AP Calculus BC Practice Exam 1
Section I: 42 multiple-choice questions · 100 minutes · then section II, free response · AP score estimate · every answer solved step by step
✎ Section II: free response 🖨 Print the exam Answers and scoring guide
How it works
- Part A: multiple choice: 29 questions, 62 minutes, no calculator.
- Part B: multiple choice: 13 questions, 38 minutes, graphing calculator.
- Each question has 4 choices. A wrong answer costs nothing, so answer every question. One question at a time: Back, Next, the question list, Mark for review.
- A timer counts down for each part (you can hide it; it shows again in the last 5 minutes). In part A the calculator is off; in part B it is on. The real test has Desmos built into the Bluebook app in part B; here the calculator button gives a scientific calculator.
- At the end: right and wrong for every question, your answer, the right one, why a wrong choice is wrong, and its step-by-step solution. Then go on to section II, free response: with both sections you get an AP score estimate (1-5).
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Part A (no calculator)
1 Question 1
What is the limit of x2 + x − 6x + 3 as x approaches −3?- A) 0
- B) 5
- C) 1
- D) −5
2 Question 2
What is the limit of ln xx − 1 as x approaches 1?- A) 0
- B) 1
- C) −1
- D) infinity
3 Question 3
If f(x) = 3x, what is f′(x)?- A) −3x2
- B) −3x
- C) 3ln x
- D) 3x2
4 Question 4
If f(x) = e−x, what is f′(x)?- A) e−x
- B) −xe−x − 1
- C) −e−x
- D) −ex
5 Question 5
If f(x) = 2tan−1(x), what is f′(x)?- A) 2/(1 + x2)
- B) 2/√(1 − x2)
- C) 1/(1 + x2)
- D) −2/(1 + x2)
6 Question 6
If y = x3x for x > 0, what is dy/dx?- A) 3x × x3x − 1
- B) x3x(3ln x + 3)
- C) x3xln x
- D) 3x3xln x
7 Question 7
For y = x2 + 3x, what is the differential dy when x = 1 and dx = 0.01?- A) 0.0501
- B) 5
- C) 0.05
- D) 0.02
8 Question 8
The area of a circle grows at 20 square cm per second. How fast is the radius growing, in cm per second, when the radius is 5 cm?- A) 10/π
- B) 4/π
- C) 200π
- D) 2/π
9 Question 9
For f(x) = x2 + 2x on [−2, 0], what value of c does Rolle's Theorem give?- A) 1
- B) 0
- C) −1
- D) −2
10 Question 10
At what point does the graph of f(x) = x3 − 6x2 + 4 have a point of inflection?- A) (2, −12)
- B) (4, −28)
- C) (−2, −28)
- D) (0, 4)
11 Question 11
At what point does f(x) = x2 + 2x − 3 have a relative minimum?- A) (1, 0)
- B) (−1, −4)
- C) (−1, −1)
- D) (−2, −3)
12 Question 12
Which is ∫ x ex dx?- A) xex + ex + C
- B) ex + C
- C) x2ex/2 + C
- D) xex − ex + C
13 Question 13
What is the value of ∫1∞1x2 dx?- A) infinity
- B) 1
- C) 0
- D) −1
14 Question 14
What is the value of ∫0∞e−2x dx?- A) 12
- B) −12
- C) infinity
- D) 2
15 Question 15
Which is ∫ e5x dx?- A) 5e5x + C
- B) e6x6 + C
- C) e5x5 + C
- D) e5x + C
16 Question 16
Which is ∫ 3x2(x3 + 4)5 dx?- A) x3(x3 + 4)6/6 + C
- B) (x3 + 4)6 + C
- C) 5(x3 + 4)4 + C
- D) (x3 + 4)6/6 + C
17 Question 17
Which function y solves dy/dx = 4y with y(0) = −2?- A) −2ex
- B) e4x − 3
- C) −2e4x
- D) 2e4x
18 Question 18
Euler's method with step 0.5 is used on dy/dx = −y, starting at y(0) = 4. What is the estimate of y(1)?- A) 2
- B) 0
- C) 9
- D) 1
19 Question 19
What is the length of the curve y = (43)x from x = 0 to x = 3?- A) 5
- B) 3
- C) 4
- D) 7
20 Question 20
What is the area of the region between y = 1 and y = x2?- A) 23
- B) 43
- C) 2
- D) 83
21 Question 21
A curve is given by x = 3t and y = t2 − 4. What is dy/dx at t = 1?- A) 32
- B) 2
- C) 23
- D) −3
22 Question 22
What is the area of the region inside the polar curve r = 3?- A) 9π
- B) 3π
- C) 6π
- D) 9π/2
23 Question 23
What is the area of one petal of the polar curve r = cos(3θ)?- A) π/6
- B) π/12
- C) π/3
- D) π/24
24 Question 24
Which is true about the series Σ from n = 1 to ∞ of (n + 4)/n?- A) converges
- B) converges to 4
- C) converges to 1
- D) diverges
25 Question 25
Which is true about the series Σ from n = 1 to ∞ of (−1)n+1/√n?- A) converges absolutely
- B) converges conditionally
- C) diverges
- D) converges to 0
26 Question 26
What is the interval of convergence of the power series Σ from n = 0 to ∞ of (x − 2)n/3n?- A) [−1, 5]
- B) (−3, 3)
- C) (−1, 5)
- D) (1, 3)
27 Question 27
What is the Maclaurin polynomial of degree 5 for sin x?- A) x − x36 + x5120
- B) x + x36 + x5120
- C) 1 − x22 + x424
- D) x − x33 + x55
28 Question 28
Which is true about the series Σ from n = 1 to ∞ of n/(n2 + 1)?- A) converges
- B) converges to 12
- C) converges absolutely
- D) diverges
29 Question 29
Which is true about the series Σ from n = 1 to ∞ of 2nn!?- A) diverges
- B) diverges to infinity
- C) converges
- D) cannot be decided by the ratio test
Part B (calculator)
1 Question 1
What is the limit of 8x3 − 12x3 + x as x increases without bound?- A) 0
- B) 4
- C) −1
- D) 8
2 Question 2
If f(x) = 5x, what is f′(x)?- A) x × 5x − 1
- B) 5x
- C) 5x ln 5
- D) 5x/ln 5
3 Question 3
For the curve x2 + y2 = 100, what is the slope of the tangent line at the point (6, −8)?- A) 34
- B) 43
- C) −34
- D) −43
4 Question 4
What is the limit of (1 − cos x)/x2 as x approaches 0?- A) infinity
- B) 0
- C) 1
- D) 12
5 Question 5
At what point does f(x) = −x2 − 6x have a relative maximum?- A) (−3, 9)
- B) (3, −27)
- C) (−3, −9)
- D) (0, 0)
6 Question 6
Two positive numbers have a sum of 20. What is the largest possible value of their product?- A) 20
- B) 100
- C) 99
- D) 400
7 Question 7
The area under y = x2 + 1 from x = 0 to x = 2 is estimated with 2 rectangles of width 1, using right endpoints. What is the estimate?- A) 3
- B) 143
- C) 7
- D) 5
8 Question 8
Which is ∫ x2 ln x dx, for x > 0?- A) x33 ln x + C
- B) x33 ln x + x39 + C
- C) 2x ln x + x + C
- D) x33 ln x − x39 + C
9 Question 9
A substance decays by dm/dt = −km. It starts at 80 grams and its half-life is 5 years. How many grams are left after 12 years, to the nearest hundredth?- A) 40
- B) 15.16
- C) 20
- D) 10
10 Question 10
The region under y = 3 from x = 0 to x = 2 is turned around the x-axis. What is the volume of the solid, to the nearest hundredth?- A) 56.55
- B) 18.85
- C) 18
- D) 226.19
11 Question 11
A curve is given by x = t3 and y = 6t2. What is dy/dx at t = 2?- A) 12
- B) 24
- C) 4
- D) 2
12 Question 12
What is the length of the curve y = (23)x3/2 from x = 0 to x = 8?- A) 32√23
- B) 8
- C) 523
- D) 40
13 Question 13
Which is true about the series Σ from n = 1 to ∞ of n2n?- A) diverges
- B) diverges to infinity
- C) cannot be decided by the ratio test
- D) converges
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.