expression (even if evaluation is wrong), correct answer with supporting work, correct use of theorems, and correct units. Points are withheld for: unsupported answers ("magic numbers"), incorrect notation, and contradicting earlier correct work. Write legibly, label your work clearly, and never erase a correct setup just because you cannot complete the computation.