Everything in Calculus AB plus sequences, series, and parametric/polar calculus.
Study guide last reviewed September 25, 2026 · see what changed
Format reflects the current published structure — always confirm against AP Central before test day, since formats occasionally change.
| Section | Details | Time | Weight |
|---|---|---|---|
| Section I – Multiple Choice | 42 questions (29 no-calculator, 13 calculator) | 1 hr 40 min | 50% |
| Section II – Free Response | 6 questions (2 calculator, 4 no-calculator) | 1 hr 30 min | 50% |
Straight answers about the AP Calculus BC exam — format, credit, dates, and this site.
Section I – Multiple Choice: 42 questions (29 no-calculator, 13 calculator) (1 hr 40 min, 50% of the total score); Section II – Free Response: 6 questions (2 calculator, 4 no-calculator) (1 hr 30 min, 50% of the total score). See the Exam Format tab above for the full breakdown, and always double-check the current year's format on AP Central before test day.
It's multiple-choice questions and written free-response or essay questions.
Partly — this exam has both a calculator-allowed portion and a no-calculator portion. Check the Exam Format tab above for exactly which section allows one.
It depends entirely on the college — most award credit for a 3 or higher, but many selective schools require a 4 or a 5, and some don't grant credit at all for certain scores. Always check your target college's own AP credit policy page rather than assuming a 3 will count everywhere.
AP exams are administered during the College Board's annual testing window in the first two full weeks of May, with the exact date and time for each subject set by the College Board and announced the preceding fall. Check AP Central or your school's AP coordinator for this year's exact date.
Yes. The unit-by-unit study guide, worked examples, practice quiz, and full-length timed practice exam on this page are all free, with no login, signup, or paywall.
Unit-by-unit breakdown of what the course actually covers, in teaching order.
The same limit and continuity foundation as AB, covered at a faster pace since it underpins the additional BC-only content, including more rigorous treatment of limits involving infinity and indeterminate forms.
AB's full differentiation toolkit (power, product, quotient, chain rules, implicit differentiation, inverse functions) mastered quickly as a prerequisite for BC's additional applications in parametric, polar, and series contexts.
Related rates, motion analysis, and optimization at AB's level, extended to include parametrically-defined curves later in the course where derivatives are found via dy/dx = (dy/dt)/(dx/dt).
The Fundamental Theorem of Calculus and u-substitution from AB, plus additional integration techniques BC requires for series and parametric/polar applications, including improper integrals with infinite bounds or discontinuities.
Separable differential equations from AB, plus the BC-specific logistic growth model (dy/dt = ky(1 - y/L)) which produces an S-shaped curve approaching a carrying capacity L, contrasting with unbounded exponential growth.
Area and volume applications from AB, plus the BC-specific addition of arc length calculations (integrating √(1 + (dy/dx)²)) for both rectangular and parametric curves.
Calculus operations on parametrically-defined curves (dy/dx and arc length via the parameter), polar curve area calculations (½∫r² dθ), and vector-valued functions where derivatives represent velocity and acceleration vectors — entirely new content not covered in AB.
Convergence tests (ratio test, comparison tests, integral test, alternating series test) for determining whether an infinite series converges, Taylor and Maclaurin series representing functions as infinite polynomials, and the Lagrange error bound for estimating how accurate a Taylor polynomial approximation is — widely considered the most distinctively BC unit and a major FRQ source.
Concrete, fully worked examples — this is what separates a page reviewers see as genuinely useful from one that just lists topics.
Test for convergence: does ∑ (n / 2n) converge?
Apply the ratio test: L = lim(n→∞) |an+1/an| = lim [(n+1)/2n+1] ÷ [n/2n] = lim (n+1)/(2n) = 1/2. Since L = 1/2 < 1, the series converges. The ratio test is the default first move whenever a series mixes a polynomial-type term with an exponential term.
Lagrange error bound: find the 3rd-degree Taylor polynomial for f(x) = ex centered at 0, and bound the error at x = 1.
Step 1: P3(x) = 1 + x + x²/2 + x³/6.
Step 2: The Lagrange error is R3(1) = f(4)(z)/4! · 14 for some z between 0 and 1. Since f(4)(x) = ex is increasing, its max on [0,1] is e ≈ 2.718.
Step 3: |R3(1)| ≤ 2.718/24 ≈ 0.113. Using the max of the derivative over the whole interval (not just at x itself) is the step students skip.
Arc length: find the length of x = t², y = t³ for t in [0, 2].
Step 1: dx/dt = 2t, dy/dt = 3t².
Step 2: Arc length = ∫₀² √[(2t)² + (3t²)²] dt = ∫₀² t√(4 + 9t²) dt.
Step 3: Let u = 4 + 9t², du = 18t dt. Evaluating gives (1/27)[401.5 − 41.5] ≈ 9.07.
Polar area: find the area enclosed by the cardioid r = 2 + 2cosθ over [0, 2π].
Step 1: Area = ½∫₀2π r² dθ = ½∫ (4 + 8cosθ + 4cos²θ) dθ.
Step 2: Over a full period, ∫8cosθ dθ = 0, and ∫4cos²θ dθ = 4π. The constant term gives ∫4 dθ = 8π.
Step 3: Total = ½(8π + 0 + 4π) = 6π. Watch for curves where r goes negative — you can accidentally trace part of the region twice.
Logistic growth: dy/dt = 0.5y(1 − y/200), y(0) = 20. What happens as t → ∞, and when is growth fastest?
As t → ∞, y approaches the carrying capacity L = 200. Growth is fastest at the inflection point, where y = L/2 = 100 — a fact BC tests directly rather than requiring you to solve the full differential equation.
Vector-valued motion: r(t) = ⟨t², ln t⟩ for t > 0. Find velocity, acceleration, and speed at t = 1.
v(t) = r′(t) = ⟨2t, 1/t⟩, so v(1) = ⟨2, 1⟩.
a(t) = r″(t) = ⟨2, −1/t²⟩, so a(1) = ⟨2, −1⟩.
Speed = |v(1)| = √(4+1) = √5. Speed is always the magnitude of the velocity vector, never the velocity vector itself.
The errors that cost the most points across the whole exam, not just one unit.
Unofficial practice questions written for this study guide — not official AP content and not from College Board. Use them to self-check, then use the real past free-response questions below for authentic exam practice.
Unofficial practice, timed to feel like the real thing. Section I pulls from this page's full multiple-choice bank; Section II gives you free-response prompts to write out and self-score against a rubric. Bank size is still growing toward full exam length — timing is paced per-question, not pulled from this subject's official duration.