AP Calculus AB 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 − 9x − 3 as x approaches 3?- A) 6
- B) 3
- C) 9
- D) 0
2 Question 2
What is the limit of 5x3 − 2x2x3 + 7 as x increases without bound?- A) 0
- B) infinity
- C) 57
- D) 52
3 Question 3
What is the limit of √x − 2x − 4 as x approaches 4?- A) 0
- B) 4
- C) 14
- D) 12
4 Question 4
The function f is defined by f(x) = x2 for x < 2 and f(x) = 3x − 2 for x ≥ 2. What is the limit of f(x) as x approaches 2?- A) 2
- B) 4
- C) 8
- D) 6
5 Question 5
What is the limit of 3x2 + 1x3 − 4 as x increases without bound?- A) 0
- B) 3
- C) −14
- D) infinity
6 Question 6
If f(x) = 2x2 − 3x, what is the limit of (f(x + h) − f(x))/h as h approaches 0?- A) 2x − 3
- B) 4x
- C) 0
- D) 4x − 3
7 Question 7
If f(x) = 6x2 − 4x + 9, what is f′(x)?- A) 6x − 4
- B) 12x + 5
- C) 12x − 4
- D) 2x3 − 2x2 + 9x
8 Question 8
If f(x) = (x2 + 1)(3x − 2), what is f′(x)?- A) 6x
- B) 9x2 − 4x + 3
- C) 6x2 − 4x + 3
- D) 9x2 + 3
9 Question 9
If f(x) = 3x − 1x + 2, what is f′(x)?- A) 3
- B) −7/(x + 2)2
- C) 5/(x + 2)2
- D) 7/(x + 2)2
10 Question 10
If f(x) = 2sin x − 4cos x, what is f′(x)?- A) 2cos x + 4sin x
- B) 2cos x − 4sin x
- C) −2cos x + 4sin x
- D) −2cos x − 4sin x
11 Question 11
If f(x) = (2x + 1)5, what is f′(x)?- A) 5(2x + 1)4
- B) 10(2x + 1)4
- C) 10(2x + 1)5
- D) 2(2x + 1)4
12 Question 12
For the curve x2 + y2 = 25, what is dy/dx at the point (3, 4)?- A) 34
- B) −43
- C) −34
- D) 43
13 Question 13
If f(x) = ln(x2 + 1), what is f′(x)?- A) 1/(x2 + 1)
- B) 2x/(x2 + 1)
- C) 2x
- D) 1/(2x)
14 Question 14
For the curve x2 − y2 = 9, what is the slope of the tangent line at the point (5, 4)?- A) 45
- B) −54
- C) 54
- D) 1
15 Question 15
The tangent line to f(x) = √x at x = 9 is used to estimate √9.2. What is the estimate?- A) 3.2
- B) 3.1
- C) 3
- D) 3.033
16 Question 16
What is the limit of sin(3x)/x as x approaches 0?- A) 3
- B) 0
- C) 1
- D) 13
17 Question 17
The function f(x) = x3 − x satisfies the Mean Value Theorem on [0, 2]. What value of c in (0, 2) does the theorem give?- A) 1
- B) 43
- C) 2
- D) 2/√3
18 Question 18
At what point does f(x) = −x2 + 4x + 1 have a relative maximum?- A) (−2, −11)
- B) (2, 5)
- C) (2, −3)
- D) (4, 1)
19 Question 19
At what point does the graph of f(x) = x3 − 9x2 + 24x have a point of inflection?- A) (3, 18)
- B) (2, 20)
- C) (4, 16)
- D) (0, 0)
20 Question 20
Two positive numbers have a product of 49. What is the smallest possible value of their sum?- A) 7
- B) 50
- C) 14
- D) 49
21 Question 21
Which is ∫ (6x2 − 4x + 1) dx?- A) 12x − 4 + C
- B) 6x3 − 4x2 + x + C
- C) 2x3 − 4x2 + x + C
- D) 2x3 − 2x2 + x + C
22 Question 22
What is the value of ∫02(3x2 + 1) dx?- A) 8
- B) 10
- C) 13
- D) 12
23 Question 23
Which is ∫ 2x(x2 + 1)3 dx?- A) (x2 + 1)4 + C
- B) x2(x2 + 1)4/4 + C
- C) (x2 + 1)4/4 + C
- D) 3(x2 + 1)2 + C
24 Question 24
What is the derivative with respect to x of ∫1x(t3 + 1) dt?- A) x3 + 1
- B) x44 + x
- C) 3x2
- D) x3
25 Question 25
The area under y = 9 − x2 from x = 0 to x = 3 is estimated with 3 rectangles of equal width, using right endpoints. What is the estimate?- A) 22
- B) 8
- C) 18
- D) 13
26 Question 26
Which function y solves dy/dx = 2xy with y(0) = 3?- A) ex2 + 2
- B) 3ex2
- C) 3e2x
- D) 3x2
27 Question 27
Which is the general solution of dy/dx = 3x2 − 2?- A) 6x + C
- B) x3 − 2 + C
- C) x3 − 2x + C
- D) 3x3 − 2x + C
28 Question 28
What is the average value of f(x) = x2 on the interval [0, 3]?- A) 3
- B) 4.5
- C) 9
- D) 27
29 Question 29
What is the area of the region between y = x and y = x2?- A) 12
- B) 16
- C) 56
- D) 13
Part B (calculator)
1 Question 1
The radius of a circle grows at 2 cm per second. How fast is the area growing when the radius is 5 cm, in square cm per second, to the nearest hundredth?- A) 31.42
- B) 62.83
- C) 78.54
- D) 12.57
2 Question 2
A 10-foot ladder leans against a wall. The top slides down the wall at 2 feet per second. How fast, in feet per second, is the bottom moving away from the wall when the top is 8 feet above the floor?- A) 2
- B) 1.5
- C) 1.6
- D) 2.67
3 Question 3
Air flows into a balloon at a rate of r(t) = 2t + 10 litres per second. How many litres flow in from t = 2 to t = 6 seconds?- A) 22
- B) 40
- C) 72
- D) 76
4 Question 4
The function f(x) = −x2 + 3x satisfies the Mean Value Theorem on [−1, 2]. What value of c does the theorem give?- A) 12
- B) 32
- C) 0
- D) −12
5 Question 5
A rectangular pen with an area of 72 square feet is fenced on three sides; a barn wall is the fourth side. What is the least length of fence needed, in feet?- A) 30
- B) 36
- C) 24
- D) 18
6 Question 6
At what point does the graph of f(x) = x3 + 3x2 − 2 have a point of inflection?- A) (0, −2)
- B) (−2, 2)
- C) (1, 2)
- D) (−1, 0)
7 Question 7
What is the value of ∫01(x2 + 1) dx?- A) 43
- B) 13
- C) 1
- D) 2
8 Question 8
The area under y = 2x + 1 from x = 0 to x = 4 is estimated with 4 rectangles of width 1, using left endpoints. What is the estimate?- A) 24
- B) 16
- C) 20
- D) 12
9 Question 9
A population P grows by dP/dt = 0.05P, with P(0) = 200. What is P(10), to the nearest hundredth?- A) 300
- B) 210
- C) 329.74
- D) 543.66
10 Question 10
What is the average value of f(x) = 3x2 − 2x on the interval [1, 3]?- A) 18
- B) 9
- C) 6
- D) 11
11 Question 11
The region between y = x and y = x2 is turned around the x-axis. What is the volume of the solid, to the nearest hundredth?- A) 0.42
- B) 0.10
- C) 1.05
- D) 0.13
12 Question 12
What is the area of the region between y = x2 + 1 and y = 2x + 4?- A) 9
- B) 24
- C) −323
- D) 323
13 Question 13
What is the area of the region between y = x2 − 1 and the x-axis from x = 0 to x = 3?- A) 6
- B) 203
- C) 223
- D) 8
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.