268 CHAPTER 3 Differentiation Rules 65. At what points on the curve y = sin x + cos x, 82. (a) Graph the function f(x) = x - 2 sin x in the viewing rectangle [0, 8] by [ -2, 8]. O a, is used to model the concen- ( b) If P(x) = f(x)g(x), find P'(2). tration at time t of a drug injected into the bloodstream. (c) If Q(x) = f(x) /g(x), find Q'(1). (a) Show that lim,-. C(t) = 0. (d) If C(x) = f(g(x)), find C'(2). (b) Find C'(t), the rate of change of drug concentration in 70. If f and g are the functions whose graphs are shown, let the blood. P (x) = f(x)g(x), e(x) = f(x)/g(x), and C(x) = f(g(x)). (c) When is this rate equal to O? Find (a) P'(2), (b) Q'(2), and (c) C'(2). 87. An equation of motion of the form s = Ae-"cos(wt + 8) represents damped oscillation of an object. Find the veloci and acceleration of the object. 88. A particle moves along a horizontal line so that its coor- dinate at time t is x = Vb2 + c2t2 , t > 0, where b and c are positive constants. (a) Find the velocity and acceleration functions. (b) Show that the particle always moves in the positive upo bait ro-da direction. 89. A particle moves on a vertical line so that its coordinate 71-78 Find f' in terms of g'. .0a time t is y = 13 - 12t + 3, t 2 0. 71. f (x) = x2g(x) 72. f(x) = g(x2) (a) Find the velocity and acceleration functions. 73. f (x) = [g(x) 12 74. f(x) = g(g(x)) (b) When is the particle moving upward and when is it moving downward? 75. f (x) = g(ex ) 76. f(x) = eg(x) (c) Find the distance that the particle travels in the tim 77. f(x) = In |g(x) | 78. f(x) = g(In x) interval 0 = t = 3. A (d) Graph the position, velocity, and acceleration fun for 0 = t = 3. 79-81 Find h' in terms of f' and g'. (e) When is the particle speeding up? When is it slov down? 79. h(x) f(x)g(x) 80. h(x) = f(x) f(x) + g(x) g(x) 90. The volume of a right circular cone is V = 3Trzh, w r is the radius of the base and h is the height. 81. h(x) = f(g(sin 4x)) (a) Find the rate of change of the volume with resp the height if the radius is constant