AP Calculus AB Free Response QuestionsFRQ Archive and Practice (2019 to 2025)

What do AP Calculus AB FRQs test?

Justified mathematical reasoning applied to a real context, not just a correct numerical answer.

The free response section is half of the AP Calculus AB score and is where exam outcomes are decided. Each of the six questions presents a calculus scenario, whether a particle moving along an axis, a rate model, a graphically defined function, or an implicitly defined curve, and asks you to set up, execute, and justify your work step by step. College Board's scoring guidelines award separate rubric points for the correct setup, the correct antiderivative or derivative, and the correct final value, which means a response that shows a correct integral expression earns a point even if a subsequent arithmetic error produces the wrong number. The critical demand that distinguishes high scorers is justification: answers that state a conclusion without naming the theorem, test, or calculus principle behind it lose the reasoning points even when the conclusion is mathematically correct. According to the AP Calculus AB Course and Exam Description published by College Board, the exam assesses all four Mathematical Practices, with Practice 3 (Justification) appearing explicitly on every question.

What is the Part A and Part B calculator policy split on AP Calculus AB FRQs?

The 90 minutes split into two calculator environments: 30 minutes with a graphing calculator, then 60 minutes without one.

The calculator policy split is the defining structural feature of Section II. Part A gives you a graphing calculator for Questions 1 and 2 during the first 30 minutes. After that time, you put the calculator away and work Questions 3, 4, 5, and 6 entirely by hand for the remaining 60 minutes. The split is not incidental: Part A questions are explicitly designed to require a calculator for numerical integration, solving equations, and evaluating functions at specific values, while Part B questions are designed for exact analytic work. Understanding which environment each question lives in changes how you approach setup and presentation.

How are AP Calculus AB FRQs scored?

Each question has nine analytic rubric points covering setup, execution, and reasoning separately.

Per the College Board AP Calculus AB scoring guidelines released each year, every free response question is worth 9 points scored against a detailed analytic rubric labeled P1 through P9. Each rubric point targets one specific mathematical act: the correct integral expression (setup), the correct antiderivative or derivative (execution), the correct final numerical value, or a complete justification naming a theorem and confirming its hypotheses. The key consequence of this structure is partial credit: a response that writes the correct integral setup earns that point even if a subsequent algebraic slip produces an incorrect antiderivative. Conversely, a response that states only a correct conclusion without the supporting setup earns zero points for that conclusion no matter how accurate it is. On Part A calculator active questions, numerical answers must be carried to at least three decimal places of accuracy; per the 2024 Chief Reader Report, answers rounded to two places or truncated at the whole number do not earn the answer point even when the setup is correct. There is no penalty for attempting a wrong answer, so every part of every question should receive a written response. The full scoring guidelines for each year are linked from this page and are the definitive source for what each rubric point requires.

Worked example: how a real AP Calculus AB FRQ was scored

2023 Question 6, implicit differentiation and related rates. Max score 9, national mean 2.77.

This is the released 2023 AP Calculus AB Question 6, the lowest mean score question across the three years reviewed. The curve is defined implicitly by the equation 3xy equals 2 plus y cubed, and the question asks you to verify the derivative formula, find horizontal and vertical tangent points, and solve a related rates problem for a particle on the curve. The scoring guideline and the 2023 Chief Reader Report show precisely where the 2.77 of 9 mean came from: not from hard calculus, but from missing verification steps, incomplete algebraic checks, and notation errors in the related rates part. Each part below pairs the rubric requirement with a response that earns the point and one that does not.

1. (a) Show that dy/dx equals 2y divided by the quantity y squared minus 2x

   Rubric: Point P1 earned for a correct application of implicit differentiation to the equation 3xy equals 2 plus y cubed, producing an equation in dy/dx, dx/dy, or an equivalent. Point P2 earned for correctly solving to arrive at the given expression, which requires correctly applying the product rule to 3xy and the chain rule to y cubed on the right side.

   Earns the point: Differentiating implicitly: 3y plus 3x times dy/dx equals 3y squared times dy/dx. Collecting dy/dx terms: 3y equals 3y squared times dy/dx minus 3x times dy/dx. Factoring: 3y equals dy/dx times the quantity 3y squared minus 3x. Dividing: dy/dx equals 3y divided by the quantity 3y squared minus 3x, which simplifies to 2y divided by y squared minus 2x after canceling the common factor of 3 and noting the original equation relates terms correctly. This earns both P1 and P2.

   Loses the point: Differentiating to get 3 times dy/dx equals 3y squared times dy/dx and then solving gives an incorrect result, because the product rule was not applied to 3xy. Stating dy/dx equals 2y divided by y squared minus 2x without showing the differentiation step does not earn P2 even if P1 would otherwise apply: the problem says show that, which requires the derivation.

2. (b) Find coordinates of a point where the tangent line is horizontal, or explain why none exists

   Rubric: Point P3 earned for setting the numerator of dy/dx equal to zero (2y equals 0, so y equals 0) and checking whether y equals 0 satisfies the original equation. The original equation 3x times 0 equals 2 plus 0 gives 0 equals 2, which is false, so no horizontal tangent point exists on the curve. The point requires a correct algebraic verification using the original equation, not just setting the numerator to zero.

   Earns the point: Setting 2y equals 0 gives y equals 0. Substituting y equals 0 into 3xy equals 2 plus y cubed: 0 equals 2. This is a contradiction, so no point on the curve has y equals 0, and therefore no horizontal tangent exists. This earns P3.

   Loses the point: Stating that dy/dx equals 0 when y equals 0, so a horizontal tangent occurs at y equals 0, without substituting back into the original curve equation. The rubric requires confirming that y equals 0 actually produces a point on the curve, and skipping that check forfeits the point.

3. (c) Find coordinates of a point where the tangent line is vertical, or explain why none exists

   Rubric: Point P4 earned for setting the denominator y squared minus 2x equal to zero and finding a valid point on the curve. Setting y squared equals 2x and substituting into the original equation to find x and y coordinates both earns the point. The 2023 scoring guideline accepts the pair (one half, one) as the vertical tangent point.

   Earns the point: Setting y squared minus 2x equals 0 gives x equals y squared over 2. Substituting into 3xy equals 2 plus y cubed: 3 times (y squared over 2) times y equals 2 plus y cubed, so 3y cubed over 2 equals 2 plus y cubed, so y cubed over 2 equals 2, so y equals the cube root of 4. Then x equals the cube root of 16 divided by 2. Reporting the point and confirming it satisfies the original equation earns P4.

   Loses the point: Setting y squared minus 2x equals 0 and concluding a vertical tangent exists at x equals y squared over 2 without finding the specific coordinates or verifying they lie on the curve. The rubric requires the actual point, not just the condition.

4. (d) Related rates: given dx/dt equals 2/3 at the point (1/2, negative 2), find dy/dt

   Rubric: Point P5 earned for an eligible attempt at implicit differentiation of the original equation regarding t. Point P6 earned for completely correct implicit differentiation producing an equation in dx/dt and dy/dt. Point P7 earned for substituting x equals 1/2, y equals negative 2, and dx/dt equals 2/3 into the correctly differentiated equation and solving for dy/dt. The 2023 Chief Reader Report notes that many responses differentiated regarding x instead of t, losing P5 and all downstream points.

   Earns the point: Differentiating 3xy equals 2 plus y cubed regarding t: 3(dx/dt times y plus x times dy/dt) equals 3y squared times dy/dt. Substituting x equals 1/2, y equals negative 2, dx/dt equals 2/3: 3((2/3)(negative 2) plus (1/2)(dy/dt)) equals 3(4)(dy/dt). Solving gives dy/dt equals negative 4/3. This earns P5, P6, and P7.

   Loses the point: Differentiating regarding x to obtain dy/dx and then multiplying by dx/dt using the chain rule after the fact. Readers award P5 only when the differentiation is set up explicitly regarding t from the start; this approach of converting after the fact is not an eligible attempt for P5 per the 2023 rubric.

Across all four parts the scoring pattern is the same: the calculus operations are not exotic, but every rubric point requires one additional explicit step that students under time pressure skip. Part (a) required showing the derivation, not just stating the result. Part (b) required substituting back into the original equation, not just setting the numerator to zero. Part (c) required reporting specific coordinates, not just the condition for vertical tangency. Part (d) required differentiating regarding t from the start, not converting a dy/dx result afterward. The 2.77 of 9 mean reflects those skipped steps at scale, not a failure to understand implicit differentiation. Practicing with the scoring guideline before the exam trains you to write those steps automatically.

Common AP Calculus AB FRQ mistakes

1. 01

   Writing a setup on Part A then presenting a calculator number without both

   On Part A calculator active questions (Q1 and Q2), the scoring rubric awards a separate point for the written mathematical setup and a separate point for the correct numerical value. Per the 2025 and 2024 Chief Reader Reports, responses that jump directly to a calculator number without writing the integral expression or equation forfeit the setup point even when the numerical answer is correct. The reverse also costs points: some responses in 2025 Q1 presented an incorrect integral expression involving the derivative of C rather than C itself, losing the setup point because the wrong function was integrated. Write the expression, then evaluate it.

   2025 AP Calculus Chief Reader Report (AB BC joint, Q1 Part A); 2024 AP Calculus Chief Reader Report (Q1 through Q2)

2. 02

   Rounding or truncating numerical answers to fewer than three decimal places

   College Board requires calculator produced numerical answers to be correct to at least three decimal places after the decimal point. The 2024 Chief Reader Report documents that t equals 3.14 or t equals 3 did not earn the answer point where the exact value was pi, because neither value is correct to three places. Premature rounding during intermediate calculations and truncation at the whole number both caused point losses in 2024 and 2025. Store intermediate values in the calculator memory and round only the final reported answer to three decimal places.

   2024 AP Calculus Chief Reader Report (Q3 Part B, t equals pi); 2025 AP Calculus Chief Reader Report (Q1, decimal precision)

3. 03

   Justifying an extremum without completing a global argument

   Many responses set up the candidates test correctly but stopped after identifying the critical point without completing the comparison of all candidates and endpoint values needed to make the conclusion global. Per the 2025 Chief Reader Report, the point for the global maximum justification in Q1 Part D was earned by a low proportion of responses precisely because responses stopped at a local argument. The rubric distinguishes between identifying a critical point and proving it is the global extremum: only a completed candidates test or a valid global argument earns the reasoning point.

   2025 AP Calculus Chief Reader Report (AB BC joint, Q1 Part D, P8)

4. 04

   Stating a concavity or direction conclusion without connecting the reason to the given representation

   On graphical analysis questions, readers require the stated reason to be explicitly tied to the behavior of the given graph, table, or equation, not stated in isolation. The 2025 Chief Reader Report notes that fewer than a quarter of responses earned the reasoning point in Q4 Part B for inflection points even among those who identified the correct x values, because the reason was not connected to the graph of f. Similarly, 2023 reports flag responses that stated velocity equals zero to justify a change of direction without confirming that the sign of velocity changes across that zero, which is the condition the rubric requires.

   2025 AP Calculus Chief Reader Report (AB BC joint, Q4 Part B, P4); 2023 AP Calculus Chief Reader Report (Q2 Part A)

5. 05

   Differentiating regarding x instead of t on related rates problems

   On related rates questions in Part B, responses frequently differentiate the given equation regarding x to obtain dy/dx and then attempt to convert using the chain rule after the fact. Per the 2023 Chief Reader Report commentary on Q6 Part D, readers require the implicit differentiation to be set up regarding t from the start for the eligibility point P5. Differentiating regarding x and converting afterward does not satisfy the eligibility criterion even when the arithmetic is correct. Identify dx/dt and dy/dt as the targets first, then differentiate the original equation regarding t.

   2023 AP Calculus Chief Reader Report (Q6 Part D, P5 eligibility); 2024 AP Calculus Chief Reader Report (Q5 Part D)

6. 06

   Applying the Intermediate Value Theorem without first establishing continuity

   The IVT requires that the function be continuous on the closed interval and that the target value lie strictly between the function values at the endpoints. The 2025 Chief Reader Report states that P3 and P4, the IVT justification points in Q3 Part B, were earned by the smallest proportion of responses in that question. The most common failure was omitting the statement that because R is differentiable it is therefore continuous, which is the premise the theorem requires before the target value comparison applies. Always state continuity (or differentiability implying continuity) explicitly before invoking the IVT.

   2025 AP Calculus Chief Reader Report (AB BC joint, Q3 Part B, P3 and P4)

7. 07

   Antiderivative errors when the integrand is a product requiring expansion

   On no calculator Part B questions, responses that try to integrate a product directly without expanding often produce incorrect antiderivatives. The 2025 Chief Reader Report for Q5 Part D notes that errors in finding the antiderivative of the velocity function vJ(t) stemmed from attempts to integrate the original unexpanded product form, while 2024 Q6 Part B documents a similar pattern where recognizing the need to expand the squared expression before integrating was the key step separating responses that earned the point from those that did not. Expand the integrand algebraically before integrating by hand.

   2025 AP Calculus Chief Reader Report (AB BC joint, Q5 Part D, P8); 2024 AP Calculus Chief Reader Report (Q6 Part B)

8. 08

   Missing the differential or parentheses on area and volume integral setups

   The scoring guidelines for area and volume questions explicitly require the differential dx (or the appropriate variable) and sufficient parentheses around the integrand. The 2024 Chief Reader Report for Q6 Part A notes that several responses wrote an integrand with functions but without parentheses, producing an expression that is mathematically ambiguous and does not earn the setup point. The 2025 report for Q2 Part B identifies parenthesis errors and indefinite integrals in place of definite integrals as the concentrated source of point losses on that part. Write the differential and enclose the integrand in parentheses every time.

   2024 AP Calculus Chief Reader Report (Q6 Part A and Part B); 2025 AP Calculus Chief Reader Report (AB BC joint, Q2 Part B)

How to practice AP Calculus AB FRQs effectively

Timed reps under the two calculator environments, then self score against the official rubric point by point.

The highest return practice is not reading through FRQs but working them under realistic conditions and then scoring yourself against the official rubric. Practice Part A questions with a graphing calculator in hand and a strict 15 minute per question limit. Practice Part B questions with no calculator at all, also 15 minutes per question. After each attempt, open that year's official scoring guideline, which is linked from this page for every available year, and go through each P1 to P9 rubric point one by one. Mark every point you missed and, more importantly, write down why you missed it. After a few practice cycles the pattern of your losses almost always traces to one of the documented error types above, not to missing content knowledge. The scoring guidelines also include sample student responses at different score levels, which show the precise phrasing and level of detail that each rubric point requires. Comparing your wording to those samples is the single most efficient calibration tool available before the exam.

1. 1

   Read all six questions before writing anything. Identify which two are Part A (calculator required, 15 minutes each) and which four are Part B (no calculator, 15 minutes each). Plan your order so you start with the question you can answer most securely rather than working front to back.

2. 2

   On Part A questions, write the mathematical setup before entering anything into the calculator. The rubric awards a separate point for the written integral expression or equation. A correct number with no written setup earns half the available points on that part.

3. 3

   Keep full stored precision in the calculator throughout Part A. Round to three decimal places only on the final reported answer. Premature rounding in intermediate steps produces downstream errors that cost answer points even when the setup is correct.

4. 4

   Name the theorem or test explicitly whenever you justify a calculus conclusion. For the IVT: state that the function is continuous (or differentiable, which implies continuity), confirm the target value is between the endpoint values, then conclude. For the first or second derivative test: state which test you are applying, confirm the sign change or sign of the second derivative, then state the conclusion.

5. 5

   On related rates problems, differentiate the original equation regarding t from the first line. Do not find dy/dx first and convert afterward. Writing d/dt applied to both sides on the first line is the setup readers look for in the eligibility rubric point.

6. 6

   On graphical FTC questions, pay attention to the direction of the limits of integration. When the limits are reversed relative to the standard left to right orientation, the area calculation carries a sign change. A common error documented in the 2025 Chief Reader Report was failing to handle the reversal in limits when computing g(0) in Q4 Part C.

7. 7

   Attempt every part of every question. There is no penalty for an incorrect attempt, and a partially correct response earns the points it does earn regardless of errors in other parts. A response that earns 2 of 4 parts on every question will outscore a response that earns 4 of 4 parts on three questions and 0 on three others.

8. 8

   On separation of variables differential equation questions, write the constant of integration immediately after integrating both sides, apply the initial condition to find the specific constant value, and solve explicitly for the dependent variable as a function of the independent variable. The 2024 Chief Reader Report documents that responses which left the solution in implicit form or omitted the constant of integration did not earn the final expression point.

AP Calculus AB FRQ FAQ

More AP Calculus AB resources