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AP Calculus AB practice test

StudySim's free AP Calculus AB practice test has 24 sample questions with answer explanations. Work through limits, derivatives, integrals, and differential equations with worked solutions. Use the answer explanations to understand the reasoning, then revisit the topics that need more work.

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Sample AP Calculus AB questions

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AP Calculus AB Sample Questions

Practice quiz

1. What is lim as x → 3 of (x² − 9)/(x − 3)?

2. Let f(x) = (x² − 4)/(x − 2) for x ≠ 2 and f(2) = k. Which k makes f continuous at 2?

3. A function has left-hand limit 5 and right-hand limit −1 at x = 2. What is its two-sided limit there?

4. A continuous function satisfies f(1) = −2 and f(4) = 7. Which statement must be true?

5. For f(x) = x³ − 4x, what is f′(2)?

6. What is the derivative of x² sin x?

7. If y = e^(3x²), what is dy/dx?

8. The curve x² + y² = 25 passes through (3, 4). What is its tangent slope at that point?

9. A particle has position s(t) = t³ − 6t² + 9t for t ≥ 0. At which times is it at rest?

10. A spherical balloon has volume V = (4/3)πr³. Its radius grows at 2 cm/s. When r = 3 cm, what is dV/dt?

11. A differentiable function has f(1) = 2 and f(5) = 10 and is continuous on [1, 5]. What does the mean value theorem guarantee?

12. If f′(x) = (x − 1)(x + 2), which describes x = 1?

13. A rectangle has perimeter 40 m. What is the largest possible area?

14. For f(x) = x⁴ − 4x³, where is its graph concave up?

15. What is ∫ from 0 to 2 of (3x² + 1) dx?

16. If F(x) = ∫ from 1 to x² of cos t dt, what is F′(x)?

17. A tank initially holds 12 L. Water enters at 5 L/min and leaves at 2t L/min for 0 ≤ t ≤ 3. How much water is present at t = 3?

18. A velocity is v(t) = t − 2 m/s on 0 ≤ t ≤ 4. What is the total distance traveled?

19. What is the average value of f(x) = x² on [0, 3]?

20. Which expression gives the volume when the region below y = x and above y = 0, for 0 ≤ x ≤ 2, rotates about the x-axis?

21. Solve dy/dx = 2xy with y(0) = 3.

22. Use one Euler step of width 0.2 for y′ = x + y, starting at (0, 1). What is the approximate value of y(0.2)?

23. Using f(2) = 5 and f′(2) = −3, what does linearization at x = 2 predict for f(2.04)?

24. What is lim as x → 0 of sin(5x)/x, with angles in radians?

0 of 24 answered

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Exam at a glance

AP Calculus AB preparation includes Limits and continuity, Differentiation, Applications of derivatives, Integration and accumulation, Applications of integration, Differential equations. This StudySim bank provides original multiple-choice skill practice. Consult the official framework for the complete exam format, required tasks, and current policies.

Detail	Information
Content framework	College Board: AP Calculus AB course framework

Source: College Board: AP Calculus AB course framework · Last reviewed Oct 27, 2026

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AP Calculus AB content areas

Limits and continuity

Practice limits and continuity with original questions and explanations. Topic labels describe this bank, rather than official section weights.

Differentiation

Practice differentiation with original questions and explanations. Topic labels describe this bank, rather than official section weights.

Applications of derivatives

Practice applications of derivatives with original questions and explanations. Topic labels describe this bank, rather than official section weights.

Integration and accumulation

Practice integration and accumulation with original questions and explanations. Topic labels describe this bank, rather than official section weights.

Applications of integration

Practice applications of integration with original questions and explanations. Topic labels describe this bank, rather than official section weights.

Differential equations

Practice differential equations with original questions and explanations. Topic labels describe this bank, rather than official section weights.

Study tips for AP Calculus AB

Explain your answer

Before opening the explanation, identify the evidence, calculation, or rule that supports your choice. Explain why the other choices do not fit.

Practice a weak topic

Use the topic and difficulty filters in exam prep, then review your missed questions. These difficulty labels are editorial estimates, not calibrated exam statistics.

Use official exam materials too

Pair these skill checks with official practice materials for pacing, extended responses, and any tasks not represented by multiple-choice practice.

A few useful details

AP Calculus AB FAQ

Is this an official AP Calculus AB exam?

No. StudySim writes original practice questions independently. This is not a full-length exam, an official past paper, or a predictor of an official exam score.

Can I practice without signing up?

Yes. These prepared practice questions and their explanations are free to use without an account. They do not require an AI generation request.

How should I use the practice results?

Use the raw number correct and topic breakdown to decide what to review. Difficulty labels are editorial estimates. Results are not converted into an official scaled score.

Does this include every exam task?

No. This bank focuses on multiple-choice skills. Official exams may also require essays, free responses, performances, audio, projects, or other task formats; use official resources for those components.

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