Can you please help answer all of the questions. Thanks so much! :)

144 CHAPTER 3 DIFFERENTIATION 62. Compute the derivative of f(x) = x/ using the limit definition. 70. Match functions (A)-(C) with their derivatives ()-(Ill) in Figue Hint: Multiply the numerator and denominator in the difference quotient fat . 63. In each case use the limit definition to compute f'(0). and then find (A) the equation of the tangent line at x = 0. (a) S(x ) = xet b) fux)= xe 64. The average speed (in meters per second) of a gas molecule is 8RT V TM where 7 is the temperature (in kelvins), M is the molar mass (in kilograms per mole), and R = 8.31. Calculate dung/d at T = 300 K for oxygen, which has a molar mass of 0.032 kg/mol. 65. The brightness b of the sun (in watts per square meter) at a distance of d meters from the sun is expressed as an inverse-square law in the form b= where L is the luminosity of the sun and equals 3.9 x 1026 watts. What is the derivative of b with respect to d at the earth's distance from (C) (III) the sun (1.5 x 10 m)? FIGURE 16 66.) A power law model relating the kidney mass K in mammals (in kilo- sams) to the body mass m (in kilograms) is given by X = 0.007m0.85 71. Make a rough sketch of the graph of the derivative of the funchi Calculate d K /dm at m = 68. Then calculate the derivative with respect to Figure 17(A). m of the relative kidney-to-mass ratio K/m at m = 68. 72. Graph the derivative of the function in Figure 17(B). omitting p where the derivative is not defined. 67. The Clausius-Clapeyron Law relates the vapor pressure of water P (in atmospheres) to the temperature 7 (in kelvins): dp aT where & is a constant. Estimate d P/dT for 7 = 303, 313. 323, 333, 343 using the data and the symmetric difference approximation 0 (A) (B) dP P(7 + 10) - P(T - 10) FIGURE 17 dT 20 73. Sketch the graph of f(x) = x |.x|. Then show that f'(0) exists. 74. Determine the values of x at which the function in Figure 18 is: (2) T (K) 293 303 313 323 333 343 353 continuous and (b) nondifferentiable. P (atm) 0.0278 0.0482 0.0808 0.1311 0.2067 0.3173 0.4754 Do your estimates seem to confirm the Clausius-Clapeyron Law? What is the approximate value of k? 68. Let L be the tangent line to the hyperbola xy = 1 at x = a, where a > 0. Show that the area of the triangle bounded by L and the coordinate axes does not depend on a. 3 69. In the setting of Exercise 68. show that the point of tangency is the midpoint of the segment of L lying in the first quadrant. FIGURE 18SECTION 3.2 The Derivative as a Function 145 GU In Exercises 75-80, zoom in on a plot of f at the point (a, f(a)) 83. Calculate the subtangent of and state whether or not f appears to be differentiable at x = a. If it is nondifferentiable, state whether the tangent line appears to be vertical or f (x) = x2 +3x atx=2 does not exist. 84. Show that the subtangent of f(x) = e is everywhere equal to 1. 75. f(x) = (x - 1)|x|. a=0 76. f(x) = (x - 3)5/3. a = 3 85. Prove in general that the subnormal at P is If'(x) f (.x)1. 77. f(x) = (x - 3)1/3, a = 3 78. f(x) = sin(x'/3), a =0 86. Show that PQ has length If (x)IVI + f'(x)-2. 79. f(x) = |sinx|, a = 0 80. f(x) = |x - sinxl. a = 0 81. Find the coordinates of the point P in Figure 19 at which the tangent y = f(x)/ line passes through (5, 0). f (x) = 9 - x2 P = (x. f (x)) Tangent line FIGURE 20 FIGURE 19 87. Prove the following theorem of Apollonius of Perga (the Greek math- ematician born in 262 BCE who gave the parabola, ellipse. and hyperbola 82. (GU Plot the derivative f' of f(x) = 2x3 - 10x- for x > 0 and their names): The subtangent of the parabola y = x2 at x = a is equal observe that f'(x) > 0. What does the positivity of f'(x) tell us about the to a /2. graph of f itself? Plot f and confirm this conclusion. 88. Show that the subtangent to y = x at x = a is equal to a/3. Exercises 83-86 refer to Figure 20. Length OR is called the subtangent at 89. Formulate and prove a generalization of Exercises 87 and 88 for P, and length RT is called the subnormal. y= x. Further Insights and Challenges 90. Two small arches have the shape of parabolas. The first is the graph of f(x) = 1 - x2 for -1 0) or FIGURE 21 right (if c 0 and f'(x) 0 such that the tangent line to the graph of (b) Calculate =0 dR f (x) = red atx = a (c) If Couture can implement only one leg (A or B) of its 1 = 0, which choice will grow revenue most rapidly? passes through the origin (Figure 5). 60. The tip speed ratio of a turbine is the ratio R = T / W. speed of the tip of a blade and W is the speed of the wind have found empirically that a turbine with " blades extragy power from the wind when R = 2x.) Calculate d R/d; ( if W = 35 km/h and W decreases at a rate of 4 km/h per mi tip speed has constant value 7 = 150 km/h. 61. The curve y = 1/(x2 + 1) is called the witch of Agnesi (F the Italian mathematician Maria Agnesi (1718-1799). This . is the result of a mistranslation of the Italian word la versie, "that which turns." Find equations of the tangent lines at x FIGURE 5 58. Current / (amperes), voltage V (volts), and resistance R (ohms) in a circuit are related by Ohm's Law. I = V/ R. di (a) Calculate dR R=6 if V is constant with value V = 24. dv FIGURE 6 The witch of Agnesi. (b) Calculate dR R=6 if / is constant with value / = 4. 62. Let f(x) = 8(x) = x. Show that (f/8)' # f'/8'. 59. The revenue per month earned by the Couture clothing chain at time ? is R(1) = N(1)S(1). where N() is the number of stores and S(1) is average 63. Use the Product Rule to show that ( f2) = 2ff'. revenue per store per month. Couture embarks on a two-part campaign: (A) to build new stores at a rate of five stores per month, and (B) to use 64. Show that (f 3) = 3 f2 f'. advertising to increase average revenue per store at a rate of $10.000 per month. Assume that N(0) = 50 and S(0) = $150,000. (a) Show that total revenue will increase at the rate dR di = 5S(1) + 10,000N(1) Further Insights and Challenges 65. Let f. g. h be differentiable functions. Show that (fgh)'(x) is equal to 68. Prove the Quotient Rule using Eq. (7) and the Product Rule f' (x)(x)h(x) + f(x)s'(x)h(x)+f(x)(x)h'(x) 69. Use the limit definition of the derivative to prove the follow case of the Product Rule: Hint: Write fgh as f(gh). ( xf ( x ) ) = f (x ) + xf' ( x ) 66. Prove the Quotient Rule using the limit definition of the derivative. 67. Derivative of the Reciprocal Use the limit definition to prove 70. Use the limit definition of the derivative to prove the follow case of the Quotient Rule: S'(x) f = ( x ) 7 ( 1() ) = xf' () - 1(x) 7 2 Hint: Show that the difference quotient for 1/f(x) is equal to 71. The Power Rule Revisited If you are familiar with prod f (x) - f(x +h) tion, use induction to prove the Power Rule for all whole numb that the Power Rule holds for n = 1: then write r" as r . *- hf(x)f( x+h) Product Rule.S E CT I 0 N 3.4 Rates 01 Change 153 Eta-rises 72 and 73: A basic fact of algebra states that r: is a roar afa poly- "nmlui I t'fand only lfftx) 2 (x c)g(.r) for some polynomial 3. We say r is a multiple mat it Its) = (.l' c)2lt(.r). where it is a polynomial. Show that c is a multiple root off item! only if: is a root of both f f'. 1'3. Use Exercise 72 to determine whether :- = 1 is a multiple rout. (a)_r5+lx"4x38x2x+2 .r (b) 1:" +x' 5x2 - 3x + 2 74- IEEi Figure 7 is the graph of a polynomial with roots at A. B. FIGURE 1 and C. Which of these is a multiple mot? Explain your reasoning using Exercise 72. 3.4 Rates of Change In this section. we pause from building tools for computing the derivative and instead focus on the derivative as a rate of change. panicularly in applied settings. Recall the notation for the average rate of change of a functiOn y = f(x) over an interval [10.11]: Ay = change in y = f(.r1) fire) A.\\' = change in x = x] .ro A ' (xi) (x0) average rate of change = J- : L-f Ax x1 .m We usually omit the word "instantaneous" In our prior discussion in Section 2.1, limits and derivatives had not yet been introduced. and refer to the derivative Simply as the Now that we have them at our disposal. we can dene the instantaneous rate of change rate of change. This is shorter and also of y with respect '0 x at .t' = In: more accurate when applied to general rates. because the term \"instantaneous" would seem to refer only to rates with f . fi-'Jtl fun) respect to time. _ m _ Keep in mind the geometric interpretations: The average rate of change is the slope of the secant line (Figure l). and the instantaneous rate of change is the slope of the tangent line (Figure 2). (1th.\" Val) it" FlGURE t The average rate of change over FIGURE 2 The instantaneous rate of 941.11] is the slope of the secant line. change 1\" It) is the Sl013': 0f the tangent line