AP Calculus BC practice test
Practice questions for all ten AP Calculus BC units, with extra weight on the BC-only topics: integration by parts, partial fractions, improper integrals, Euler's method, logistic growth, parametric and polar curves, and series. Answer in your browser and read the working for each answer.
Question 1 of 25
Evaluate lim (x→0) of sin(5x)/x.
Question 2 of 25
f(x) = (x² − 9)/(x − 3) for x ≠ 3. What value of f(3) makes f continuous at x = 3?
Question 3 of 25
What is the derivative of f(x) = x·eˣ?
Question 4 of 25
What is d/dx of arctan x?
Question 5 of 25
f is differentiable and one-to-one, with f(2) = 5 and f′(2) = 3. What is (f⁻¹)′(5)?
Question 6 of 25
Use L'Hospital's rule to evaluate lim (x→0) of (1 − cos x)/x².
Question 7 of 25
Where does f(x) = x³ − 3x have a relative maximum?
Question 8 of 25
Find ∫ x·eˣ dx.
Question 9 of 25
Find ∫ 1/(x(x + 1)) dx.
Question 10 of 25
Evaluate the improper integral ∫ from 1 to ∞ of 1/x² dx.
Question 11 of 25
dy/dx = x + y with y(0) = 1. Using Euler's method with one step of size 0.1, what is the approximation of y(0.1)?
Question 12 of 25
A population satisfies dP/dt = 0.2P(1 − P/500). At what population is it growing fastest?
Question 13 of 25
For the logistic model dP/dt = 0.2P(1 − P/500) with P(0) = 50, what is lim (t→∞) P(t)?
Question 14 of 25
Which integral gives the length of the curve y = f(x) from x = a to x = b?
Question 15 of 25
A curve is given by x = t², y = t³. What is dy/dx at t = 2?
Question 16 of 25
For a parametric curve, how do you find d²y/dx²?
Question 17 of 25
A particle moves with x′(t) = 3 and y′(t) = 4. What is its speed?
Question 18 of 25
Which integral gives the area enclosed by a polar curve r = f(θ) for α ≤ θ ≤ β?
Question 19 of 25
What is the sum of the geometric series Σ from n = 0 to ∞ of (1/3)ⁿ?
Question 20 of 25
Which series converges?
Question 21 of 25
What is the radius of convergence of Σ xⁿ/3ⁿ?
Question 22 of 25
What is the Maclaurin series for eˣ?
Question 23 of 25
What is the second-degree Taylor polynomial for cos x centered at x = 0?
Question 24 of 25
The alternating series Σ (−1)ⁿ⁺¹/n is estimated by its first 4 terms. What bound does the alternating series error bound give?
Question 25 of 25
For a series Σ aₙ, lim (n→∞) |aₙ₊₁/aₙ| = 1/2. What does the ratio test tell you?
AP Calculus BC key terms
Flip through these flashcards to review the terms that come up most often. Use the arrow keys or click a card to flip it.
| Integration by parts | ∫ u dv = uv − ∫ v du |
| Partial fractions (distinct linear factors) | 1/((x − a)(x − b)) = A/(x − a) + B/(x − b) |
| Improper integral | An integral with an infinite limit or an unbounded integrand; evaluate as a limit. |
| Euler's method | yₙ₊₁ = yₙ + h · f(xₙ, yₙ) |
| Logistic growth | dP/dt = kP(1 − P/L); P → L, fastest growth at P = L/2 |
| Arc length | ∫ √(1 + (dy/dx)²) dx; parametric: ∫ √((dx/dt)² + (dy/dt)²) dt |
| Parametric dy/dx | (dy/dt) / (dx/dt) |
| Polar area | (1/2) ∫ r² dθ |
| Geometric series | Σ arⁿ = a/(1 − r) when |r| < 1 |
| p-series | Σ 1/nᵖ converges if p > 1, diverges if p ≤ 1 |
| nth term test | If lim aₙ ≠ 0, the series diverges. (If the limit is 0, the test says nothing.) |
| Ratio test | L = lim |aₙ₊₁/aₙ|: L < 1 converges absolutely, L > 1 diverges, L = 1 inconclusive |
| Taylor series | Σ f⁽ⁿ⁾(a)(x − a)ⁿ / n! |
| Maclaurin series for sin x | x − x³/3! + x⁵/5! − … |
| Alternating series error bound | |error| ≤ the first omitted term |
| Lagrange error bound | |Rₙ(x)| ≤ max|f⁽ⁿ⁺¹⁾| · |x − a|ⁿ⁺¹ / (n + 1)! |
See all 16 cards
Integration by parts∫ u dv = uv − ∫ v du
Partial fractions (distinct linear factors)1/((x − a)(x − b)) = A/(x − a) + B/(x − b)
Improper integralAn integral with an infinite limit or an unbounded integrand; evaluate as a limit.
Euler's methodyₙ₊₁ = yₙ + h · f(xₙ, yₙ)
Logistic growthdP/dt = kP(1 − P/L); P → L, fastest growth at P = L/2
Arc length∫ √(1 + (dy/dx)²) dx; parametric: ∫ √((dx/dt)² + (dy/dt)²) dt
Parametric dy/dx(dy/dt) / (dx/dt)
Polar area(1/2) ∫ r² dθ
Geometric seriesΣ arⁿ = a/(1 − r) when |r| < 1
p-seriesΣ 1/nᵖ converges if p > 1, diverges if p ≤ 1
nth term testIf lim aₙ ≠ 0, the series diverges. (If the limit is 0, the test says nothing.)
Ratio testL = lim |aₙ₊₁/aₙ|: L < 1 converges absolutely, L > 1 diverges, L = 1 inconclusive
Taylor seriesΣ f⁽ⁿ⁾(a)(x − a)ⁿ / n!
Maclaurin series for sin xx − x³/3! + x⁵/5! − …
Alternating series error bound|error| ≤ the first omitted term
Lagrange error bound|Rₙ(x)| ≤ max|f⁽ⁿ⁺¹⁾| · |x − a|ⁿ⁺¹ / (n + 1)!
About the AP Calculus BC exam
- Section I: 42 multiple-choice questions in 100 minutes, 50% of the score. Part A is 29 questions in 62 minutes with no calculator; Part B is 13 questions in 38 minutes with a graphing calculator.
- Section II: 6 free-response questions in 90 minutes, 50% of the score. Part A is 2 questions in 30 minutes with a calculator; Part B is 4 questions in 60 minutes without one.
- Multiple choice is answered in the Bluebook app; free-response answers are handwritten in a paper booklet. The question counts and timing are new for May 2027.
- The BC exam includes an AB subscore, so the AB units (1–8) still matter as much as series and parametric work.
Questions
Are these real AP Calculus BC exam questions?
No. They are original practice questions written to match the topics and style of the AP Calculus BC course. For released exam questions, see AP Central from the College Board.
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