How it works
- Part A: free response: 2 questions, 30 minutes, graphing calculator.
- Part B: free response: 4 questions, 60 minutes, no calculator.
- On the real exam you write your answers by hand in a paper booklet; here you work on paper or type below. Each question is worth 9 points.
- Math parts (✎): type your working one line at a time. When the section ends, every line is marked like an exam: working marks for right lines, the last mark for the right answer.
- Parts that need words, a reason or a drawing: write your answer, then, after the section, open our solution and its scoring guide and tick the points your answer earns.
- At the end: your section II points, and with section I your AP score estimate (1-5).
Your written answers are kept in this browser. No sign-up.
Question 1: Sand on a pile (9 points)
Sand is added to a pile at the rate r(t) = 16 − t2 cubic metres per hour, for 0 ≤ t ≤ 4. At t = 0 the pile holds 20 cubic metres of sand. No sand is taken away.
- (a) How many cubic metres of sand are added to the pile from t = 0 to t = 3? Show the integral you use. (2 points)
- (b) Find r′(2). Using correct units, explain what r′(2) means in this story. (2 points)
- (c) What is the average rate, in cubic metres per hour, at which sand is added over 0 ≤ t ≤ 4? (3 points)
- (d) How many cubic metres of sand are in the pile at t = 4? (2 points)
Question 2: A particle on a line (9 points)
A particle moves along the x-axis. Its velocity is v(t) = t2 − 5t + 4 metres per second, for 0 ≤ t ≤ 5. At t = 0 the particle is at x = 3.
- (a) At what times t is the particle at rest? Show how you find them. (2 points)
- (b) Is the particle speeding up or slowing down at t = 3? Give a reason. (2 points)
- (c) Find the displacement and the total distance the particle travels from t = 0 to t = 5. (3 points)
- (d) Where is the particle at t = 5? (2 points)
Question 3: A cubic function (9 points)
The function f is defined by f(x) = x3 + 3x2 − 9x − 4.
- (a) Find f′(x). (2 points)
- (b) Find the x-values where f has a relative maximum and a relative minimum. Justify your answers. (3 points)
- (c) Find the x-value of the point of inflection of the graph of f. Give a reason. (2 points)
- (d) Find the absolute maximum and the absolute minimum values of f on the closed interval [−4, 2]. (2 points)
Question 4: A differential equation (9 points)
Consider the differential equation dy/dx = 4x3 y.
- (a) Find d²y/dx² in terms of x and y. (2 points)
- (b) A solution curve goes through the point (1, 2). Write an equation of its tangent line at x = 1, and use it to estimate y at x = 1.1. (2 points)
- (c) Find y = f(x), the particular solution of the differential equation with f(0) = 3. (5 points)
Question 5: A region and its solids (9 points)
R is the region between the graph of y = 3x − x2 and the x-axis.
- (a) Find the area of R. (3 points)
- (b) R is turned around the x-axis. Find the volume of the solid made. (3 points)
- (c) R is the base of another solid. Its cross sections at right angles to the x-axis are squares. Write, but do not work out, an integral for its volume. (3 points)
Question 6: An accumulation function (9 points)
The function g is defined by g(x) = ∫2x(t2 − 9) dt.
- (a) Find g′(x). (2 points)
- (b) For 0 < x < 6, at what x-value does g have a relative minimum? Justify your answer. (2 points)
- (c) Find g(5), that is, ∫25(x2 − 9) dx. (2 points)
- (d) Write an equation of the line tangent to the graph of g at x = 5. (3 points)
Steply AP practice exams are our own questions, written in the style of the AP exams for May 2027; they are not official, and the AP score is our own rough estimate. AP® is a trademark registered by the College Board, which is not affiliated with, and does not endorse, this site. Official practice: the College Board's AP Students site and its free-response questions from past years.
