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150 multiple-choice questions and 212 flashcards, written to the College Board AP Calculus BC Course and Exam Description blueprint. Every question carries a full rationale.
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College Board AP Calculus BC Course and Exam Description — 10 CED units (extends AB's 8 with parametric/polar/vector functions and sequences/series)
CoStudy's AP Calculus BC bank holds 362 items. Every multiple-choice question carries a written rationale explaining why the correct answer is correct and why each distractor is tempting but wrong.
A sample of 12 multiple-choice questions from the bank, with the full rationale shown.
If Σ aₙ converges absolutely with sum S, then:
Answer: D — Σ |aₙ| ≥ |S| with equality possible — rearrangements preserve sum
D) For absolutely convergent series: Σ |aₙ| ≥ |Σ aₙ| (triangle inequality), with equality if all aₙ same sign. Rearrangements preserve sum. C) Riemann rearrangement applies only to CONDITIONALLY convergent series. A/B) Wrong.
The series Σ_{n=1}^∞ (−1)^n / n^(1/3) is:
Answer: C — Conditionally convergent — alternating series test passes; Σ 1/n^(1/3) diverges (p = 1/3 < 1)
C) 1/n^(1/3) decreases to 0 → alternating series converges. Absolute series Σ 1/n^(1/3) is p-series with p = 1/3 < 1 → diverges. Conditionally convergent. A) No, absolute series diverges. B) Test passes. D) Determinable.
Implicit differentiation: if x² + y² = 25, then dy/dx =
Answer: B — −x/y (differentiate: 2x + 2y·y' = 0 → y' = −x/y)
Differentiate both sides w.r.t. x: 2x + 2y·(dy/dx) = 0 → dy/dx = −x/y. Circle of radius 5: makes sense geometrically (perpendicular to radius). Foundational AB topic.
Apply the ratio test to Σ n!/n^n. The limit L equals:
Answer: D — 1/e — converges since 1/e < 1
D) L = lim |((n+1)!/(n+1)^(n+1)) · (n^n/n!)| = lim (n^n/(n+1)^n) = lim 1/((n+1)/n)^n = 1/e ≈ 0.368 < 1. Series converges. A/B/C) Wrong limit.
A particle moves so that its position is given by parametric equations x(t) = t² and y(t) = 2t³. The speed at t = 1 is:
Answer: B — √(4 + 36) = √40 ≈ 6.32
For parametric motion, speed = √[(dx/dt)² + (dy/dt)²]. dx/dt = 2t = 2; dy/dt = 6t² = 6. Speed = √(4 + 36) = √40 = 2√10 ≈ 6.32.
∫ x² dx =
Answer: C — x³/3 + C
C) Power rule for integration. A) Differentiation. B/D) Wrong exponent or coefficient.
For x(t) = t² − 1, y(t) = t³ − 3t, find dy/dx at t = 2.
Answer: A — 9/4 — (3t² − 3)/(2t) = 9/4 at t = 2
A) dy/dx = (dy/dt)/(dx/dt) = (3t² − 3)/(2t) = (12 − 3)/4 = 9/4. B/D) Used wrong derivative for numerator. C) Forgot the −3.
The derivative of ln(x²+1) is:
Answer: B — 2x/(x²+1) (chain rule: d/dx[ln u] = u'/u with u = x²+1)
d/dx[ln u] = u'/u. u = x²+1, u' = 2x. Result: 2x/(x²+1). (A) Forgot the chain rule. (C,D) Misapplied.
L'Hôpital's rule applies when a limit takes the indeterminate form:
Answer: A — 0/0 or ∞/∞ — then take the limit of f'/g'
A) Both classic indeterminate cases. B) Not indeterminate. C/D) Too narrow.
The polar equation r = 2 cos θ represents:
Answer: D — A circle of radius 1 centered at (1, 0) in Cartesian coordinates
D) Standard polar-to-Cartesian conversion. A/B/C) Each is incorrect.
∫₀¹ x · ln(x) dx (taking the limit at 0):
Answer: D — Converges to −1/4 — computable via integration by parts
D) Standard improper integral. A/B/C) Each is incorrect.
If position s(t) = t³ − 6t² + 9t, when is the particle at rest?
Answer: D — t = 1 and t = 3 (v = s' = 3t² − 12t + 9 = 3(t−1)(t−3) = 0)
Velocity v(t) = s'(t) = 3t² − 12t + 9. Factor: 3(t² − 4t + 3) = 3(t−1)(t−3). Zero at t = 1 and t = 3. Common kinematics question. AP commonly asks about direction changes too (sign of v).
6 sample cards from the 212 in the bank.
Common error: forgetting to test endpoints of interval of convergence?
Power series may converge or diverge at endpoints — must check separately to determine interval (closed, half-open, open).
Geometric series sum (|r| < 1)?
S = a₁/(1 − r). Diverges if |r| ≥ 1.
FTC Part 2?
If g(x) = ∫_a^x f(t) dt, then g'(x) = f(x).
Find dy/dx if y = ln(sin x).
(1/sin x)·cos x = cot x.
Solve dy/dx = y² with y(0) = 1.
Separable: −1/y = x + C; y(0) = 1 → C = −1; y = 1/(1−x).
Common BC FRQ topic: 'particle moves on parametric curve…'?
Asks for speed, distance traveled (∫|v(t)|dt), positions, slope of path (dy/dx), etc.
These samples are a small slice. The full bank runs flashcards, multiple choice and timed mock exams with per-chapter progress tracking, on the web and in the iOS app.
The AP Calculus BC bank holds 362 items: 150 multiple-choice questions, 212 flashcards. 18 of them are on this page to read free, with no signup.
Yes. Every multiple-choice item carries a written rationale that states the controlling principle behind the correct answer and then addresses each wrong option in turn — why it tempts and precisely where it fails. Knowing why the plausible answer was wrong is worth more than knowing which letter was right.
College Board AP Calculus BC Course and Exam Description — 10 CED units (extends AB's 8 with parametric/polar/vector functions and sequences/series)
The samples on this page are free to read in full, rationales included, with no account. The complete 362-item bank, the timed mock exams and per-chapter progress tracking are part of CoStudy on the web and in the iOS app.
Last reviewed 2026-08-22. Banks are written against the certifying body's published exam outline and re-checked when that outline changes — exams get renumbered, retired and reweighted, and a bank written to a superseded outline teaches the wrong proportions. Figures that are re-indexed annually are deliberately not asserted as rules; the questions test the governing principle instead.
This bank is written against the College Board's published exam material. Check the AP Course and Exam Descriptions for the current outline, fees and eligibility rules — those change, and the certifying body is the only authority on them. CoStudy is not affiliated with the College Board.