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: Water in a tank (9 points)
Water is pumped into a tank at the rate r(t) = 3t2 + 2t litres per minute, for 0 ≤ t ≤ 3. At the same time water drains out of the tank at a constant rate of 10 litres per minute. At t = 0 the tank holds 50 litres.
- (a) How many litres of water are pumped into the tank from t = 0 to t = 3? Show the integral you use. (2 points)
- (b) Is the amount of water in the tank increasing or decreasing at t = 1? Give a reason. (2 points)
- (c) Write an expression for A(t), the litres of water in the tank at time t. Use it to find A(3). (3 points)
- (d) At what time t, for 0 < t < 3, is the amount of water in the tank least? Justify your answer. (2 points)
Question 2: A particle on a line (9 points)
A particle moves along the x-axis. Its velocity is v(t) = t2 − 4t + 3 feet per second, for 0 ≤ t ≤ 5. At t = 0 the particle is at x = 2.
- (a) At what times in 0 < t < 5 is the particle at rest? Show how you find them. (2 points)
- (b) Is the particle speeding up or slowing down at t = 4? Give a reason. (2 points)
- (c) Find the displacement of the particle from t = 0 to t = 5. Show the integral you use. (3 points)
- (d) Find the position of the particle at t = 5 and the total distance it travels from t = 0 to t = 5. (2 points)
Question 3: Analysing a cubic (9 points)
Let f(x) = x3 − 3x2 − 9x + 5.
- (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) On what interval is the graph of f concave up? Give a reason. (2 points)
- (d) Find the absolute minimum value of f on the closed interval [0, 4]. Justify your answer. (2 points)
Question 4: A differential equation (9 points)
Consider the differential equation dy/dx = 6x2 y.
- (a) Write an equation of the tangent line to the solution curve through the point (1, 2). Use it to estimate y(1.1). (2 points)
- (b) Find d²y/dx² at the point (1, 2). Is the estimate in part (a) too big or too small? Give a reason. (2 points)
- (c) Find the particular solution y = f(x) with f(0) = 2. Show your separation of variables. (5 points)
Question 5: Area and volume (9 points)
Let R be the region between the graph of y = 4x − 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. (3 points)
- (c) The vertical line x = k divides R into two regions of equal area. Write, but do not solve, an equation for k. (3 points)
Question 6: An accumulation function (9 points)
Let g(x) = ∫1x(t2 − 4) dt.
- (a) Find g′(x). (2 points)
- (b) For 0 < x < 5, at what x-value does g have a relative minimum? Justify your answer. (2 points)
- (c) Find g(3). (2 points)
- (d) Find the limit of g(x)/(x − 1) as x approaches 1. (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.
