AP Calculus AB past papers and free-response questions
College Board publishes the AP Calculus AB free-response questions every year, with scoring guidelines, sample student responses and a chief reader report. This page collects every year it currently makes public, 2023 to 2026, to preview, open or download. The exam has 6 free-response questions: 2 with a graphing calculator in 30 minutes, then 4 without a calculator in 60 minutes. Each question is worth 9 points.
On this page
AP Calculus AB free-response questions by year
Each file opens from College Board's official site. Preview shows it here, Open in new tab gives you the full screen, and Download saves a copy to print.
2026
- Free-response questionsPreviewOpen in new tabDownload
- Scoring guidelinesPreviewOpen in new tabDownload
- Sample responses and chief reader reportCollege Board usually adds these in the autumn after the exam
2025
- Free-response questionsPreviewOpen in new tabDownload
- Scoring guidelinesPreviewOpen in new tabDownload
- Chief reader reportPreviewOpen in new tabDownload
- Scoring statisticsPreviewOpen in new tabDownload
- Score distributionsPreviewOpen in new tabDownload
- Sample student responsesQ1Q2Q3Q4Q5Q6
2024
- Free-response questionsPreviewOpen in new tabDownload
- Scoring guidelinesPreviewOpen in new tabDownload
- Chief reader reportPreviewOpen in new tabDownload
- Scoring statisticsPreviewOpen in new tabDownload
- Score distributionsPreviewOpen in new tabDownload
- Sample student responsesQ1Q2Q3Q4Q5Q6
2023
- Free-response questionsPreviewOpen in new tabDownload
- Scoring guidelinesPreviewOpen in new tabDownload
- Chief reader reportPreviewOpen in new tabDownload
- Scoring statisticsPreviewOpen in new tabDownload
- Score distributionsPreviewOpen in new tabDownload
- Sample student responsesQ1Q2Q3Q4Q5Q6
Source: AP Calculus AB past exam questions on AP Central. All papers are © College Board.
What each AP Calculus AB free-response question tests
Use this index to find past questions on the topics you need to practise. The descriptions are our own summaries.
2026
| Q1 | Rate of arrival from a table: Riemann sums and accumulation Also on the BC paper |
|---|---|
| Q2 | Area between an exponential and a rational curve, including integrating with respect to y |
| Q3 | Differential equation for a cooling pie: tangent line approximation and separation of variables Also on the BC paper |
| Q4 | Graph of f′: derivative of a log composite, extrema and concavity Also on the BC paper |
| Q5 | Toy car with piecewise velocity: displacement, total distance and acceleration |
| Q6 | Table of values: limits and the chain rule for a composite f(f(x)) |
2025
| Q1 | Spread of an invasive plant modelled with arctan: rates and accumulation (calculator) Also on the BC paper |
|---|---|
| Q2 | Area between curves and a solid with known cross-sections |
| Q3 | Reading rate from a table: Riemann sums, mean value theorem and average value Also on the BC paper |
| Q4 | Accumulation function from a graph of semicircles and a segment Also on the BC paper |
| Q5 | Two particles on the x-axis: velocity and direction of motion |
| Q6 | Implicit differentiation: tangent line approximation and tangent lines |
2024
| Q1 | Coffee cooling from a table: average rate of change, Riemann sums and average value Also on the BC paper |
|---|---|
| Q2 | Particle motion with a log velocity: at rest, direction and position |
| Q3 | Seawater depth: slope field and separable differential equation Also on the BC paper |
| Q4 | Accumulation function g from the graph of f: fundamental theorem of calculus Also on the BC paper |
| Q5 | Implicit curve: tangent line approximation and horizontal tangents |
| Q6 | Area between two curves and a volume |
2023
| Q1 | Rate of flow from a table: interpreting units and Riemann sums Also on the BC paper |
|---|---|
| Q2 | Swimmer's velocity: speed, direction and distance travelled |
| Q3 | Warming milk: tangent line approximation and separable differential equation Also on the BC paper |
| Q4 | Graph of f′: relative extrema, increasing intervals and concavity Also on the BC paper |
| Q5 | Table of f and g: chain rule and derivatives of composites |
| Q6 | Implicit differentiation: horizontal and vertical tangents |
AP Calculus AB exam format
| Section | Questions | Time | Score |
|---|---|---|---|
| Section I: multiple choice | 42 questions | 1 hour 40 minutes | 50% |
| Part A | 29 questions, no calculator | 62 minutes | |
| Part B | 13 questions, graphing calculator required | 38 minutes | |
| Section II: free response | 6 questions | 1 hour 30 minutes | 50% |
| Part A | 2 questions, graphing calculator required | 30 minutes | |
| Part B | 4 questions, no calculator | 60 minutes |
This is the format from the May 2027 exam onwards. The exam is hybrid digital: multiple-choice answers go into the Bluebook app, and free-response answers are handwritten in a paper booklet. Before 2027 the multiple-choice section had 45 questions in 1 hour 45 minutes. The free-response section has not changed, so every past FRQ on this page is exactly the kind of question you will meet.
How to use past free-response questions
- Start with the newest papers. They match the current exam most closely.
- Do one question at a time, timed. Use the timings in the exam format above, with no notes open.
- Mark yourself strictly with the scoring guidelines, point by point, before reading anything else.
- Compare with the sample responses to see exactly what earned or lost each point.
- Read the chief reader report for the mistakes most students made that year, and keep a list of your own.
- Go back to weak topics using the topic index, then redo the question a week later.
Tips specific to AP Calculus AB:
- Write every answer to three decimal places on calculator questions, and show the setup (the integral or equation) before the number. A bare number loses the setup point.
- Justify with the right language: for an extremum, name the sign change of f′; for the mean value theorem, state that the function is continuous and differentiable.
- Include units when the question gives a context. Interpretation parts lose the point without them.
- Read the chief reader report after marking yourself: it lists the exact mistakes that cost students points that year.
Free AP Calculus AB practice questions
Ten original multiple-choice questions written by our tutors in the style of the exam. Choose an answer to see if you are right and why.
Answered 0 of 10 · 0 correct
1. What is limx→3 (x² − 9)/(x − 3)?
Show answer
C. Factor: (x − 3)(x + 3)/(x − 3) = x + 3 for x ≠ 3, so the limit is 3 + 3 = 6.
2. If y = sin(3x²), what is dy/dx?
Show answer
B. Chain rule: the derivative of sin(u) is cos(u)·u′, and u′ = 6x.
3. f(x) = x³ − 6x² + 9x + 1. At which x does f have a relative maximum?
Show answer
A. f′(x) = 3(x − 1)(x − 3). f′ changes from positive to negative at x = 1, so there is a relative maximum there.
4. What is ∫02 (3x² + 1) dx?
Show answer
C. The antiderivative is x³ + x. Evaluated from 0 to 2: 8 + 2 = 10.
5. A particle has velocity v(t) = t² − 4t + 3. At what times is it at rest?
Show answer
C. v(t) = (t − 1)(t − 3) = 0 when t = 1 or t = 3.
6. Using the tangent line to y = √x at x = 4, what is the approximate value of √4.2?
Show answer
B. f(4) = 2 and f′(4) = 1/(2√4) = 1/4, so √4.2 ≈ 2 + (1/4)(0.2) = 2.05.
7. What is d/dx ∫1x² cos t dt?
Show answer
B. By the fundamental theorem of calculus with the chain rule: cos(x²)·2x.
8. For the circle x² + y² = 25, what is dy/dx at the point (3, 4)?
Show answer
B. 2x + 2y·dy/dx = 0, so dy/dx = −x/y = −3/4.
9. What is the average value of f(x) = x² on the interval [0, 3]?
Show answer
B. (1/3)∫03 x² dx = (1/3)(9) = 3.
10. What is ∫ 2x ex² dx?
Show answer
A. Substitute u = x², du = 2x dx: ∫ eu du = ex² + C.
Frequently asked questions
Are AP Calculus AB past papers free?
Yes. College Board publishes the AP Calculus AB free-response questions, scoring guidelines and sample responses free on AP Central. This page links to those official files so you can preview, open or download them.
Why are there no older AP Calculus AB papers before 2023?
College Board currently makes the most recent years public on AP Central and removed older years from public access. Teachers can find many older released questions in AP Classroom.
Can I get full AP Calculus AB past exams with the multiple-choice section?
College Board does not publish full past multiple-choice sections publicly; complete practice exams are released to teachers through AP Classroom. The free-response questions, which are half of your score, are public, and this page has 10 original multiple-choice questions to practise with.
How is the AP Calculus AB free-response section scored?
Each free-response question is worth 9 points, so the section is out of 54.