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

Section 2.4 • Differentiating the Product and the Quotient of Two Functions; Higher-Order Derivatives 217
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Preparing for the AP Exam
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AP Practice Problems
Multiple-Choice Questions
PAGE PAGE
207 1. What is the instantaneous rate of change at x = −2 of the 211 6. The position of an object moving along a line at
2
x − 1 time t, in seconds, is given by s(t) = 16t − 5t + 20 meters.
function f (x) = ? What is the acceleration of the object when t = 2?
2
x + 2
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1 1 1 (A) 32 m/s (B) 0 m/s 2 (C) 32 m/s 2 (D) 64 m/s 2
(A) − (B) (C) (D) −1
6 9 2
PAGE x − 3
PAGE , x 6= −3, the instantaneous rate of change of y
207 2. An equation of the line tangent to the graph 207 7. If y = x + 3
5x − 3
of f (x) = at the point (3, 4) is with respect to x at x = 3 is
3x − 6
1 1 1
(A) 7x + 3y = 37 (B) 7x + 3y = 33 (A) − (B) (C) (D) 1
6 6 36
(C) 7x − 3y = 9 (D) 13x + 3y = 51
PAGE
PAGE 207 8. Find an equation of the line tangent to the graph of the function
207 3. If f , g, and h are nonzero differentiable functions of x,
x 2

d gh f (x) = at x = 1.
then = x + 1
dx f
(A) 8x + 6y = 11 (B) −8x + 6y = −5
′
f gh + f g h − f gh g h − gh f ′
′
′ ′
′
(A) (B)
f 2 f 2 (C) −3x + 4y = −1 (D) 3x + 4y = 5
′
′
′
′
′
gh + g h f gh + f g h + f gh PAGE
x
(C) (D) 210 9. If y = xe , then the nth derivative of y is
f ′ f 2
dy
n x
PAGE 3 x (A) e x (B) (x + n)e x (C) ne x (D) x e
205 4. If y = x e + ln x, then =
dx
1 1 2
2 x 2 x x + x + 4
(A) 3x e + (B) 3x + e − PAGE . Find f (c) if f (c) = 0 and c < 0.
′′
′
x x 210 10. f (x) = x
1
2 x
2 x
(C) 3x e (x + 1) + x (D) x e (x + 3) + (A) −4 (B) −2 (C) −1 (D) 1
x

PAGE d 2 1 1 PAGE
x
′′
208 5. t − 2 + at t = 2 is 210 11. f (x) = 3 ln x − 4x + 5e . Find f (1).
dt t t
7 9 9 (A) −4 + 5e (B) −3 + 5e (C) 5e − 1 (D) 4
(A) (B) (C) (D) 4
2 2 4
Retain Your Knowledge
Multiple-Choice Questions Free-Response Question
sin(4x)  √
2
1. lim =  3x + 4 if x < 2
x→0 sin x
4. Given the function f (x) = 4 if x = 2
1  5x − 6 if x > 2
(A) 0 (B) (C) 4 (D) ∞
4
(a) Determine the domain of f .
2
(t − 4) 3
2. Find lim if it exists. (b) Determine whether f is continuous at x = 2.
2
t→2 t + t − 6 (c) Is f continuous on its domain? Justify your answer.
8 4
(A) − (B) 0 (C) (D) The limit does not exist.
5 5
f (x) − f (c) √
3. Find lim if f (x) = 3x and c = 4.
x→c x − c
√
√ 1 3
(A) 3 (B) √ (C) (D) ∞
3 4
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