AP Calculus free-response questions: how to earn every point

The free-response section, AB and BC

Source: AP Central exam pages for Calculus AB and BC, checked 7 October 2026. The section is 50% of the exam score. Multiple choice is answered in the Bluebook app and free-response answers are handwritten.

What the questions test

Free-response questions come in a few recurring shapes: a rate problem in context (water, people, cars), a function given as a table of values, a function given as a graph of its derivative, a particle moving along a line, a differential equation with a slope field or separation of variables, and an area or volume question. BC papers add parametric, polar or vector motion and a Taylor series question.

Each question is split into parts (a), (b), (c) and often (d), and points are awarded per part. That matters for strategy: a student who cannot do part (a) can still earn every point in parts (b) to (d), because later parts are often independent or can use your earlier answer.

Two habits separate full-credit answers from partial ones. First, write the mathematical setup before the number: an integral with limits, or a derivative expression, then the value. Second, justify with a reason that names the function: 'g has a relative minimum at x = 3 because g'(x) = f(x) changes from negative to positive there'. Saying 'the graph goes down then up' does not earn the point.

Worked example 1 (calculator, Part A style): a tank of water

Water flows into a tank at R(t) = 12 + 4 sin(t/3) liters per minute and is pumped out at a constant 13 liters per minute, for 0 ≤ t ≤ 30 minutes. At t = 0 the tank holds 200 liters. (a) How much water flows in over the 30 minutes? (b) Is the amount of water increasing or decreasing at t = 20? Justify. (c) How much water is in the tank at t = 30? (d) At what time is the amount of water least? Justify.

1. (a) Total inflow = integral from 0 to 30 of R(t) dt = 382.069 liters (calculator). Write the integral, then the value, then the units.
2. (b) The rate of change of the amount is A'(t) = R(t) - 13. A'(20) = 12 + 4 sin(20/3) - 13 = 0.497 > 0, so the amount is increasing at t = 20 because the inflow rate exceeds the outflow rate.
3. (c) A(30) = 200 + integral from 0 to 30 of (R(t) - 13) dt = 200 + 382.069 - 390 = 192.069 liters.
4. (d) Candidates test. A'(t) = 0 when sin(t/3) = 0.25, giving t = 0.758, 8.667, 19.608 and 27.516 in the interval. Evaluate A at these and at the endpoints: A(0) = 200, A(0.758) = 199.623, A(8.667) = 214.952, A(19.608) = 180.773, A(27.516) = 196.103, A(30) = 192.069. The least amount is about 180.773 liters at t = 19.608 minutes.
5. Note on rounding: keep full calculator precision until the final answer and report three decimal places, which is the AP convention.

Worked example 2 (no calculator, Part B style): extrema and justification

Let f(x) = x^3 - 6x^2 + 9x + 1 on the closed interval [0, 4]. (a) Find the x-coordinates of the relative extrema and classify each. Justify. (b) Find the absolute maximum and minimum values. (c) Where is the graph concave up?

1. (a) f'(x) = 3x^2 - 12x + 9 = 3(x - 1)(x - 3), so f'(x) = 0 at x = 1 and x = 3. f' changes from positive to negative at x = 1, so f has a relative maximum there; f' changes from negative to positive at x = 3, so f has a relative minimum there.
2. (b) Evaluate at critical points and endpoints: f(0) = 1, f(1) = 1 - 6 + 9 + 1 = 5, f(3) = 27 - 54 + 27 + 1 = 1, f(4) = 64 - 96 + 36 + 1 = 5. The absolute maximum value is 5 (at x = 1 and x = 4) and the absolute minimum value is 1 (at x = 0 and x = 3).
3. (c) f''(x) = 6x - 12, which is positive for x > 2, so the graph is concave up on (2, 4). It changes concavity at x = 2, which is a point of inflection.
4. What earns the points: the derivative, the sign-change reason in words, and a table of values for the absolute extrema. A sign chart alone, without a sentence, may not be accepted as justification.

Common ways students lose points

• Giving a number without the integral or derivative that produced it.
• Missing or wrong units, especially in rate questions (liters per minute versus liters).
• Justifying with the graph of f when the question gives the graph of f', or saying 'it goes up' instead of naming the function whose sign changes.
• Rounding intermediate calculator values, which shifts the final answer outside the accepted range.
• Forgetting endpoints in an absolute extremum question.
• Leaving parts blank. Later parts often stand alone, so attempt every part.

How one-to-one lessons help with Calculus FRQs

Free-response technique is hard to learn from answer keys, because the key shows the final answer and not why a near-miss lost its point. In lessons the tutor sets an FRQ under time, then marks it against a rubric-style point list on the shared whiteboard and shows exactly which sentence would have earned the missing point. Over a few weeks students build a bank of justification phrases and a routine for the calculator part (store values, do not round, write the setup).

College Board publishes past free-response questions with scoring guidelines on AP Central, and a tutor can use them to practice in the real format. Good preparation starts in the spring semester, once the integration units have been taught, with timed full sections in the final weeks.

Self-check

• Can you write a one-sentence justification for a relative minimum using the sign of the derivative?
• Do you know when to use the candidates test instead of the first derivative test?
• Given a rate in gallons per hour, can you say the units of its integral and of its derivative?
• Can you evaluate a definite integral on your calculator and report three decimal places?
• BC only: can you write the first four nonzero terms of a Maclaurin series and bound the error with the alternating series error bound?