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AP Calculus AB multiple-choice practice 24 questions with answers

Written by STEM-Tutor's tutors · Updated October 2026 · 24 questions

Free AP Calculus AB 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
  1. How the section works
  2. Part A: no calculator
  3. Part B: calculator
  4. Tips
  5. After the questions
  6. FAQ

How the AP Calculus AB 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 of AB. Give yourself about two minutes a question, and do the no-calculator ones by hand.

Part A: no calculator (21 questions)

Q1Warm-upNo calculator
A function f is continuous at x = a when:

Hint

Continuity needs three things to hold together.

Worked solution
  1. f(a) must exist, the limit must exist, and they must be equal.
  2. Option C includes all three conditions.
  3. Differentiability is stronger than continuity, so it is not required. The answer is C.
Q2Exam-styleNo calculator
f is continuous on [1, 3], with f(1) = −2 and f(3) = 5. Which must be true?

Hint

Which theorem guarantees a value between f(1) and f(3)?

Worked solution
  1. f is continuous, and 0 lies between f(1) = −2 and f(3) = 5.
  2. By the intermediate value theorem, f(c) = 0 for some c in (1, 3).
  3. The other options need more information. The answer is A.
Q3Warm-upNo calculator
limh→0 [(2 + h)2 − 4]/h equals:

Hint

This is the definition of a derivative. Of which function, at which point?

Worked solution
  1. It is f'(2) for f(x) = x2, since f(2) = 4.
  2. f'(x) = 2x, so f'(2) = 4.
  3. The answer is C.
Q4Exam-styleNo calculator
For f(x) = |x| at x = 0, which is true?

Hint

Look at the slope from each side.

Worked solution
  1. The graph is unbroken at 0, so f is continuous.
  2. The slope is −1 from the left and +1 from the right: a corner, so no derivative.
  3. The answer is B. Differentiable always implies continuous, but not the other way round.
Q5Warm-upNo calculator
d/dx ln(x2 + 1) equals:

Hint

d/dx ln u = u'/u.

Worked solution
  1. u = x2 + 1, so u' = 2x.
  2. The derivative is 2x/(x2 + 1). The answer is B.
Q6Warm-upNo calculator
V(t) is the volume of water in a tank, in litres, t minutes after it starts filling. What are the units of V'(t)?

Hint

A derivative's units are the output's units divided by the input's units.

Worked solution
  1. V is in litres and t is in minutes.
  2. So V'(t) is in litres per minute. The answer is C.
Q7Exam-styleNo calculator
For the particle with v(t) = t2 − 4t + 3, at t = 0.5 the particle is:

Hint

Compare the signs of v(0.5) and a(0.5).

Worked solution
  1. v(0.5) = 0.25 − 2 + 3 = 1.25 > 0.
  2. a(0.5) = 1 − 4 = −3 < 0.
  3. Opposite signs mean it is slowing down: B.
Q8Warm-upNo calculator
If f''(x) > 0 on an interval, the graph of f there is:

Hint

The second derivative describes the bending.

Worked solution
  1. f'' > 0 means f' is increasing, so the slopes are getting larger.
  2. That is concave up. The answer is C.
Q9Warm-upNo calculator
f'(x) changes from positive to negative at x = c. Then f has:

Hint

Think about what f does before and after c.

Worked solution
  1. f' > 0 before c: f increases. f' < 0 after c: f decreases.
  2. So f has a local maximum at c. The answer is B.
Q10Warm-upNo calculator
d/dx ∫0x cos(t2) dt equals:

Hint

This is the fundamental theorem, part 1.

Worked solution
  1. The derivative of an accumulation function is the integrand at the upper limit.
  2. So the answer is cos(x2): B.
Q11Warm-upNo calculator
Which function is a solution of dy/dx = 2y?

Hint

Substitute each one: is its derivative twice itself?

Worked solution
  1. For y = e2x, dy/dx = 2e2x = 2y.
  2. None of the others satisfies the equation. The answer is C.
Q12Warm-upNo calculator
dy/dt = ky with k < 0 models:

Hint

The solution is y = y0ekt. What happens when k is negative?

Worked solution
  1. y = y0ekt with k < 0 shrinks by the same factor each unit of time.
  2. That is exponential decay: B.
Q13Warm-upNo calculator
For a particle with velocity v(t), the total distance travelled from t = a to t = b is:

Hint

Distance counts movement in both directions as positive.

Worked solution
  1. ∫ v(t) dt is the displacement, where backward motion cancels forward motion.
  2. Distance needs |v(t)|. The answer is B.
Q14Warm-upNo calculator
If f(x) = x3 − 4x, what is f′(2)?

Hint

Differentiate term by term, then substitute.

Worked solution
  1. f′(x) = 3x2 − 4.
  2. f′(2) = 3(4) − 4 = 8.
  3. The answer is B.
Q15Warm-upNo calculator
What is ∫02 (3x2 + 1) dx?

Hint

Find an antiderivative and use the fundamental theorem.

Worked solution
  1. An antiderivative is x3 + x.
  2. At 2: 8 + 2 = 10. At 0: 0.
  3. The integral is 10. The answer is B.
Q16Warm-upNo calculator
What is the derivative of ln(x2 + 1) at x = 1?

Hint

Chain rule: the derivative of ln(u) is u′/u.

Worked solution
  1. d/dx ln(x2 + 1) = 2x/(x2 + 1).
  2. At x = 1: 2/2 = 1.
  3. The answer is B.
Q17Warm-upNo calculator
If y = sin(3x), what is dy/dx at x = 0?

Hint

Chain rule: multiply by the derivative of the inside.

Worked solution
  1. dy/dx = 3 cos(3x).
  2. At x = 0: 3 cos 0 = 3.
  3. The answer is C.
Q18Warm-upNo calculator
What is the area of the region between y = x and y = x2 for 0 ≤ x ≤ 1?

Hint

Integrate top minus bottom. Which curve is on top between 0 and 1?

Worked solution
  1. On (0, 1), x > x2, so the area is ∫01 (x − x2) dx.
  2. That is 1/2 − 1/3.
  3. The area is 1/6. The answer is A.
Q19Exam-styleNo calculator
The curve x2 + y2 = 25 passes through (3, 4). What is dy/dx there?

Hint

Differentiate implicitly with respect to x.

Worked solution
  1. 2x + 2y dy/dx = 0, so dy/dx = −x/y.
  2. At (3, 4): dy/dx = −3/4.
  3. The answer is A.
Q20Exam-styleNo calculator
Using the tangent line to y = √x at x = 4, what is the approximate value of √4.1?

Hint

The tangent line is y = f(4) + f′(4)(x − 4).

Worked solution
  1. f(4) = 2 and f′(x) = 1/(2√x), so f′(4) = 1/4.
  2. √4.1 ≈ 2 + (1/4)(0.1) = 2.025.
  3. The answer is A.
Q21Exam-styleNo calculator
dy/dx = 2y and y(0) = 3. What is y(1)?

Hint

Separate the variables, or recognise exponential growth.

Worked solution
  1. The solution of dy/dx = ky is y = Cekx, so y = Ce2x.
  2. y(0) = 3 gives C = 3, so y = 3e2x.
  3. y(1) = 3e2. The answer is A.

Part B: calculator (3 questions)

Q22Exam-styleCalculator
What is the maximum value of f(x) = x e−x for x > 0?

Hint

Set f′(x) = 0, then evaluate f there.

Worked solution
  1. f′(x) = e−x(1 − x), which is zero at x = 1.
  2. f′ changes from positive to negative at x = 1, so it is a maximum.
  3. f(1) = 1/e ≈ 0.368. The answer is A.
Q23Exam-styleCalculator
A particle moves along a line with velocity v(t) = t2 − 4t + 3 for 0 ≤ t ≤ 4. What is the total distance it travels?

Hint

Total distance is the integral of |v(t)|, so split the interval where v changes sign.

Worked solution
  1. v(t) = (t − 1)(t − 3), which is zero at t = 1 and t = 3.
  2. On [0, 1], [1, 3] and [3, 4] the distances are 4/3, 4/3 and 4/3.
  3. Total distance = 4. The displacement, 4/3, is a different quantity. The answer is C.
Q24Exam-styleCalculator
What is the average value of f(x) = sin x on the interval [0, π]?

Hint

Average value = (1/(b − a)) ∫ab f(x) dx.

Worked solution
  1. ∫0π sin x dx = 2.
  2. Divide by the length of the interval, π: 2/π.
  3. 2/π ≈ 0.637. The answer is A.

Tips for the AP Calculus AB multiple-choice section

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 AB score calculator. For the free-response section, use the official AP Calculus AB past papers.

Frequently asked questions

Are these real AP Calculus AB 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 AB past papers.

How many multiple-choice questions are on the AP Calculus AB 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.