188Chapter 3Applications of Differentiation71. Anaf haFindingTechnology In Exercises 49-54, (a) use a computer algebraan to diferentiate the function, (b) sketch the graphs of2 3PEA om the same set of coordinate axes over the givenSma (c) find the critical numbers of f in the open interval,18 Gad chhe h.terval(s) on which f' is positive and the)or which f'is negative. Compare the behavior of fca he ign offUsingandAnalyzingDerivativesEXPLORING CONCEPTSTransformations of Functions In Exercises 63-66,assume that f is differentiable for all x. The signs of fexinare as follows.72. Anf'(x) > 0 on (-0, -4)f'(x) < 0 on (-4, 6)f'(x)> 0 on (6, oo)48 FO) = 2x/9- x, [-3, 3]0 /) = 10(5 - 3x+ 16),ThinkdifferSupply the appropriate inequality sign for the indicated[0, 5]value of c.SL f(D =2 sin t, [0, 2]seleSign of g'(c)Function52. f)=2-+cos [0, 47]the+COSg'(0)1063. g(x) f(x) +573.fisg'(-5)53. f(x) 3 sin064. g(x) 3f(x) - 3[0, 67]g'(-6)065. g(x)f(x)54. f(x) 2 sin 3x + 4 cos 3x, [0, 7T]'(0)066. g(x) = f(x - 10)Comparing FunctionsIn Exercises 55 and 56, usesymmetry, extrema, and zeros to sketch the graph of f. Howdo the functions f and g differ?Sketch the graph of the arbitrary67. Sketchinga Graphfunction f such that-4x3 3x0,x < 455. f(x) =f'(x){undefined,x2- 1x = 4.74. f 4g(x) = x(x2 - 3)56. f(t) cos2t- sin2 tIs the sum of two increasing68. Increasing Functionsfunctions always increasing? Explain.11 - 2 sin2 tg(t)69. Increasing Functions Is the product of twoincreasing functions always increasing? Explain.Think About It In Exercises 57-62, the graph of f is shownin the figure. Sketch a graph of the derivative of f. To print anenlarged copy of the graph, go to MathGraphs.com57.y58.y. HOW DO YOU SEE IT? Use the graph off' to (a) identify the critical numbers of f(b) identify the open intervals on which f isincreasing or decreasing, and (c) determinewhether f has a relative maximum, a relativeminimum, or neither at each critical number.75.42ff1++2 --2 -112 31++-2 -1+1 2(i)59.y60.(ii)f'2.f624-224+-4-22-2-6 8A 76.-6-4-4+4 6-4-6+2-4-2-2+61.62.y(ii)6-(iv)422 4-4 -22 4-22 462 4-2 +-6-4-2-6+4t22

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Asked Oct 21, 2019
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188
Chapter 3
Applications of Differentiation
71. Ana
f ha
Finding
Technology In Exercises 49-54, (a) use a computer algebra
an to diferentiate the function, (b) sketch the graphs of
2 3PEA om the same set of coordinate axes over the given
Sma (c) find the critical numbers of f in the open interval,
18 Gad chhe h.terval(s) on which f' is positive and the
)or which f'is negative. Compare the behavior of f
ca he ign off
Using
and
Analyzing
Derivatives
EXPLORING CONCEPTS
Transformations of Functions In Exercises 63-66,
assume that f is differentiable for all x. The signs of f
exin
are as follows.
72. An
f'(x) > 0 on (-0, -4)
f'(x) < 0 on (-4, 6)
f'(x)> 0 on (6, oo)
48 FO) = 2x/9- x, [-3, 3]
0 /) = 10(5 - 3x+ 16),
Think
differ
Supply the appropriate inequality sign for the indicated
[0, 5]
value of c.
SL f(D =2 sin t, [0, 2]
sele
Sign of g'(c)
Function
52. f)=2
-+cos [0, 47]
the
+COS
g'(0)
10
63. g(x) f(x) +5
73.fis
g'(-5)
53. f(x) 3 sin
0
64. g(x) 3f(x) - 3
[0, 67]
g'(-6)
0
65. g(x)f(x)
54. f(x) 2 sin 3x + 4 cos 3x, [0, 7T]
'(0)
0
66. g(x) = f(x - 10)
Comparing FunctionsIn Exercises 55 and 56, use
symmetry, extrema, and zeros to sketch the graph of f. How
do the functions f and g differ?
Sketch the graph of the arbitrary
67. Sketchinga Graph
function f such that
-4x3 3x
0,
x < 4
55. f(x) =
f'(x){undefined,
x2- 1
x = 4.
74. f
<0,
x > 4
g(x) = x(x2 - 3)
56. f(t) cos2t- sin2 t
Is the sum of two increasing
68. Increasing Functions
functions always increasing? Explain.
1
1 - 2 sin2 t
g(t)
69. Increasing Functions Is the product of two
increasing functions always increasing? Explain.
Think About It In Exercises 57-62, the graph of f is shown
in the figure. Sketch a graph of the derivative of f. To print an
enlarged copy of the graph, go to MathGraphs.com
57.
y
58.
y
. HOW DO YOU SEE IT? Use the graph of
f' to (a) identify the critical numbers of f
(b) identify the open intervals on which f is
increasing or decreasing, and (c) determine
whether f has a relative maximum, a relative
minimum, or neither at each critical number.
75.
4
2
f
f
1
++
2 -
-2 -1
1
2 3
1
++
-2 -1
+
1 2
(i)
59.
y
60.
(ii)
f'
2.
f
6
2
4
-2
2
4
+
-4-2
2
-2-
6 8
A 76.
-6-4
-4+
4 6
-4
-6+
2
-4
-2
-2+
61.
62.
y
(ii)
6-
(iv)
4
2
2 4
-4 -2
2 4
-2
2 46
2 4
-2 +
-6-4
-2
-6+
4t
2
2
help_outline

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188 Chapter 3 Applications of Differentiation 71. Ana f ha Finding Technology In Exercises 49-54, (a) use a computer algebra an to diferentiate the function, (b) sketch the graphs of 2 3PEA om the same set of coordinate axes over the given Sma (c) find the critical numbers of f in the open interval, 18 Gad chhe h.terval(s) on which f' is positive and the )or which f'is negative. Compare the behavior of f ca he ign off Using and Analyzing Derivatives EXPLORING CONCEPTS Transformations of Functions In Exercises 63-66, assume that f is differentiable for all x. The signs of f exin are as follows. 72. An f'(x) > 0 on (-0, -4) f'(x) < 0 on (-4, 6) f'(x)> 0 on (6, oo) 48 FO) = 2x/9- x, [-3, 3] 0 /) = 10(5 - 3x+ 16), Think differ Supply the appropriate inequality sign for the indicated [0, 5] value of c. SL f(D =2 sin t, [0, 2] sele Sign of g'(c) Function 52. f)=2 -+cos [0, 47] the +COS g'(0) 10 63. g(x) f(x) +5 73.fis g'(-5) 53. f(x) 3 sin 0 64. g(x) 3f(x) - 3 [0, 67] g'(-6) 0 65. g(x)f(x) 54. f(x) 2 sin 3x + 4 cos 3x, [0, 7T] '(0) 0 66. g(x) = f(x - 10) Comparing FunctionsIn Exercises 55 and 56, use symmetry, extrema, and zeros to sketch the graph of f. How do the functions f and g differ? Sketch the graph of the arbitrary 67. Sketchinga Graph function f such that -4x3 3x 0, x < 4 55. f(x) = f'(x){undefined, x2- 1 x = 4. 74. f <0, x > 4 g(x) = x(x2 - 3) 56. f(t) cos2t- sin2 t Is the sum of two increasing 68. Increasing Functions functions always increasing? Explain. 1 1 - 2 sin2 t g(t) 69. Increasing Functions Is the product of two increasing functions always increasing? Explain. Think About It In Exercises 57-62, the graph of f is shown in the figure. Sketch a graph of the derivative of f. To print an enlarged copy of the graph, go to MathGraphs.com 57. y 58. y . HOW DO YOU SEE IT? Use the graph of f' to (a) identify the critical numbers of f (b) identify the open intervals on which f is increasing or decreasing, and (c) determine whether f has a relative maximum, a relative minimum, or neither at each critical number. 75. 4 2 f f 1 ++ 2 - -2 -1 1 2 3 1 ++ -2 -1 + 1 2 (i) 59. y 60. (ii) f' 2. f 6 2 4 -2 2 4 + -4-2 2 -2- 6 8 A 76. -6-4 -4+ 4 6 -4 -6+ 2 -4 -2 -2+ 61. 62. y (ii) 6- (iv) 4 2 2 4 -4 -2 2 4 -2 2 46 2 4 -2 + -6-4 -2 -6+ 4t 2 2

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