AP Calculus BC Practice Exam 1: Free Response

Section II · 6 questions · 90 minutes · math parts marked line by line · scoring guide for the rest

▶ Section I: multiple choice 🖨 Print the exam Answers and scoring guide

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: A particle in the plane (9 points)

A particle moves in the xy-plane. Its position at time t is x(t) = t² + 1 and y(t) = t³ − 3t, for 0 ≤ t ≤ 2.

(a) Find dy/dx at t = 2. (2 points)
(b) Find the speed of the particle at t = 2. (2 points)
(c) Find the total distance the particle travels from t = 0 to t = 2. (3 points)
(d) At what point (x, y) is the particle moving straight up or straight down? Give a reason. (2 points)

Question 2: A polar curve (9 points)

The curve C is given by the polar equation r = 2 + 2cos θ, for 0 ≤ θ ≤ 2π.

(a) Find the area of the region inside C. (3 points)
(b) Find dr/dθ at θ = π/2. What does it say about the point on C as θ grows past π/2? (2 points)
(c) Find the slope dy/dx of C at θ = π/2. (2 points)
(d) Find the area of the region inside C and outside the circle r = 2. (2 points)

Question 3: A Taylor polynomial (9 points)

Let f(x) = e^−x.

(a) Find the Maclaurin polynomial of degree 3 for f. (3 points)
(b) Use your polynomial to estimate f(0.1). (2 points)
(c) The Maclaurin series of f at x = 0.1 is alternating, with terms that shrink to 0. Show that the estimate in (b) is within 0.00001 of f(0.1). (2 points)
(d) Write the first four nonzero terms of the Maclaurin series of x e^−x. (2 points)

Question 4: Euler's method (9 points)

Let y = f(x) be the solution of dy/dx = x + y with f(0) = 1.

(a) Use Euler's method, starting at x = 0 with two steps of equal size, to estimate f(1). Show the work that leads to your answer. (3 points)
(b) Find d²y/dx² in terms of x and y. Is the solution curve concave up or concave down at (0, 1)? (2 points)
(c) Show that y = 2e^x − x − 1 is the solution with f(0) = 1. (2 points)
(d) Find f(1) exactly. Is the Euler estimate too big or too small? Give a reason. (2 points)

Question 5: Integration by parts and an improper integral (9 points)

This question has three integrals.

(a) Find ∫ x cos x dx. (3 points)
(b) Find ∫₀^∞e^−3x dx, or show that it diverges. (3 points)
(c) Does ∫₁^∞1x dx converge or diverge? Give a reason. (3 points)

Question 6: A power series (9 points)

Consider the power series Σ from n = 1 to ∞ of (x − 1)ⁿ/(n × 2ⁿ).

(a) Find the interval of convergence. Check the endpoints. (4 points)
(b) At x = −1 the series converges. Does it converge absolutely or conditionally? Give a reason. (3 points)
(c) At x = 2, find the sum of the first three terms of the series. (2 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.