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