AP Calculus AB Practice Exam 2
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 − 2 as x approaches 2?- A) 0
- B) 5
- C) 4
- D) 3
2 Question 2
What is the limit of 4x2 − x2x2 + 3 as x increases without bound?- A) 0
- B) −13
- C) 2
- D) infinity
3 Question 3
What is the limit of √x − 3x − 9 as x approaches 9?- A) 16
- B) 0
- C) 6
- D) 13
4 Question 4
The function f is defined by f(x) = x2 + 1 for x < 1 and f(x) = 3 − x for x ≥ 1. What is the limit of f(x) as x approaches 1?- A) 1
- B) 4
- C) 3
- D) 2
5 Question 5
Using the definition of the derivative, f(x) = x2 + 5x has f′(x) equal to which of these?- A) 2x
- B) 2x + 5
- C) x + 5
- D) 0
6 Question 6
If f(x) = 4x3 − 5x2 + 7, what is f′(x)?- A) 12x2 − 10x
- B) 12x2 − 10x + 7
- C) 4x2 − 5x
- D) 12x3 − 10x2
7 Question 7
If f(x) = (2x + 1)(x − 4), what is f′(x)?- A) 2
- B) 2x − 8
- C) 4x − 7
- D) 4x − 8
8 Question 8
If f(x) = x + 2x − 1, what is f′(x)?- A) 1
- B) 3/(x − 1)2
- C) 1/(x − 1)2
- D) −3/(x − 1)2
9 Question 9
Which is an equation of the line tangent to the graph of y = x3 − 3x at x = 2?- A) y = 9x
- B) y = 9x − 16
- C) y = 3x − 4
- D) y = 9x + 20
10 Question 10
If f(x) = 6ex + 2ln x, what is f′(x)?- A) 6ex + 2x
- B) 6ex + 2x
- C) 6xex − 1 + 2x
- D) 6ex + 1/(2x)
11 Question 11
If f(x) = (x2 − 5)3, what is f′(x)?- A) 3(x2 − 5)2
- B) 6x(x2 − 5)3
- C) 6x(x2 − 5)2
- D) 2x(x2 − 5)2
12 Question 12
If f(x) = cos(x2), what is f′(x)?- A) −sin(x2)
- B) 2x sin(x2)
- C) −2x sin(x2)
- D) −sin(2x)
13 Question 13
The curve x2 + xy + y2 = 7 goes through (1, 2). What is the slope of the curve at that point?- A) 45
- B) −45
- C) −54
- D) −12
14 Question 14
If f(x) = tan⁻¹(3x), what is f′(x)?- A) 1/(1 + 9x2)
- B) 3/(1 + 3x2)
- C) 3/(1 + 9x2)
- D) 3/(1 + x2)
15 Question 15
What is the limit of (e3x − 1)/x as x approaches 0?- A) 0
- B) 1
- C) 13
- D) 3
16 Question 16
What is the limit of (1 − cos(2x))/x2 as x approaches 0?- A) 0
- B) 12
- C) 2
- D) 4
17 Question 17
The line tangent to y = √x at x = 16 is used to estimate √16.4. What is the estimate?- A) 4.05
- B) 4.4
- C) 4.2
- D) 4
18 Question 18
The radius of a circle grows at 3 cm per second. How fast is the area growing, in square cm per second, when the radius is 10 cm?- A) 60π
- B) 30π
- C) 100π
- D) 6π
19 Question 19
The function f(x) = x2 − 6x + 2 has a relative minimum. At which point is it?- A) (3, −7)
- B) (−3, 29)
- C) (3, 2)
- D) (6, 2)
20 Question 20
At what point does the graph of f(x) = x3 − 6x2 + 5 have its point of inflection?- A) (0, 5)
- B) (4, −27)
- C) (2, −11)
- D) (−2, −27)
21 Question 21
The function f(x) = x2 + 2x satisfies the Mean Value Theorem on [0, 4]. What value of c does the theorem give?- A) 6
- B) 2
- C) 4
- D) 3
22 Question 22
Two numbers add up to 30. What is the largest possible value of their product?- A) 225
- B) 15
- C) 900
- D) 200
23 Question 23
Two positive numbers have a product of 64. What is the smallest possible value of their sum?- A) 8
- B) 65
- C) 16
- D) 20
24 Question 24
What is ∫ (8x3 − 6x + 5) dx?- A) 24x2 − 6 + C
- B) 2x4 − 3x2 + 5x + C
- C) 8x4 − 6x2 + 5x + C
- D) 2x4 − 3x2 + C
25 Question 25
What is the value of ∫12(4x − 1) dx?- A) 7
- B) 6
- C) 5
- D) 2
26 Question 26
What is ∫ 3x2(x3 − 2)4 dx?- A) (x3 − 2)5/5 + C
- B) (x3 − 2)5 + C
- C) 4(x3 − 2)3 + C
- D) x3(x3 − 2)5/5 + C
27 Question 27
What is the derivative, with respect to x, of ∫0x√(t2 + 4) dt?- A) √(x2 + 4) − 2
- B) x/√(x2 + 4)
- C) √(x2 + 4)
- D) √(t2 + 4)
28 Question 28
A left Riemann sum with 4 equal parts is used to estimate the area under y = x2 from x = 0 to x = 4. What is the estimate?- A) 30
- B) 14
- C) 643
- D) 13
29 Question 29
The function y satisfies dy/dx = −2y and y(0) = 5. Which is y?- A) y = 5e2x
- B) y = e−2x + 4
- C) y = 5e−2x
- D) y = 5 − 2x
Part B (calculator)
1 Question 1
A 13-foot ladder leans against a wall. The bottom slides away from the wall at 2 feet per second. How fast, in feet per second, is the top sliding down the wall when the bottom is 5 feet from the wall?- A) 0.83
- B) 2
- C) 4.8
- D) 0.77
2 Question 2
Air is pumped into a round balloon at 100 cubic cm per second. How fast, in cm per second, is the radius growing when the radius is 5 cm?- A) 1.27
- B) 0.32
- C) 3.18
- D) 0.19
3 Question 3
The area of a circle grows at 30 square cm per second. How fast, in cm per second, is the radius growing when the radius is 3 cm?- A) 3.18
- B) 5
- C) 1.59
- D) 1.06
4 Question 4
A population P grows so that dP/dt = 0.03P, where t is in years. At t = 0 there are 800. To the nearest hundredth, what is P after 5 years?- A) 920
- B) 824.36
- C) 927.42
- D) 929.47
5 Question 5
A cup of tea cools in a 20° room so that dT/dt = −0.1(T − 20), where t is in minutes. At t = 0 the tea is 90°. To the nearest hundredth, what is its temperature after 10 minutes?- A) 33.11
- B) 45.75
- C) 20
- D) 53.11
6 Question 6
What is the value of ∫132x dx, to the nearest thousandth?- A) 1.099
- B) 1.333
- C) 4
- D) 2.197
7 Question 7
The region under y = x2 from x = 0 to x = 2 is turned around the x-axis. To the nearest hundredth, what is the volume of the solid?- A) 20.11
- B) 8.38
- C) 6.4
- D) 2.67
8 Question 8
The function f(x) = x3 satisfies the Mean Value Theorem on [0, 3]. To the nearest hundredth, what value of c does the theorem give?- A) 1.5
- B) 3
- C) 1.73
- D) 9
9 Question 9
What is the area of the region between y = x2 and y = x + 2?- A) 7.5
- B) 3
- C) 10.5
- D) 4.5
10 Question 10
What is the value of ∫016e3x dx, to the nearest thousandth?- A) 38.171
- B) 114.513
- C) 40.171
- D) 3.437
11 Question 11
Water flows into a tank at a rate of r(t) = 3t2 + 2 litres per minute. How many litres flow in from t = 1 to t = 3 minutes?- A) 29
- B) 26
- C) 33
- D) 30
12 Question 12
What is the limit of x2 + 2x − 3x2 − 1 as x approaches 1?- A) 0
- B) 2
- C) 1
- D) 4
13 Question 13
What is the average value of f(x) = 3x2 + 1 on the interval [0, 2]?- A) 10
- B) 5
- C) 7
- D) 4
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.