Free AP Calculus AB practice questions

Calculus AB rewards a handful of moves done cleanly — differentiate, take a limit, integrate — and the AP exam keeps testing them. Try the AP-style samples below, see each one fully explained (including why every wrong choice is wrong), then keep practicing by unit. No credit card.

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AP Calculus AB practice: Show the actual product action implied by this brief, using one dominant interface or physical workflow rather than several students watching a screen

What the exam looks like

What the AP Calculus AB exam looks like

So you know what these samples are preparing you for. AP Calculus AB is a college-level introductory calculus course built on eight units — limits and continuity, differentiation, applications of the derivative, integration and accumulation of change, differential equations, and applications of integration.

Section I · Multiple choice45 questions, in two parts — one allows a graphing calculator and one doesn't. Discrete items on limits, derivatives, integrals and analysis of functions. Worth half your score.

Section II · Free response6 multi-step questions, again split into calculator and no-calculator parts. They reward correct setup, full work and clear justification. The other half of your score.

Timing3 hours 15 minutes total across the two sections, so pacing on both the MCQ and the free-response matters.

ScoringOne score from 1 to 5; the two sections are weighted equally. A 3 is the common "qualified" line and a 4 or 5 is strong — but each college sets its own credit policy, so confirm yours.

Format and scoring can change year to year — always confirm current details on College Board — AP Calculus AB. The questions below are AP-style practice questions, not real or official AP exam questions.

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AP-style practice questions

Three to warm up on. Pick an answer, submit, and read the full explanation — the correct choice, why the main traps fail, and the skill it tests.

Question 1 of 3

Differentiation · Power ruleEasy

What is the derivative of f(x) = x³ − 4x evaluated at x = 2?

Explanation — Correct: A

Differentiate term by term with the power rule: the derivative of x³ is 3x², and the derivative of −4x is −4. So f′(x) = 3x² − 4. Substitute x = 2: 3(2²) − 4 = 3(4) − 4 = 12 − 4 = 8.

Why C is wrong: 12 drops the −4 term entirely and reports only 3x² at x = 2. The −4x term still has a derivative (−4) — you can't ignore it.

Why B is wrong: 4 comes from mishandling the constant — for example computing 3·2 − 2 or otherwise losing the squaring in 3x². Apply the exponent first (3x²), then subtract 4.

Skill: differentiate a polynomial with the power rule term by term, then evaluate at the point. Don't drop the derivative of the linear term.

Question 2 of 3

Limits & ContinuityMedium

Evaluate lim_(x→3) (x² − 9)/(x − 3).

Explanation — Correct: B

Direct substitution gives 0/0, which is indeterminate — a signal to simplify, not to stop. Factor the numerator: x² − 9 = (x − 3)(x + 3). The (x − 3) cancels, leaving x + 3. Now substitute x = 3: 3 + 3 = 6.

Why C is wrong: "Does not exist" treats the zero in the denominator as a dead end. This is a removable discontinuity — the factor cancels, so the limit exists even though the function is undefined exactly at x = 3.

Why A is wrong: 0 assumes that 0/0 simply equals 0. The 0/0 form is indeterminate — it tells you nothing until you simplify.

Skill: when substitution yields 0/0, factor and cancel before re-substituting. Recognize a removable discontinuity instead of declaring the limit gone.

Question 3 of 3

Integration · Definite integralEasy

Evaluate the definite integral ∫₀² 2x dx.

Explanation — Correct: C

Find an antiderivative of 2x: that's x² (since d/dx of x² is 2x). By the Fundamental Theorem, evaluate x² from 0 to 2: (2²) − (0²) = 4 − 0 = 4.

Why A is wrong: 2 comes from integrating to x instead of x² — that is, treating the antiderivative of 2x as if the 2 just stayed. The antiderivative of 2x is x², not 2x or x.

Why B is wrong: 8 uses 2² · 2 — squaring the bound and then multiplying by the coefficient again, double-counting the 2. Once you antidifferentiate to x², the coefficient is already built in.

Skill: antidifferentiate first, then plug the upper bound minus the lower bound. Confirm your antiderivative by differentiating it back.

Applications of derivativesMedium

A particle moves so that its position is s(t) = t³ − 6t² + 9t. At what time(s) is the particle at rest…

At = 1 only

Bt = 1 and t = 3

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Free questions that actually teach

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A full explanation on every answer

Not just an answer key — a four-part breakdown of why the right answer is right, why each wrong choice fails, the skill it tests, and the trap to avoid. You learn the reasoning, not just the letter.

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Built to match the AP exam

Written to the format and difficulty of AP Calculus AB and aligned to the College Board's course framework, with calculator and no-calculator style items across all eight units.

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Practice unit by unit

Work one unit at a time, then mix units the way the exam does, and let the mistake bank resurface what you miss so weak spots actually close.

Practice it, unit by unit.

Reading about calculus isn't the same as answering AP-style questions and learning from every miss. Start free and practice by unit.

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FAQ

Free AP Calculus AB practice, answered

Are these AP Calculus AB practice questions free?

Yes. The sample questions on this page are free to try with full explanations, and no credit card is required. A free account adds daily AP Calculus AB practice by unit; a subscription removes the daily limit at $9.90/month, $19.90 for three months, or $39.90 for lifetime access.

Are these like the real AP Calculus AB exam?

They are written to match the format and difficulty of the AP Calculus AB exam and are aligned to the College Board's course framework across limits, derivatives, integrals and the Fundamental Theorem. They are original, AP-style practice questions, not real, official or past AP exam questions.

Do I need to sign up to practice AP Calculus AB questions?

No sign-up is needed to try the samples here. To keep practicing across the full question bank, unit by unit with a full explanation on every question, you create a free account. Daily free to start.

What topics do the AP Calculus AB practice questions cover?

The full set spans all eight units of AP Calculus AB — limits and continuity, differentiation, applications of derivatives, integration and accumulation of change, differential equations, and applications of integration — with both calculator and no-calculator style items.

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