AP Calculus BC multiple-choice practice 18 questions with answers

Written by STEM-Tutor's tutors · Updated October 2026 · 18 questions

Free AP Calculus BC multiple-choice questions in the style of the real exam. Click an answer and it is marked straight away; then open the hint or the worked solution, one step at a time.

On this page

How the section works
Part A: no calculator
Part B: calculator
Tips
After the questions
FAQ

How the AP Calculus BC multiple-choice section works

From May 2027 the multiple-choice section has 42 questions: 29 questions in 62 minutes without a calculator, then 13 questions in 38 minutes with one. It is worth 50% of your AP score, there is no penalty for a wrong answer, and each question has four options.

The questions below cover every unit, with the BC-only units (parametric, polar and vector functions, and series) and BC topics from other units such as Euler's method and integration by parts. Give yourself about two minutes a question, and do the no-calculator ones by hand.

Part A: no calculator (14 questions)

Q1Warm-upNo calculator

The harmonic series Σ 1/n:

Hint

Compare with the integral of 1/x, or group the terms.

Worked solution

It is a p-series with p = 1, and p-series converge only for p > 1.
So it diverges, even though its terms go to 0: C.

Q2Warm-upNo calculator

Apply the ratio test to Σ n/2ⁿ. The series:

Hint

Find lim |a_n+1/a_n|.

Worked solution

a_n+1/a_n = [(n + 1)/2ⁿ⁺¹] × [2ⁿ/n] = (n + 1)/(2n).
This tends to 1/2 < 1, so the series converges: B.

Q3Warm-upNo calculator

A curve is given by x = t², y = t³. What is dy/dx at t = 2?

Hint

dy/dx = (dy/dt)/(dx/dt).

Worked solution

dy/dt = 3t² and dx/dt = 2t.
dy/dx = 3t²/(2t) = 3t/2.
At t = 2: 3. The answer is A.

Q4Warm-upNo calculator

What is the sum of the series ∑_n=0^∞ (2/3)ⁿ?

Hint

A geometric series with |r| < 1 sums to a/(1 − r).

Worked solution

The first term is a = 1 and the ratio is r = 2/3.
Sum = 1/(1 − 2/3) = 1/(1/3).
The sum is 3. The answer is B.

Q5Warm-upNo calculator

What is the radius of convergence of ∑_n=0^∞ xⁿ/3ⁿ?

Hint

Use the ratio test, or see it as a geometric series in x/3.

Worked solution

The series is geometric with ratio x/3.
It converges when |x/3| < 1, that is |x| < 3.
The radius is 3. The answer is C.

Q6Exam-styleNo calculator

In the Maclaurin series for e^2x, what is the coefficient of x³?

Hint

Replace x by 2x in the series for e^x.

Worked solution

e^u = ∑ uⁿ/n!, and with u = 2x the x³ term is (2x)³/3!.
That is 8x³/6.
The coefficient is 4/3. The answer is A.

Q7Warm-upNo calculator

Which of these series converges?

Hint

Think about p-series, and the nth term test.

Worked solution

∑ 1/n^p converges only when p > 1, so 1/n (p = 1) and 1/√n (p = 1/2) diverge.
n/(n + 1) tends to 1, not 0, so that series diverges by the nth term test.
∑ 1/n² is a p-series with p = 2 > 1, so it converges. The answer is C.

Q8Exam-styleNo calculator

The alternating series ∑_n=1^∞ (−1)ⁿ⁺¹/n² is approximated by its first 3 terms. What upper bound does the alternating series error bound give for the error?

Hint

For an alternating series that meets the test's conditions, the error is at most the first omitted term.

Worked solution

The terms 1/n² decrease to 0, so the alternating series error bound applies.
The first omitted term is the 4th: 1/4².
The error is at most 1/16. The answer is B.

Q9Exam-styleNo calculator

dy/dx = x + y with y(0) = 1. Using Euler's method with two steps of size 0.5, what is the estimate of y(1)?

Hint

Each step: y_new = y + (step)(slope at the current point).

Worked solution

Step 1: slope at (0, 1) is 1, so y(0.5) ≈ 1 + 0.5(1) = 1.5.
Step 2: slope at (0.5, 1.5) is 2, so y(1) ≈ 1.5 + 0.5(2) = 2.5.
The answer is A.

Q10Exam-styleNo calculator

What is ∫₀¹ x e^x dx?

Hint

Integration by parts with u = x and dv = e^x dx.

Worked solution

∫ x e^x dx = x e^x − e^x + C.
At 1: e − e = 0. At 0: 0 − 1 = −1.
The integral is 0 − (−1) = 1. The answer is A.

Q11Warm-upNo calculator

What is ∫₁^∞ 1/x² dx?

Hint

Write it as a limit of ∫₁^b as b → ∞.

Worked solution

∫₁^b x⁻² dx = 1 − 1/b.
As b → ∞, 1/b → 0.
The integral converges to 1. The answer is B.

Q12Exam-styleNo calculator

A population satisfies dP/dt = 0.4P(1 − P/500) with P(0) = 100. At what population is P growing fastest?

Hint

For logistic growth, the growth rate is greatest at half the carrying capacity.

Worked solution

The carrying capacity is 500.
dP/dt = 0.4P − 0.0008P² is a quadratic in P with its maximum at P = 0.4/0.0016.
That is P = 250, half the carrying capacity. The answer is B.

Q13ChallengeNo calculator

What is the arc length of y = (2/3)x^3/2 from x = 0 to x = 3?

Hint

Arc length = ∫ √(1 + (dy/dx)²) dx.

Worked solution

dy/dx = x^1/2, so the integrand is √(1 + x).
∫₀³ (1 + x)^1/2 dx = (2/3)(4^3/2 − 1) = (2/3)(7).
The length is 14/3. The answer is A.

Q14Exam-styleNo calculator

Using the first two nonzero terms of the Maclaurin series for sin x, what is the approximate value of sin(0.5)?

Hint

sin x = x − x³/6 + …

Worked solution

The first two nonzero terms are x − x³/6.
At x = 0.5: 0.5 − 0.125/6 = 0.5 − 0.02083.
That is 0.4792. The answer is A.

Part B: calculator (4 questions)

Q15Warm-upCalculator

A particle has position x = t², y = 2t. What is its speed at t = 1?

Hint

Speed = √((dx/dt)² + (dy/dt)²).

Worked solution

dx/dt = 2t = 2 and dy/dt = 2 at t = 1.
Speed = √(4 + 4) = √8.
√8 ≈ 2.828. The answer is A.

Q16Exam-styleCalculator

What is the area of the region inside the polar curve r = 3 for 0 ≤ θ ≤ π/2?

Hint

Polar area = (1/2) ∫ r² dθ.

Worked solution

Area = (1/2) ∫₀^π/2 9 dθ.
= (1/2)(9)(π/2) = 9π/4.
9π/4 ≈ 7.07, a quarter of the circle's area. The answer is A.

Q17ChallengeCalculator

What is ∫₂³ 1/(x(x − 1)) dx?

Hint

Use partial fractions: 1/(x(x − 1)) = 1/(x − 1) − 1/x.

Worked solution

The antiderivative is ln|x − 1| − ln|x|.
At 3: ln 2 − ln 3. At 2: 0 − ln 2.
The integral is 2 ln 2 − ln 3 = ln(4/3) ≈ 0.288. The answer is A.

Q18Exam-styleCalculator

A particle has position vector ⟨t², sin(πt)⟩. What is its speed at t = 1?

Hint

Differentiate each component, then find the magnitude.

Worked solution

Velocity = ⟨2t, π cos(πt)⟩ = ⟨2, −π⟩ at t = 1.
Speed = √(4 + π²).
That is about 3.724. The answer is A.

Tips for the AP Calculus BC multiple-choice section

BC-only topics are only part of the exam. Series, parametric, polar and vector questions sit alongside AB content, so also practise the AB multiple-choice questions.
Series questions reward knowing the tests cold: geometric, p-series, ratio, comparison and alternating series, with their conditions.
Know the standard Maclaurin series for e^x, sin x, cos x and 1/(1 − x), and how to build new ones from them.
Polar area needs the half: (1/2) ∫ r² dθ. It is the most common slip on polar questions.

After the questions

Note every question you missed and its topic, then practise that unit: AP Calculus practice by unit. For a full timed test, take the practice exam, then put your score into our AP Calculus BC score calculator. For the free-response section, use the official AP Calculus BC past papers.

Frequently asked questions

Are these real AP Calculus BC exam questions?

No. They are original questions written by our tutors in the style of the exam, so each one can have a full worked solution. For real exam questions, use the official AP Calculus BC past papers.

How many multiple-choice questions are on the AP Calculus BC exam?

From May 2027, 42: 29 questions in 62 minutes without a calculator, then 13 questions in 38 minutes with one. The multiple-choice section is 50% of your score.

Is there a penalty for wrong answers?

No. Only correct answers count, so answer every question, even if you have to guess.