Can you please help me solve 66, 73, 87 on 3.2 and 3 on 3.3. 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)