AP Calc AB past FRQs decoded: a year-by-year walkthrough

The AP (Advanced Placement) Calculus AB free-response section is 6 questions in 90 minutes: 2 calculator-active in the first 30 minutes, 4 no-calculator in the remaining 60. The questions rotate through a fairly predictable set of task types (rate in-rate out, particle motion, area and volume, differential equations, tables and graphs) and AP Central posts the free-response questions and scoring guidelines for roughly the three most recent exams.

This guide walks through several recent past papers (2019, 2022 and 2023) with what each paper tested, which questions students most commonly lost marks on, and how to approach the tougher parts. If you'd rather practice on the papers themselves with worked solutions, Cognito's AP Calc AB past papers group them by year.

Question 3 was a graph question, no calculator: f is continuous on [−6, 5] with a given graph, and the parts asked students to use one given definite integral to find another, evaluate an integral involving f'(x), find the absolute maximum of an accumulation function g, and evaluate a limit.

Where students lost points: on the accumulation-function part, justifying the absolute maximum rather than just naming an x-value.

Question 5 was the area-and-volume question, no calculator, on a region R bounded by two curves. It asked for the area of R, the volume of a solid whose base is R with cross sections of a specified shape, and an integral expression for the volume of the solid formed by rotating R about the line y = 6.

Where students lost points: on the axis of rotation. Rotating about y = 6 rather than the x-axis means both radii are measured from y = 6, and forgetting that shift is a common way to lose the volume points.

Question 4 was a related-rates problem, no calculator: an ice sculpture melts while keeping a conical shape, and the parts asked students to relate the rates at which the height, radius and volume change.

Where students lost points: on differentiating the cone volume formula with respect to time while treating both r and h as functions of t.

Question 5 was the differential equation, no calculator, and the particular-solution part needed a u-substitution before applying the initial condition.

Where students lost points: on the constant of integration — after integrating both sides, students frequently forgot to apply the initial condition to find C before solving for y.

Question 6 was a particle-motion problem with particles P and Q moving on two different axes, asking about velocity, speed and position.

Where students lost points: on 'speeding up versus slowing down'. A particle speeds up when velocity and acceleration have the same sign; students often check only one of the two.

Question 2 was a particle-motion problem, calculator-active: Stephen swims back and forth along a 50-metre pool for 90 seconds with velocity modelled by an exponential-sine function. The parts asked when he changes direction, his acceleration at t = 60 and whether he is speeding up or slowing down, the distance between his positions at t = 20 and t = 80, and the total distance he swims.

Where students lost points: on total distance, integrating velocity instead of the absolute value of velocity. Displacement and total distance are different quantities.

Question 5 was a table question, no calculator, with values and derivatives of two twice-differentiable functions f and g at selected x-values. It asked for the derivative of a composite h(x) = f(g(x)), the concavity of a function k defined by k'(x) = (f(x))²·g(x), the value of an integral-defined function m at x = 2, and whether m is increasing or decreasing there.

Where students lost points: on reading the right row of the table and on justifying concavity from the sign of k''.