138
7.
8.
9.
Chapter 3 • Derivatives
33–42. Derivatives and tangent lines
An equation of the line tangent to the graph of ƒ at the point 12, 72
is y = 4x - 1. Find ƒ122 and ƒ′122.
a. For the following functions and values of a, find ƒ′1a2.
b. Determine an equation of the line tangent to the graph of ƒ at the
point 1a, ƒ1a22 for the given value of a.
An equation of the line tangent to the graph of g at x = 3 is
y = 5x + 4. Find g132 and g′132.
If h112 = 2 and h′112 = 3, find an equation of the line tangent to
the graph of h at x = 1.
10. If ƒ′1- 22 = 7, find an equation of the line tangent to the graph of
ƒ at the point 1- 2, 42.
35. ƒ1x2 = 4x 2 + 2x; a = - 2
36. ƒ1x2 = 2x 3; a = 10
1
1
;a =
4
1x
38. ƒ1x2 =
1
;a = 1
x2
11. Use definition (1) (p. 133) to find the slope of the line tangent to
the graph of ƒ1x2 = - 5x + 1 at the point 11, -42.
39. ƒ1x2 = 12x + 1; a = 4
40. ƒ1x2 = 13x; a = 12
41. ƒ1x2 =
42. ƒ1x2 =
Practice Exercises
43–46. Derivative calculations Evaluate the derivative of the following functions at the given point.
13–14. Velocity functions A projectile is fired vertically upward into
the air; its position (in feet) above the ground after t seconds is given
by the function s1t2. For the following functions, use limits to determine
the instantaneous velocity of the projectile at t = a seconds for the
given value of a.
43. ƒ1t2 =
47. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Use definition (1) (p. 133) to find the slope of the line tangent to the
graph of ƒ at P.
b. Determine an equation of the tangent line at P.
c. Plot the graph of ƒ and the tangent line at P.
a. For linear functions, the slope of any secant line always equals
the slope of any tangent line.
b. The slope of the secant line passing through the points P and Q
is less than the slope of the tangent line at P.
c. Consider the graph of the parabola ƒ1x2 = x 2. For a 7 0 and
h 7 0, the secant line through 1a, ƒ1a22 and 1a + h, ƒ1a + h22
always has a greater slope than the tangent line at 1a, ƒ1a22.
15. ƒ1x2 = x 2 - 5; P13, 42
16. ƒ1x2 = - 3x 2 - 5x + 1; P11, -72
T
18. ƒ1x2 =
4
; P1- 1, 42
x2
T
19. ƒ1x2 = 13x + 3; P12, 32
48–51. Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.
48. When a faucet is turned on to fill a bathtub, the volume of water in
gallons in the tub after t minutes is V1t2 = 3t. Find V′1122.
49. An object dropped from rest falls d1t2 = 16t 2 feet in t seconds.
Find d′142.
2
20. ƒ1x2 =
; P14, 12
1x
50. The gravitational force of attraction between two masses separated by a distance of x meters is inversely proportional to the
square of the distance between them, which implies that the force
is described by the function F1x2 = k>x 2, for some constant k,
where F1x2 is measured in newtons. Find F′1102, expressing your
answer in terms of k.
21–32. Equations of tangent lines by definition (2)
a. Use definition (2) (p. 135) to find the slope of the line tangent to the
graph of ƒ at P.
b. Determine an equation of the tangent line at P.
21. ƒ1x2 = 2x + 1; P10, 12
22. ƒ1x2 = - 7x; P1-1, 72
51. Suppose the speed of a car approaching a stop sign is given by
v1t2 = 1t - 52 2, for 0 … t … 5, where t is measured in seconds
and v1t2 is measured in meters per second. Find v′132.
23. ƒ1x2 = 3x 2 - 4x; P11, -12 24. ƒ1x2 = 8 - 2x 2; P10, 82
1
; P11, 12
x
25. ƒ1x2 = x 2 - 4; P12, 02
26. ƒ1x2 =
27. ƒ1x2 = x 3; P11, 12
1
28. ƒ1x2 =
; P10, 12
2x + 1
29. ƒ1x2 =
1
1
; P a - 1, b 30. ƒ1x2 = 1x - 1; P12, 12
3 - 2x
5
31. ƒ1x2 = 1x + 3; P11, 22
M03_BRIG3644_03_SE_C03_131-240.indd 138
1
;a = 1
t + 1
46. ƒ1r2 = pr 2; a = 3
15–20. Equations of tangent lines by definition (1)
1
; P1- 1, - 12
x
1
;a = 2
3x - 1
45. ƒ1s2 = 21s - 1; a = 25
14. s1t2 = - 16t 2 + 128t + 192; a = 2
17. ƒ1x2 =
1
;a = 5
x + 5
44. ƒ1t2 = t - t 2; a = 2
13. s1t2 = - 16t 2 + 100t; a = 1
T
34. ƒ1x2 = x 2; a = 3
37. ƒ1x2 =
12. Use definition (2) (p. 135) to find the slope of the line tangent to
the graph of ƒ1x2 = 5 at P11, 52.
T
33. ƒ1x2 = 8x; a = - 3
32. ƒ1x2 =
x
; P1-2, 22
x + 1
T
52. Population of Las Vegas Let p1t2 represent the population of the
Las Vegas metropolitan area t years after 1970, as shown in the
table and figure.
a. Compute the average rate of growth of Las Vegas from 1970
to 1980.
b. Explain why the average rate of growth calculated in part (a)
is a good estimate of the instantaneous rate of growth of
Las Vegas in 1975.
06/07/17 1:47 PM
3.2 The Derivative as a Function
16. Matching derivatives with functions Match graphs a–d of derivative functions with possible graphs A–D of the corresponding
functions.
y
Derivatives
20. Use the graph of g in the figure to do the following.
a. Find the values of x in 1-2, 22 at which g is not continuous.
b. Find the values of x in 1-2, 22 at which g is not differentiable.
y
2
a
1
y 5 g(x)
b
22
149
1
21
c
2
x
21
d
22
22
1
21
2
x
Practice Exercises
y
21–30. Derivatives
2
a. Use limits to find the derivative function ƒ′ for the following
functions f.
b. Evaluate ƒ′1a2 for the given values of a.
D
B
1
Functions
22
1
21
C
2
x
21. ƒ1x2 = 5x + 2; a = 1, 2
22. ƒ1x2 = 7; a = -1, 2
23. ƒ1x2 = 4x 2 + 1; a = 2, 4
24. ƒ1x2 = x 2 + 3x; a = -1, 4
21
25. ƒ1x2 =
22
A
27. ƒ1t2 =
1
1
;a = - ;5
x + 1
2
26. ƒ1x2 =
1
1
; a = 9,
4
1t
28. ƒ1w2 = 14w - 3; a = 1, 3
17–18. Sketching derivatives Reproduce the graph of ƒ and then
sketch a graph of ƒ′ on the same axes.
29. ƒ1s2 = 4s3 + 3s; a = - 3, - 1
17.
30. ƒ1t2 = 3t 4; a = - 2, 2
y
2
18.
31–32. Velocity functions A projectile is fired vertically upward into
the air, and its position (in feet) above the ground after t seconds is
given by the function s1t2.
y 5 f (x)
a. For the following functions s1t2, find the instantaneous velocity
function v1t2. (Recall that the velocity function v is the derivative of
the position function s.)
b. Determine the instantaneous velocity of the projectile at t = 1 and
t = 2 seconds.
x
2
22
y
5
y 5 f (x)
31. s1t2 = -16t 2 + 100t
33. Evaluate
1
x
1 2
x
; a = -1, 0
x + 2
19. Use the graph of ƒ in the figure to do the following.
a. Find the values of x in 1- 2, 22 at which ƒ is not continuous.
b. Find the values of x in 1- 2, 22 at which ƒ is not differentiable.
y
34. Evaluate
32. s1t2 = - 16t 2 + 128t + 192
dy
dy
x + 1
and `
if y =
.
dx
dx x = 2
x + 2
ds
ds
and `
if s = 11t 3 + t + 1.
dt
dt t = -1
35–40. Tangent lines
a. Find the derivative function ƒ′ for the following functions ƒ.
b. Find an equation of the line tangent to the graph of ƒ at 1a, ƒ1a22
for the given value of a.
35. ƒ1x2 = 3x 2 + 2x - 10; a = 1
y 5 f (x)
22
M03_BRIG3644_03_SE_C03_131-240.indd 149
21
36. ƒ1x2 = 5x 2 - 6x + 1; a = 2
1
2
37. ƒ1x2 = 13x + 1; a = 8
38. ƒ1x2 = 1x + 2; a = 7
39. ƒ1x2 =
40. ƒ1x2 =
2
; a = -1
3x + 1
1
; a = -5
x
x
06/07/17 1:47 PM
160
Chapter 3 • Derivatives
9–11. Let F1x2 = ƒ1x2 + g1x2, G1x2 = ƒ1x2 - g1x2, and H1x2 =
3ƒ1x2 + 2g1x2, where the graphs of ƒ and g are shown in the figure.
Find each of the following.
35. g1w2 = 2w3 + 3w2 + 10w 36. s1t2 = 41t 37. ƒ1x2 = 3ex + 5x + 5
y
39. ƒ1x2 = b
y 5 g(x))
8
y 5 f (x)
40. g1w2 = b
6
2
0
2
4
6
x
8
12–14. Use the table to find the following derivatives.
12.
14.
1
2
3
4
5
ƒ′ 1x 2
3
5
2
1
4
2
4
3
1
5
g′ 1x 2
d
1ƒ1x2 + g1x22 `
dx
x=1
d
12x - 3g(x22 `
dx
x=4
13.
d
11.5ƒ1x22 `
dx
x=2
15. If ƒ1t2 = t 10, find ƒ′1t2, ƒ″1t2, and ƒ‴1t2.
16. Find an equation of the line tangent to the graph of y = ex at
x = 0.
1
17. The line tangent to the graph of ƒ at x = 5 is y =
x - 2. Find
10
d
14ƒ1x22 `
.
dx
x=5
18. The line tangent to the graph of ƒ at x = 3 is y = 4x - 2 and the
line tangent to the graph of g at x = 3 is y = - 5x + 1. Find the
values of 1ƒ + g2132 and 1ƒ + g2′132.
Practice Exercises
19–40. Derivatives Find the derivative of the following functions. See
Example 2 of Section 3.2 for the derivative of 1x.
19. y = x 5
20. ƒ1t2 = t
21. ƒ1x2 = 5
22. g1x2 = e3
23. ƒ1x2 = 5x 3
24. g1w2 =
25. h1t2 =
t2
+ 1
2
5 12
w
6
26. ƒ1v2 = v 100 + ev + 10
27. p1x2 = 8x
28. g1t2 = 61t
29. g1t2 = 100t 2
30. ƒ1s2 =
31. ƒ1x2 = 3x 4 + 7x
5
32. g1x2 = 6x 5 - x 2 + x + 5
2
33. ƒ1x2 = 10x 4 - 32x + e2
34. ƒ1t2 = 61t - 4t 3 + 9
1s
4
if x … 0
if x 7 0
if w … 1
if w 7 1
a. Compute d′1t2. What units are associated with the derivative
and what does it measure?
b. If it takes 6 s for a stone to fall to the ground, how high is the
ledge? How fast is the stone moving when it strikes the ground
(in miles per hour)?
F′122 10. G′162 11. H′122
x
w + 5ew
2w3 + 4w + 5
38. g1w2 = ew - e2 + 8
41. Height estimate The distance an object falls (when released from
rest, under the influence of Earth’s gravity, and with no air resistance) is given by d1t2 = 16t 2, where d is measured in feet and
t is measured in seconds. A rock climber sits on a ledge on a vertical wall and carefully observes the time it takes for a small stone
to fall from the ledge to the ground.
4
9.
x2 + 1
2x 2 + x + 1
1 4
t + t + 1
4
T
42. Projectile trajectory The position of a small rocket that is
launched vertically upward is given by s1t2 = - 5t 2 + 40t + 100,
for 0 … t … 10, where t is measured in seconds and s is measured
in meters above the ground.
a. Find the rate of change in the position (instantaneous velocity)
of the rocket, for 0 … t … 10.
b. At what time is the instantaneous velocity zero?
c. At what time does the instantaneous velocity have the greatest
magnitude, for 0 … t … 10?
d. Graph the position and instantaneous velocity, for 0 … t … 10.
43. City urbanization City planners model the size of their city using
1
the function A1t2 = - t 2 + 2t + 20, for 0 … t … 50, where A
50
is measured in square miles and t is the number of years after 2010.
a. Compute A′1t2. What units are associated with this derivative
and what does the derivative measure?
b. How fast will the city be growing when it reaches a size of
38 mi2?
c. Suppose the population density of the city remains constant
from year to year at 1000 people>mi2. Determine the growth
rate of the population in 2030.
44. Cell growth When observations begin at t = 0, a cell culture
has 1200 cells and continues to grow according to the function
p1t2 = 1200et, where p is the number of cells and t is measured
in days.
a. Compute p′1t2. What units are associated with the derivative
and what does it measure?
b. On the interval 30, 44, when is the growth rate p′1t2 the least?
When is it the greatest?
45. Weight of Atlantic salmon The weight w1x2 (in pounds) of an
Atlantic salmon that is x inches long can be estimated by the
function
0.4x - 5
w1x2 = c 0.8x - 13.4
1.5x - 35.8
if 19 … x … 21
if 21 6 x … 32
if x 7 32
Calculate w′1x2 and explain the physical meaning of this derivative. (Source: www.atlanticsalmonfederation.org)
3.3 Rules of Differentiation
46–58. Derivatives of products and quotients Find the derivative of
the following functions by first expanding or simplifying the expression.
Simplify your answers.
46. ƒ1x2 =
1 1x
+ 1 21 1x - 1 2
2
47. ƒ1x2 = 12x + 1213x + 22
48. g1r2 = 15r 3 + 3r + 121r 2 + 32
49. ƒ1w2 =
w3 - w
w
51. h1x2 = 1x 2 + 12 2
53. g1x2 =
55. y =
56. y =
12s3 - 8s2 + 12s
4s
52. h1x2 = 1x 1 1x - x 3>2 2
54. h1x2 =
x 3 - 6x 2 + 8x
x 2 - 2x
x 2 - 2ax + a2
; a is a constant.
x - a
e2w + ew
ew
58. r1t2 =
e2t + 3et + 2
et + 2
a. Find an equation of the line tangent to the given curve at a.
b. Use a graphing utility to graph the curve and the tangent line on the
same set of axes.
59. y = - 3x 2 + 2; a = 1
69. ƒ1x2 = 5x 4 + 10x 3 + 3x + 6
70. ƒ1x2 = 3x 2 + 5ex
71. ƒ1x2 =
x 2 - 7x - 8
x + 1
73. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d
11052 = 5 # 104.
dx
b. The slope of a line tangent to ƒ1x2 = ex is never 0.
d 3
d x
c.
1e 2 = e3.
d.
1e 2 = xex - 1.
dx
dx
dn
15x 3 + 2x + 52 = 0, for any integer n Ú 3.
e.
dx n
a.
a. Find an equation of the line tangent to y = g1x2 at x = 3.
b. Find an equation of the line tangent to y = h1x2 at x = 3.
75. Derivatives from tangent lines Suppose the line tangent to the
graph of ƒ at x = 2 is y = 4x + 1 and suppose the line tangent to
the graph of g at x = 2 has slope 3 and passes through 10, - 22.
Find an equation of the line tangent to the following curves at
x = 2.
a. y = ƒ1x2 + g1x2
c. y = 4ƒ1x2
60. y = x 3 - 4x 2 + 2x - 1; a = 2
62. y =
68. ƒ1x2 = 3x 3 + 5x 2 + 6x
74. Tangent lines Suppose ƒ132 = 1 and ƒ′132 = 4. Let
g1x2 = x 2 + ƒ1x2 and h1x2 = 3ƒ1x2.
59–62. Equations of tangent lines
61. y = ex; a = ln 3
68–72. Higher-order derivatives Find ƒ′1x2, ƒ″1x2, and ƒ‴1x2 for the
following functions.
72. ƒ1x2 = 10ex
x - a
; a is a positive constant.
1x - 1a
57. g1w2 =
T
x2 - 1
x - 1
50. y =
161
ex
- x; a = 0
4
63. Finding slope locations Let ƒ1x2 = x 2 - 6x + 5.
a. Find the values of x for which the slope of the curve y = ƒ1x2
is 0.
b. Find the values of x for which the slope of the curve y = ƒ1x2
is 2.
b. y = ƒ1x2 - 2g1x2
76. Tangent line Find the equation of the line tangent to the curve
y = x + 1x that has slope 2.
77. Tangent line given Determine the constants b and c such that the
line tangent to ƒ1x2 = x 2 + bx + c at x = 1 is y = 4x + 2.
78–81. Derivatives from a graph Let F = ƒ + g and G = 3ƒ - g,
where the graphs of ƒ and g are shown in the figure. Find the following
derivatives.
64. Finding slope locations Let ƒ1t2 = t 3 - 27t + 5.
y
a. Find the values of t for which the slope of the curve y = ƒ1t2
is 0.
b. Find the values of t for which the slope of the curve y = ƒ1t2
is 21.
7
y 5 f (x)
5
65. Finding slope locations Let ƒ1x2 = 2x 3 - 3x 2 - 12x + 4.
a. Find all points on the graph of ƒ at which the tangent line is
horizontal.
b. Find all points on the graph of ƒ at which the tangent line has
slope 60.
3
66. Finding slope locations Let ƒ1x2 = 2ex - 6x.
a. Find all points on the graph of ƒ at which the tangent line is
horizontal.
b. Find all points on the graph of ƒ at which the tangent line has
slope 12.
67. Finding slope locations Let ƒ1x2 = 41x - x.
a. Find all points on the graph of ƒ at which the tangent line is
horizontal.
b. Find all points on the graph of ƒ at which the tangent line has
1
slope - .
2
M03_BRIG3644_03_SE_C03_131-240.indd 161
y 5 g(x)
1
0
1
3
5
7
x
78. F′122 79. G′122 80. F′152 81. G′152
82–87. Derivatives from limits The following limits represent ƒ′1a2
for some function ƒ and some real number a.
a. Find a possible function ƒ and number a.
b. Evaluate the limit by computing ƒ′1a2.
82. lim
xS0
ex - 1
x
83. lim
xS0
x + ex - 1
x
06/07/17 1:48 PM
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