AP Calculus BC past papers and free-response questions
College Board publishes the AP Calculus BC 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. BC has 6 free-response questions, and 3 of them (questions 1, 3 and 4) are shared with the AB paper each year. The others test BC-only topics: parametric and polar curves and series.
On this page
AP Calculus BC 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 BC past exam questions on AP Central. All papers are © College Board.
What each AP Calculus BC 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 AB paper |
|---|---|
| Q2 | Polar curve: area of a region (calculator) |
| Q3 | Differential equation for a cooling pie: tangent line approximation and separation of variables Also on the AB paper |
| Q4 | Graph of f′: derivative of a log composite, extrema and concavity Also on the AB paper |
| Q5 | Area of a region and an integral of f |
| Q6 | Maclaurin series as a geometric series: sums and error |
2025
| Q1 | Spread of an invasive plant modelled with arctan: rates and accumulation (calculator) Also on the AB paper |
|---|---|
| Q2 | Polar curves: dr/dθ and area |
| Q3 | Reading rate from a table: Riemann sums, mean value theorem and average value Also on the AB paper |
| Q4 | Accumulation function from a graph of semicircles and a segment Also on the AB paper |
| Q5 | Differential equation: second derivative and a second-degree Taylor polynomial |
| Q6 | Taylor series about x = 4: ratio test and interval of convergence |
2024
| Q1 | Coffee cooling from a table: average rate of change, Riemann sums and average value Also on the AB paper |
|---|---|
| Q2 | Particle moving in the plane: speed and position from parametric derivatives |
| Q3 | Seawater depth: slope field and separable differential equation Also on the AB paper |
| Q4 | Accumulation function g from the graph of f: fundamental theorem of calculus Also on the AB paper |
| Q5 | Table of f′: an integral-defined function and its derivative |
| Q6 | Maclaurin series: convergence at an endpoint and error bounds |
2023
| Q1 | Rate of flow from a table: interpreting units and Riemann sums Also on the AB paper |
|---|---|
| Q2 | Particle moving along a curve: parametric motion and position |
| Q3 | Warming milk: tangent line approximation and separable differential equation Also on the AB paper |
| Q4 | Graph of f′: relative extrema, increasing intervals and concavity Also on the AB paper |
| Q5 | Area between curves and an improper integral |
| Q6 | Taylor polynomial built from relations between derivatives |
AP Calculus BC 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 BC:
- For series questions, name the test you use and check its conditions. A correct answer without the test named rarely earns full credit.
- Polar area needs the ½ and the correct limits; write the integral before using your calculator.
- Give three decimal places on calculator answers, and show the setup before the number.
- Questions 1, 3 and 4 also appear on the AB paper, so the AB scoring guidelines and sample responses are useful too.
Free AP Calculus BC 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. In the Maclaurin series for ex, what is the coefficient of x³?
Show answer
B. ex = Σ xⁿ/n!, so the x³ coefficient is 1/3! = 1/6.
2. What is the sum of the series Σn=0∞ (1/2)ⁿ?
Show answer
C. A geometric series with first term 1 and ratio 1/2 sums to 1/(1 − 1/2) = 2.
3. A curve is given by x = t², y = t³. What is dy/dx at t = 2?
Show answer
B. dy/dx = (3t²)/(2t) = 3t/2, which is 3 at t = 2.
4. What is the area enclosed by the polar curve r = 2 for 0 ≤ θ ≤ π?
Show answer
B. Area = ½∫0π r² dθ = ½∫0π 4 dθ = 2π.
5. What is ∫1∞ 1/x² dx?
Show answer
B. limb→∞ [−1/x]1b = 0 − (−1) = 1.
6. What is ∫ x ex dx?
Show answer
B. By parts with u = x and dv = ex dx: x ex − ∫ ex dx = (x − 1)ex + C.
7. dy/dx = x + y with y(0) = 1. Using Euler's method with one step of size 0.1, what is the estimate of y(0.1)?
Show answer
C. y(0.1) ≈ y(0) + 0.1·(0 + 1) = 1.1.
8. What is the radius of convergence of Σ xⁿ/3ⁿ?
Show answer
C. It is geometric with ratio x/3, which converges when |x/3| < 1, so |x| < 3.
9. A population satisfies dP/dt = 0.2P(1 − P/500). At what population is it growing fastest?
Show answer
B. Logistic growth is fastest at half the carrying capacity: 500/2 = 250.
10. What is the arc length of y = (2/3)x3/2 from x = 0 to x = 3?
Show answer
B. y′ = √x, so L = ∫03 √(1 + x) dx = (2/3)(43/2 − 1) = (2/3)(7) = 14/3.
Frequently asked questions
Are AP Calculus BC past papers free?
Yes. College Board publishes the AP Calculus BC 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 BC 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 BC 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 BC free-response section scored?
Each free-response question is worth 9 points, so the section is out of 54.