Calculus late transcendental 4th edition smith test bank
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Chapter 2
1. Find the equation of the tangent line to y x 2 – 6 x at x 3.
A) y –9 B) y 3 C) y –9 x D) y 3 x
Ans: A Difficulty: Moderate Section: 2.1
2. Find an equation of the tangent line to y = f(x) at x = –3.
f x x3 x 2 x
A) y = –6x + 18 B) y = 22x – 45 C) y = 6x + 18
Ans: D Difficulty: Moderate Section: 2.1
D) y = 22x + 45
3. Find an equation of the tangent line to y = f(x) at x = 4.
f ( x) 2 x 3 5
A) y = 21x – 64 B) y = –96x – 251 C) y = 96x – 251
Ans: C Difficulty: Moderate Section: 2.1
4. Find the equation of the tangent line to y
2
14
x
25
25
2
14
B)
y – x
25
25
Ans: C Difficulty: Moderate
A)
D) y = 96x + 251
2
at x 2.
x3
y
C)
D)
2
14
x
25
25
2
14
y
x
25
25
y–
Section: 2.1
5. Find the equation of the tangent line to y 4 x + 4 at x –3.
A) y 4 x + 10 B) y 2 x + 10 C) y 4 x + 20
Ans: B Difficulty: Moderate Section: 2.1
D) y 2 x + 20
6. Compute the slopes of the secant lines between the point at x = 1 and points close to it
(such as x = 0, x = 2, x = 0.9, x = 1.1) and use these results to estimate the slope of the
tangent line at x = 1. Round to two decimal places.
x 1
x 1
A) –1.30 B) –0.70 C) –0.20 D) 0.50
Ans: D Difficulty: Moderate Section: 2.1
Page 90
Chapter 2
7. Compute the slope of the secant line between the points x = 2.9 and x = 3. Round your
answer to the thousandths place.
f ( x) sin(2 x)
A) –0.981 B) 1.852 C) –4.372 D) –1.963
Ans: B Difficulty: Easy Section: 2.1
8. List the points A, B, C, D, and E in order of increasing slope of the tangent line.
A) B, C, E, D, A B) A, E, D, C, B C) E, A, D, B, C
Ans: B Difficulty: Easy Section: 2.1
D) A, B, C, D, E
9. Use the position function s(t ) 4.9t 2 8 meters to find the velocity at time t 1
seconds.
A) 3.1 m/sec B) –9.8 m/sec C) –1.8 m/sec D) –4.9 m/sec
Ans: B Difficulty: Moderate Section: 2.1
10. Use the position function s(t ) t – 1 meters to find the velocity at time t 5 seconds.
1
1
m/sec D)
m/sec
2
4
Difficulty: Moderate Section: 2.1
A) 2 m/sec B) 4 m/sec C)
Ans: D
11. Find the average velocity for an object between t = 2 sec and t = 2.1 sec if
f(t) = –16t2 + 100t + 10 represents its position in feet.
A) 34.4 ft/s B) 36 ft/s C) 32.8 ft/s D) 146 ft/s
Ans: A Difficulty: Moderate Section: 2.1
Page 91
Chapter 2
12. Find the average velocity for an object between t = –1 sec and t = –0.9 sec if
f(t) = 5sin(t) + 5 represents its position in feet. (Round to the nearest thousandth.)
A) 2.702 B) 3.108 C) 2.907 D) –2.907
Ans: C Difficulty: Moderate Section: 2.1
13. Estimate the slope of the tangent line to the curve at x = –2.
A) –1 B) –2 C) 2 D) 0
Ans: B Difficulty: Easy Section: 2.1
14. Estimate the slope of the tangent line to the curve at x = 2.
A) 2
1
1
D)
4
2
Difficulty: Easy Section: 2.1
B) –2
Ans: D
C)
Page 92
Chapter 2
15. The table shows the temperature in degrees Celsius at various distances, d in feet, from a
specified point. Estimate the slope of the tangent line at d 2 and interpret the result.
d
0
1
3
4
6
12
18
15
8
2
C
m 5; The temperature is increasing 5 C per foot at the point 2 feet from the
specified point.
m –0.67; The temperature is decreasing 0.67 C per foot at the point 2 feet from
B)
the specified point.
m –1.5; The temperature is decreasing 1.5 C per foot at the point 2 feet from the
C)
specified point.
m 18; The temperature is increasing 18 C per foot at the point 2 feet from the
D)
specified point.
Ans: C Difficulty: Moderate Section: 2.1
A)
16. The graph below gives distance in miles from a starting point as a function of time in
hours for a car on a trip. Find the fastest speed (magnitude of velocity) during the trip.
Describe how the speed during the first 2 hours compares to the speed during the last 2
hours. Describe what is happening between 2 and 3 hours.
Ans: The fastest speed occurred during the last 2 hours of the trip when the car traveled
at about 70 mph. The speed during the first 2 hours is 60 mph while the speed
from 8 to 10 hours is about 70 mph. Between 2 and 3 hours the car was stopped.
Difficulty: Moderate Section: 2.1
17. Compute f(2) for the function f ( x) 4 x3 5x .
A) 58 B) 43 C) 38 D) –43
Ans: B Difficulty: Moderate Section: 2.2
Page 93
Chapter 2
18. Compute f(5) for the function f ( x)
4
.
x 4
2
2
40
40
40
B)
C) –
D) –
5
841
841
841
Ans: D Difficulty: Moderate Section: 2.2
A)
19. Compute the derivative function f(x) of f ( x)
15
(3 x 7) 2
3
f ( x )
B)
(3 x 7) 2
Ans: A Difficulty: Moderate
A)
f ( x )
C)
D)
5
.
3x 7
5
(3 x 7) 2
15
f ( x )
(3 x 7) 2
f ( x )
Section: 2.2
20. Compute the derivative function f(x) of f ( x) 2 x 2 7 .
A)
f ( x)
B)
f ( x)
Ans: B
4 x
C)
2x 7
2x
2
2 x2 7
Difficulty: Moderate
D)
Section: 2.2
Page 94
f ( x)
2 x
2 x2 7
2 x
f ( x )
4x 7
Chapter 2
21. Below is a graph of f ( x ) . Sketch a graph of f ( x) .
Ans:
Difficulty: Moderate
9+
Section: 2.2
Page 95
Chapter 2
22. Below is a graph of f ( x ) . Sketch a graph of f ( x) .
Ans:
Difficulty: Difficult
Section: 2.2
Page 96
Chapter 2
23. Below is a graph of f ( x) . Sketch a plausible graph of a continuous function f ( x ) .
Ans: Answers may vary. Below is one possible answer.
Difficulty: Moderate
Section: 2.2
Page 97
Chapter 2
24. Below is a graph of f ( x) . Sketch a plausible graph of a continuous function f ( x ) .
Ans: Answers may vary. Below is one possible answer.
Difficulty: Difficult
Section: 2.2
25. Compute the right-hand derivative D f (0) lim
h 0
f (h) f (0)
.
h 0
h
–8 x – 9 if x 0
f ( x)
10 x – 9 if x 0
f (h) f (0)
and the left-hand
h
derivative D f (0) lim
A)
C)
D f (0) 10 , D f (0) –8
B)
D)
D f (0) –8 , D f (0) 10
Ans: A Difficulty: Moderate Section: 2.2
Page 98
D f (0) –9 , D f (0) –9
D f (0) 1, D f (0) 1
Chapter 2
26. The table below gives the position s(t) for a car beginning at a point and returning 5 hours
later. Estimate the velocity v(t) at two points around the third hour.
t (hours)
s(t)
(miles)
0
0
1
15
2
50
3
80
4
70
5
0
Ans: The velocity is the change in distance traveled divided by the elapsed time. From
hour 3 to 4 the average velocity is (70 − 80)/(4 − 3) = −10 mph. Likewise, the
velocity between hour 2 and hour 3 is about 30 mph.
Difficulty: Easy Section: 2.2
27. Use the distances f(t) to estimate the velocity at t = 2.2. (Round to 2 decimal places.)
t 1.6
f(t) 43
1.8 2 2.2 2.4 2.6
38 32.5 28 23.5 18.5
2.8
13
A) 2250.00 B) 12.73 C) –22.50 D) –25.00
Ans: C Difficulty: Easy Section: 2.2
4 x 2 + 3 x if x 0
28. For f ( x)
find all real numbers a and b such that f (0) exists.
ax b if x 0
a 8, b any real number
A)
a 11, b 0
B)
Ans: D Difficulty: Moderate
C)
D)
Section: 2.2
Page 99
a 3, b any real number
a 3, b 0
Chapter 2
29. Sketch the graph of a function with the following properties: f (0) 0, f (2) 1,
f (4) –2, f (0) 1, f (2) 0, and f (4) –3.
A)
5
4
3
2
1
y
x
-1-1
-2
-3
-4
1
2
3
4
5
B)
5
4
3
2
1
y
x
-1-1
-2
-3
-4
1
2
3
4
5
C)
5
4
3
2
1
-1-1
-2
-3
-4
y
x
1
2
3
4
5
6
Page 100
Chapter 2
D)
5
4
3
2
1
-1-1
-2
-3
-4
Ans: B
y
x
1
2
3
4
5
Difficulty: Moderate
Section: 2.2
30. Suppose a sprinter reaches the following distances in the given times. Estimate the
velocity of the sprinter at the 6 second mark. Round to the nearest integer.
t sec
f (t ) ft
5
121.7
5.5
142.5
6
158.5
6.5
174.7
7
193.9
A) 32 ft/sec B) 36 ft/sec C) 26 ft/sec D) 28 ft/sec
Ans: A Difficulty: Moderate Section: 2.2
31. Give the units for the derivative function.
c t represents the amount of a chemical present, in milligrams, at time t
seconds.
A) seconds per milligram
C)
B) seconds per milligram squared
D)
Ans: C Difficulty: Easy Section: 2.2
milligrams per second
milligrams per second squared
(2 h)3 (2 h) 10
32. lim
equals f (a ) for some function f ( x ) and some constant a.
h 0
h
Determine which of the following could be the function f ( x ) and the constant a.
A)
f ( x) x3 x and a 2
B)
f ( x) x3 x 2 and a 1
Ans: D Difficulty: Moderate
C)
D)
Section: 2.2
Page 101
f ( x) x3 x 20 and a 1
f ( x) x3 x and a 2
Chapter 2
1
1
2
(h 2) 4
33. lim
equals f (a ) for some function f ( x ) and some constant a. Determine
h 0
h
which of the following could be the function f ( x ) and the constant a.
1
and a 2
x2
2
B)
f ( x) 2 and a 2
x
Ans: A Difficulty: Moderate
A)
f ( x)
C)
D)
1
and a 3
x2
1
f ( x) 2 and a 2
x
f ( x)
Section: 2.2
34. Find the derivative of f(x) = –5x2 + 2x – 5.
A) –5x + 2 B) –10x2 – 5 C) –10x + 2 D) 10x – 2
Ans: C Difficulty: Easy Section: 2.3
35. Differentiate the function.
f (t ) 8t 3 6 t
A)
f (t ) 24t 2 12 t
C)
B)
f (t ) 24t 2 12
D)
Ans: C
Difficulty: Moderate
36. Find the derivative of f ( x)
1
+5
x2
1
B)
f ( x) – 2 + 5
x
Ans: B Difficulty: Easy
A)
24t 5/ 2 3
t
2
24t 3
f (t )
t
f (t )
Section: 2.3
1
+ 5x + 2 .
x
f ( x)
C)
D)
Section: 2.3
Page 102
1
+5
x
1
f ( x) – 2 + 10 x 2
x
f ( x) –
Chapter 2
37. Differentiate the function.
f (s) 3s3/ 2 5s 1/ 3
27 s 5 / 3 2
6s 2 / 3
27 s1/ 2 2 s1/ 3
f ( s )
B)
6
Ans: D Difficulty: Moderate
A)
f ( s )
C)
D)
27 s1/ 2 2 s 2 / 3
6
11/ 6
27 s 10
f ( s )
6s 4 / 3
f ( s )
Section: 2.3
–5 x 2 + 5 x – 2
38. Find the derivative of f ( x)
.
7x
10 x – 5
7
10 x 5
B)
f ( x)
–
7
7
Ans: C Difficulty: Moderate
A)
f ( x) –
39. Find the derivative of f ( x)
C)
D)
Section: 2.3
3x 2 + x + 1
.
x
9 x
1
1
C)
+
–
2
2 x 2 x3
12 x + 2
B)
D)
f ( x)
x
Ans: A Difficulty: Moderate Section: 2.3
A)
f ( x)
40. Differentiate the function.
f ( x) x 6 x 2 6 x
A)
f ( x) 18x2 9 x
B)
f ( x)
Ans: A
5
2
f ( x) – + 2
7 7x
5x2 5x 2
f ( x) –
+
–
7
7 7x
9 x
1
1
–
+
2
2 x 2 x3
1
1
f ( x) 9 x +
+
x
x3
f ( x)
12 x3/ 2 3
x
Difficulty: Moderate
C)
f ( x) 12 x 2 – 3 x
D)
f ( x) 12 x – 3 x
Section: 2.3
Page 103
Chapter 2
41. Find the third derivative of f ( x) 2 x5 + 5 x –
12
x4
A)
f ( x) 120 x 2 –
B)
f ( x) 120 x 2 + 5 +
Ans: D
12
x4
Difficulty: Moderate
2
.
x
C)
D)
4
x3
12
f ( x) 120 x 2 + 4
x
f ( x) 40 x3 –
Section: 2.3
42. Find the second derivative of y 6 x +
7
.
x
d2y
21
d2y
21
d2y
21
B)
C)
6
+
–
2
2
2
dx
dx
dx
4 x5
4 x5
4 x5
Ans: B Difficulty: Moderate Section: 2.3
A)
D)
d2y
21
2
dx
4 x3
43. Using the position function s(t ) –5t 3 – 3t – 3 , find the acceleration function.
A) a (t ) –15t B) a (t ) –10t C) a (t ) –30t
Ans: C Difficulty: Moderate Section: 2.3
D) a(t ) –30t – 3
2
44. Using the position function s (t ) – t + , find the velocity function.
t
A)
v (t )
1
+
2
t2
2 t
1
2
v(t ) –
– 2
B)
2 t t
Ans: B Difficulty: Moderate
C)
D)
v (t )
1
–
2
t2
2 t
1
4
v(t ) –
– 2
2 t t
Section: 2.3
4
45. Using the position function s(t ) 3t 4 + 4t 3 + , find the velocity function.
t
A)
v(t ) 12t 3 + 12t 2 –
B)
v(t ) 9t 3 + 8t 2 –
Ans: A
4
t2
4
t2
Difficulty: Moderate
4
t2
C)
v(t ) 12t 3 + 12t 2 +
D)
v(t ) –12t 3 – 12t 2 –
Section: 2.3
Page 104
4
t2
Chapter 2
46. Using the position function s (t )
A) a(t ) –
Ans: D
9
B) a(t )
3
6
– 7 , find the acceleration function.
t
C) a (t ) –
2 t5
2 t5
Difficulty: Moderate Section: 2.3
3
t3
D) a(t )
9
2 t5
47. The height of an object at time t is given by h(t ) 16t 2 – 2t + 6 . Determine the object's
velocity at t = 4.
A) 130 B) –136 C) –130 D) –66
Ans: C Difficulty: Easy Section: 2.3
48. The height of an object at time t is given by h(t ) 2t 2 + t . Determine the object's
acceleration at t = 2.
A) 10 B) 4 C) 9 D) –4
Ans: B Difficulty: Easy Section: 2.3
49. Find an equation of the line tangent to f ( x) x 2 + 3x – 9 at x = –6.
g ( x) –9 x – 45
A)
g ( x) –12 x – 45
B)
Ans: A Difficulty: Easy
C)
D)
g ( x) –9 x – 3
g ( x) –12 x – 3
Section: 2.3
50. Find an equation of the line tangent to f ( x) 7 x + 8x – 1at x = 3.
–7 3 – 48
7
C)
g ( x)
3 +1
x –
6
2
7 3 + 16
7
B)
D)
g ( x)
3 +1
x +
3
2
Ans: D Difficulty: Moderate Section: 2.3
A)
Page 105
7 3 + 24
7
g ( x)
3
x +
6
2
7 3 + 48
7
g ( x)
3 –1
x +
6
2
Chapter 2
51. Use the graph of f ( x ) below to sketch the graph of f ( x) on the same axes. (Hint:
sketch f ( x) first.)
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2-1
-1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1 2 3 4 5 6 7 8 9
A)
9 y
8
7
6
5
4
3
x
2
1
-1 1 2 3 4 5 6 7 8 9
-9-8-7-6-5-4-3-2-1
-2
-3
-4
-5
-6
-7
-8
-9
B)
9 y
8
7
6
5
4
3
x
2
1
-1 1 2 3 4 5 6 7 8 9
-9-8-7-6-5-4-3-2-1
-2
-3
-4
-5
-6
-7
-8
-9
Page 106
Chapter 2
C)
9 y
8
7
6
5
4
3
x
2
1
-1 1 2 3 4 5 6 7 8 9
-9-8-7-6-5-4-3-2-1
-2
-3
-4
-5
-6
-7
-8
-9
D)
9 y
8
7
6
5
4
3
x
2
1
-1 1 2 3 4 5 6 7 8 9
-9-8-7-6-5-4-3-2-1
-2
-3
-4
-5
-6
-7
-8
-9
Ans: A
Difficulty: Difficult
Section: 2.3
52. Determine the real value(s) of x for which the line tangent to f ( x) 7 x 2 – 9 x – 4 is
horizontal.
9 193
9
9
C) x
, x 0 B) x
14
14
14
Ans: C Difficulty: Easy Section: 2.3
A) x
D) x = 0
53. Determine the real value(s) of x for which the line tangent to f ( x) 2 x 4 – 16 x 2 – 8 is
horizontal.
A) x = –2, x = 2 B) x = 0, x = –2, x = 2 C) x = 0
Ans: B Difficulty: Easy Section: 2.3
D) x = 0, x = 2
54. Determine the value(s) of x, if there are any, for which the slope of the tangent line to
f ( x) | x 2 – 13x + 30 | does not exist.
A)
x 6.5
x –10, x –3
B)
Ans: C Difficulty: Moderate
C)
D)
Section: 2.3
Page 107
x 3, x 10
The slope exists for all values of x.
Chapter 2
55. Find the second-degree polynomial (of the form ax2 + bx + c) such that f(0) = 0, f '(0) = 5,
and f ''(0) = 1.
x2
x2
x2
x2
5 x B) 5 x C)
5 x 1 D) 5 x 1
2
2
2
2
Ans: A Difficulty: Moderate Section: 2.3
A)
56. Find a formula for the nth derivative f ( n ) ( x) of f ( x)
20n !
C)
( x + 10) n 1
2n !
f ( n ) ( x) ( 1) n 1
B)
D)
( x + 10) n
Ans: D Difficulty: Difficult Section: 2.3
A)
f ( n ) ( x) (1) n 1
2
.
x + 10
20n !
( x + 10) n
2n !
f ( n ) ( x) ( 1) n
( x + 10) n 1
f ( n ) ( x) ( 1) n
57. Find a function with the given derivative.
f ( x) 36 x8
A) f ( x) 36 x9 B) f ( x) 4 x9 C) f ( x) 36 x7
Ans: B Difficulty: Moderate Section: 2.3
D) f ( x) 288 x7
58. Let f (t ) equal the average monthly salary of families in a certain city in year t. Several
values are given in the table below. Estimate and interpret f (2010) .
t
1995
2000
2005
2010
$1700
$2000
$2200
$2450
f (t )
f (2010) 2 ; The rate at which the average monthly salary is increasing each year
in 2010 is increasing by $2 per year.
f (2010) 2 ; The average monthly salary is increasing by $2 per year in 2010.
B)
f (2010) 50 ; The rate at which the average monthly salary is increasing each
C)
year in 2010 is increasing by $50 per year.
f (2010) 50 ; The average monthly salary is increasing by $50 per year in 2010.
D)
Ans: A Difficulty: Moderate Section: 2.3
A)
Page 108
Chapter 2
59. Find the derivative of f ( x)
1
x – 5x 9 x2 – .
x
45 3/ 2
1
x + 3/ 2
2
2x
45
1
B)
f ( x) –135 x 2 + x3/ 2 + 3/ 2
2
2x
45
1
C)
f ( x) 135 x 2 + x3/ 2 – 3/ 2
2
2x
45
10
1
D)
f ( x) –135 x 2 + x3/ 2 + + 3/ 2
2
x 2x
Ans: B Difficulty: Moderate Section: 2.4
A)
f ( x) –135 x 2 –
60. Find the derivative of f ( x)
6x + 2
.
9x – 5
48
2
2
B)
C) –
2
(9 x – 5)
3
3
Ans: D Difficulty: Moderate
A)
61. Find the derivative of f ( x)
–48
(9 x – 5) 2
Section: 2.4
D)
9x
.
6 x2 – 4
–54 x 2 – 36
54 x 2 + 36
3
3
B)
C)
D)
–
2
2
2
2
2
2x 2
(6 x – 4)
(6 x – 4)
2x
Ans: A Difficulty: Moderate Section: 2.4
A)
Page 109
Chapter 2
62. Find the derivative of the function.
4x – 8 x
4x2 – 7
4 – 4 x 4 x
1/ 2
A)
C)
D)
– 7 4 x – 8 x 8x
4x2 – 7
4 – 4 x 4 x – 7 4 x – 8 x 8x
4x – 7
4 x – 8 x 8x 4 – 4 x 4 x – 7
4x – 7
4 x – 8 x 8x 4 – 4 x 4 x – 7
1/ 2
B)
2
2
2
2
Ans: B
2
1/ 2
2
1/ 2
2
2
4x2 – 7
Difficulty: Moderate
Section: 2.4
63. Find the derivative of f ( x) –8 3 x + 5 x .
32 3
x +5
3
8
B)
f ( x) – 3 x – 5
3
Ans: C Difficulty: Moderate
A)
f ( x)
C)
D)
32 3
x +5
3
16
f ( x) – 3 x + 10
3
f ( x) –
Section: 2.4
64. Find the equation of the tangent line to the graph of y = f (x) at x = –2.
x+5
f x 2
x +3
19
59
x+
49
49
19
22
B)
y x+
7
7
Ans: A Difficulty: Moderate
A)
y
C)
D)
19
17
x–
49
49
19
16
y – x+
7
7
y–
Section: 2.4
65. Find an equation of the line tangent to h( x) f ( x) g ( x) at x –2 if
f (–2) 3 , f (–2) 3 , g (–2) –2 , and g (–2) –1 .
A) y 3x – 12 B) y 3 x – 24 C) y –9 x – 24
Ans: C Difficulty: Moderate Section: 2.4
Page 110
D) y –9 x + 12
Chapter 2
f ( x)
at x –3 if
g ( x)
f (–3) –2 , f (–3) 1 , g (–3) 1 , and g (–3) 2 .
66. Find an equation of the line tangent to h( x)
A) y –3 x + 13 B) y 5 x + 13 C) y –3 x – 11
Ans: B Difficulty: Moderate Section: 2.4
D) y 5 x – 17
67. A small company sold 1000 widgets this year at a price of $10 each. If the price
increases at rate of $1.25 per year and the quantity sold increases at a rate of 250 widgets
per year, at what rate will revenue increase?
A) $312.5/year B) $3750/year C) $1250/year
Ans: B Difficulty: Moderate Section: 2.4
D) $4062.5/year
( x 2 + 2) 4
68. Find the derivative of f ( x)
.
6
2
x( x 2 + 2)3
3
1
B)
f ( x) x( x 2 + 2)3
3
Ans: C Difficulty: Moderate
A)
f ( x)
C)
D)
4
x( x 2 + 2)3
3
1
f ( x) x( x 2 + 2)3
6
f ( x)
Section: 2.5
69. Find the derivative of f ( x) x 2 – 2 .
A)
f ( x)
B)
f ( x)
Ans: D
2x
x –2
4x
2
x2 – 2
Difficulty: Moderate
C)
f ( x)
D)
f ( x)
–x
x2 – 2
x
x2 – 2
Section: 2.5
70. Differentiate the function.
f (t ) t 3 t 7 – 2
A)
f (t )
B)
f (t )
Ans: C
7t 3 – 12t 2
2 t7 – 2
3t 2
2 t7 – 2
Difficulty: Difficult
C)
f (t )
D)
f (t )
Section: 2.5
Page 111
13t 9 – 12t 2
2 t7 – 2
21t 8
2 t7 – 2
Chapter 2
x
.
x +1
71. Find the derivative of f ( x)
2
A)
1
1
2 x x2 1
B)
x
x2 1
2 x x2 1
Ans: B
3
x2 1
x
3
1
f ( x)
B)
f ( x)
3
(5 x + 9)
Difficulty: Moderate
f ( x)
B)
C)
D)
f ( x)
f ( x)
f ( x)
f ( x)
Ans: A
x3 + 5 4 x
f ( x)
D)
f ( x)
2
x3 + 5 4 x
3
16 x3 + 5 3x 2
2 x3 + 5
x3 + 5 4 x
2
2 x3 + 5 8
x3 + 5 4 x
3
2 x3 + 5 8
x3 + 5 4 x
2
Difficulty: Difficult
x2 1
C)
Section: 2.5
8 x3 + 5 3x 2
x3 + 5
1
2 x2
x2 1
2
.
(5 x 2 + 9)3
40 x
73. Differentiate the function.
A)
5x2 + 9
–20 x
2
Section: 2.5
4
72. Find the derivative of f ( x)
Ans: A
D)
Difficulty: Moderate
A)
C)
1
1
x x2 1
2 x x2 1
Section: 2.5
Page 112
20 x
(5 x 2 + 9)3
8x
(5 x 2 + 9)3
Chapter 2
1
74. Find an equation of the line tangent to f ( x)
x2 – 3
at x = 2.
A) y = –2x + 3 B) y = –2x C) y = 2x + 3 D) y = –2x + 5
Ans: D Difficulty: Moderate Section: 2.5
75. Use the position function s(t ) t 2 65 meters to find the velocity at t = 4 seconds.
4
1
2
m/s C)
m/s D)
m/s
9
9
9
Difficulty: Moderate Section: 2.5
A) 9 m/s
B)
Ans: B
76. Compute the derivative of h( x) f g ( x) at x = –8 where
f (–8) 8 , g (–8) 2 , f (–8) 4 , f (2) –7 , g (–8) 9 , and g (2) 7 .
A) h(–8) 36 B) h(–8) 72 C) h(–8) –63
Ans: C Difficulty: Moderate Section: 2.5
D) h(–8) 16
77. Find the derivative where f is an unspecified differentiable function.
f (9 x 4 )
A) 36 x3 f (9 x 4 ) B) (36 x3 9 x 4 ) f (9 x 4 ) C) f (36 x3 )
Ans: A Difficulty: Moderate Section: 2.5
D) f (36 x3 9 x 4 )
78. Find the derivative where f is an unspecified differentiable function.
4
A)
f x
1
4 3 f x
Ans: D
4
B)
f x
4 3 f x
Difficulty: Moderate
4
C)
1
4 4 f x
3
D)
f x
4 4 f x
3
Section: 2.5
79. Find the second derivative of the function.
f ( x) 100 x 2
A)
f ( x)
B)
f ( x)
Ans: C
100 x
(100 x 2 )3/ 2
x 2 100
(100 x 2 )3/ 2
Difficulty: Moderate
C)
f ( x)
100
(100 x 2 )3/ 2
D)
f ( x)
100 x
(100 x 2 )3/ 2
Section: 2.5
Page 113
Chapter 2
80. Find a function g ( x) such that g ( x) f ( x).
f ( x) x 2 + 5 (2 x)
4
5
A)
x3
x2
+
5
x
3
5
B)
g ( x) x + 5 (16 x)
2
Ans: D
3
Difficulty: Moderate
C)
g ( x) x 2 + 5
D)
x
g ( x)
2
5
+ 5
5
5
Section: 2.5
81. Use the table of values to estimate the derivative of h( x) f g ( x) at x = 6.
x
f(x)
g(x)
–1
–5
4
0
–4
2
1
–3
0
2
–4
0
3
–5
2
4
–6
4
5
–5
2
6
–3
0
7
–1
–1
A) h(6) 2 B) h(6) –3 C) h(6) –2 D) h(6) 3
Ans: A Difficulty: Moderate Section: 2.5
82. Find the derivative of f ( x) 5sin( x) – 3cos(3 x) x .
f ( x) 5cos x + 9sin 3 x 1
A)
C)
f ( x) 5cos x + 3sin 3x 1
B)
D)
Ans: A Difficulty: Easy Section: 2.6
f ( x) –5cos x – 9sin 3 x 1
f ( x) cos x – 3sin 3 x 1
83. Find the derivative of f ( x) 7sin 2 x + 9 x 2 .
f ( x) –14sin x cos x + 18 x
A)
C)
f ( x) 14sin x cos x + 9 x
B)
D)
Ans: D Difficulty: Easy Section: 2.6
84. Find the derivative of f ( x)
–9 cos x 2
.
x2
–18( x 2 sin x 2 cos x 2 )
C)
x3
18( x sin x 2 cos x 2 )
f ( x)
B)
D)
x3
Ans: C Difficulty: Moderate Section: 2.6
A)
f ( x) 14sin x + 18 x
f ( x) 14sin x cos x + 18 x
f ( x)
Page 114
18( x 2 sin x 2 cos x 2 )
x3
18( x 2 sin x 2 cos x 2 )
f ( x)
x4
f ( x)
