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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

178 Chapter 2 • The Derivative and Its Properties
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Preparing for the AP Exam
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AP Practice Problems
Multiple-Choice Questions
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169 1. The line x + y = 5 is tangent to the graph of y = f (x) at the 169 5. If x − 3y = 13 is an equation of the tangent line to the graph
point where x = 2. The values f (2) and f (2) are: of f at the point (2, 6), then f (2) =
′
′
(A) f (2) = 2; f (2) = −1 (B) f (2) = 3; f (2) = −1 1 1 13
′
′
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(A) − (B) (C) −3 (D) −
′
(C) f (2) = 2; f (2) = 1 (D) f (2) = 3; f (2) = 2 3 3 3
′
PAGE PAGE f (x) − f (−3)
173 2. The graph of the function f , given below, consists of three 173 6. If f is a function for which lim x + 3 = 0, then
x→−3
′
line segments. Find f (3).
which of the following statements must be true?
y
(A) x = −3 is a vertical asymptote of the graph.
12
(5, 11)
10 (B) The derivative of f at x = −3 exists.
(C) The function f is continuous at x = 3.
8
(D) f is not defined at x = −3.
6
(25, 4)
4 PAGE 2
171 7. If the position of an object on the x-axis at time t is 4t , then
2 the average velocity of the object over the interval 0 ≤ t ≤ 5 is
(0, 1) (15, 1)
26 24 22 2 4 6 8 10 12 14 16 x (A) 5 (B) 20 (C) 40 (D) 100
Graph of f
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175 8. The table below lists several values of a function y = f (x) that
′
(A) 1 (B) 2 (C) 3 (D) f (3) does not exist is continuous on the closed interval [−2, 6] and has a derivative
at each number in the open interval (−2, 6). Approximate the
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170 3. What is the instantaneous rate of change of the function derivative of f at 0.
2
f (x) = 3x + 5 at x = 2?
x −1 −1/2 1 3 5
(A) 5 (B) 7 (C) 12 (D) 17
f (x) −4 −1 2 1 −2
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174 4. The function f is defined on the closed interval [−2, 16]. The
′
graph of the derivative of f , y = f (x), is given below. 1 2
(A) 0 (B) (C) (D) 2
3 3
y
6 Free-Response Question
4 (6, 3) PAGE 9. A tank is filled with 80 liters of water at 7 a.m. (t = 0). Over the
y 5 f9(x) 175
2 next 12 hours the water is continuously used and no water is
added to replace it. The table below gives the amount of
22 4 6 8 12 16 x
22 water A(t) (in liters) remaining in the tank at selected times t,
where t measures the number of hours after 7 a.m.
The point (6, −2) is on the graph of y = f (x). An equation of t 0 2 5 7 9 12
the tangent line to the graph of f at (6, −2) is A(t) 80 71 66 60 54 50
(A) y = 3 (B) y + 2 = 6(x + 3) Use the table to approximate A (6). Interpret your answer in the
′
(C) y + 2 = 6x (D) y + 2 = 3(x − 6) context of the problem.
Retain Your Knowledge
Multiple-Choice Questions
2
x + 3x + 2 2
1. Given f (x) = , which of the following 3. Suppose g(x) = x − 3, x 6= 4. Find lim g(x), if it exists.
x + 1 x→4
statements, if any, must be true? (A) 4 (B) 13 (C) 19 (D) The limit does not exist.
I. f is continuous at −1.
II. lim f (x) exists. Free-Response Question
x→−1
3
III. The graph of f has a vertical asymptote at x = − 1. x − 8
4. How should the function r(x) = be defined at the
x − 2
(A) II only (B) I and II only number 2, so that r is continuous 2?
(C) II and III only (D) I, II, and III
2. lim (4x · tan x) =
π
x→
4
π
(A) 0 (B) (C) 1 (D) π
4
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