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

232 Chapter 2 • The Derivative and Its Properties

x
6. If y = ln x + xe + 6, what is the instantaneous rate of 10. Find an equation of the line tangent to the graph
change of y with respect to x at x = 5? x + 3
of f (x) = at x = 1.
2
1 x + 2
(A) 5 + 6e 5 (B) + 5e
5 (A) 5x + 9y = 17 (B) 9y − 5x = 7
1
5
(C) 5 + 5e 5 (D) 6e + (C) 5x + 3y = 9 (D) 5x + 9y = 7
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5
tan x − 1
11. lim =
7. An equation of the line tangent to the graph of π π
x→ 4 x − 4
x
f (x) = 3xe + 5 at x = 0 is
(A) 0 (B) −1
1
(A) y = 3x + 5 (B) y = − x + 5
3 (C) 2 (D) Does not exist.
1
(C) y = x + 5 (D) y = −3x + 5
3 Free-Response Questions
12. An object moves on a line according to
8. An object moves along a horizontal line so that its position the position function s = 2t − 15t + 24t + 3,
2
3
4
3
at time t is x(t) = t − 6t − 2t − 1. At what time t is the where t is measured in minutes and s in meters.
acceleration of the object zero?
(a) When is the velocity of the object 0?
(A) at 0 only (B) at 1 only
(b) Find the object’s acceleration when t = 3.
(C) at 3 only (D) at 0 and 3 only
13. Find the value of the limit below and specify
x
′
9. If f (x) = e (sin x + cos x), then f (x) = the function f for which this is the derivative.
x
x
(A) 2e (cos x + sin x) (B) e cos x [4 − 2(x + h)] − (4 − 2x) 2
2
lim
2
2
x
x
(C) 2e cos x (D) e (cos x − sin x) h→0 h
AP Cumulative Review Problems: Chapters 1–2 Preparing for the AP® Exam
R
Multiple-Choice Questions
x − 4
1. lim = 5. Suppose the function f is continuous at all real
x→4 4 − x
numbers and f (−2) = 1 and f (5) = −3. Suppose the
(A) −4 (B) −1 (C) 0 (D) The limit does not exist. function g is also continuous at all real numbers
and g(x) = f −1 (x) for all x. The Intermediate Value
3x + sin x
2. lim = Theorem guarantees that
x→0 2x
(A) g(c) = 2 for at least one c between −3 and 1.
(A) 0 (B) 1 (C) 2 (D) The limit does not exist.
(B) g(c) = 0 for at least one c between −2 and 5.
3. Let h be defined by (C) f (c) = 0 for at least one c between −3 and 1.

f (x) · g(x) if x ≤ 1 (D) f (c) = 2 for at least one c between −2 and 5.
h(x) =
k + x if x > 1
where f and g are both continuous at all real numbers. 6. The line x = c is a vertical asymptote to the graph of
If lim f (x) = 2 and lim g(x) = −2, then for what the function f . Which of the following statements
x→1 x→1 cannot be true?
number k is h continuous?
(A) lim f (x) = ∞ (B) lim f (x) = c
x→c x→∞
(A) −5 (B) −4 (C) −2 (D) 2
(C) f (c) is not defined. (D) f is continuous at x = c.
4. Which function has the horizontal asymptotes y = 1 7. The position function of an object moving along a
and y = −1?
1 3 1 2 −1
2 −1 −x straight line is s(t) = t − t + 5t . What is the
(A) f (x) = tan x (B) f (x) = e + 1 15 2
π object’s acceleration at t = 5?
2
1 − x 2 2x − 1 27 1 1 27
(C) f (x) = 2 (D) f (x) = 2 (C) (D)
1 + x 2x + x (A) − 25 (B) − 5 5 25
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