Can you please help me with the problems that are circled and also the two questions on the last scan? Thanks so much!!! :)

SECTION 3.2 The Derivative as a Function 143 46. Of the two functions f and g in Figure 12, which is the deriva- 51. Use the table of values of f to determine which of (A) or (B) in live of the other? Justify your answer. Figure 15 is the graph of f'. Explain. 0 0.5 I 1.5 2 2.5 3 3.5 4 f(x) 10 55 98 139 177 210 237 257 268 FIGURE 12 (A) (B) 47. Assign the labels y = f(x), y = g(x). and y = h(x) to the graphs in FIGURE 15 Which is the graph of f'? Figure 13 in such a way that f'(x) = g(x) and g'(x) = h(x). 52. Let R be a variable and r a constant. Compute the derivatives: (a) d - R d (b) JR' (c) Compute the derivatives, where c is a constant. (a) d (b) "(5z + 4cz?) (A) (B) (C) (9cy) - 24c) dy FIGURE 13 54. Find the points on the graph of f(x) = 12x - x where the tangent line is horizontal. 48. Prove each of the following using the definition of the derivative. (a) The First-Power Rule: 7 x = 1 55. Find the points on the graph of y = x2 + 3x - 7 at which the slope of the tangent line is equal to 4. (b) The Constant Rule: fc = 0 49. Use the rules in Exercise 48 and the Linearity Rules to prove the first 56. Find the values of x where y = x and y = x2 + 5.x have parallel tan- part of Theorem 1 in Section 3.1. gent lines. 50. According to the Peak Oil Theory. first proposed in 1956 by geophysi 57. Determine a and b such that cist M. Hubbert, the total amount of crude oil @(1) produced worldwide up to time / has a graph like that in Figure 14 p (x ) = x2+ ax + b a) Sketch the derivative Q'() for 1900 r s 2150. What does O'(t) epresent? satisfies p(1) = 0 and p'(1) = 4. (b) In which year (approximately) does Q'(t) take on its maximum value? What is L = lim O()? And what is its interpretation? 58. Find all values of x such that the tangent line to What is the value of lim O'(r)? ) = 4x7 + Ilx + 2 is steeper than the tangent line to y = x3. Q (trillions of barrels) 2.3- 59. Let f(x) = x3 - 3.x + 1. Show that f'(x) 2 -3 for all x and that, for 20 every m > -3, there are precisely two points where f'(x) = m. Indicate the position of these points and the corresponding tangent lines for one 1.5 value of m in a sketch of the graph of f. 1.0 60. Show that the tangent lines to y = jx - x- at x = a and at x = b are parallel if a = bora + b = 2. 05 61. Compute the derivative of f(x) = x3/ using the limit definition. Hint: Show that 1900 1950 2000 2050 2100 2150 1 (year) f( x + h) - f(x)_(xth )3-x3 h FIGURE 14 Total oil production up to time t.144 CHAPTER 3 DIFFERENTIATION 62. Compute the derivative of f(x) = x1/) using the limit definition. 70. Match functions (A)-(C) with their derivatives (1)-n); Hint: Multiply the numerator and denominator in the difference quotient ( + 1) 2/3+ (x+ 1)1/3x 1/3+x213 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) f(x) = xe' (b) f ( x ) = xe' P 64. The average speed (in meters per second) of a gas molecule is 8RT where T is the temperature (in kelvins). M is the molar mass (in kilograms per mole). and R = 8.31. Calculate dung/dT 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 = why where L is the luminosity of the sun and equals 3.9 x 106 watts. What is the derivative of b with respect to d at the earth's distance from (C) the sun (1.5 x 10" m)? FIGURE 16 66. A power law model relating the kidney mass A" in mammals (in kilo- seems) to the body mass m (in kilograms) is given by A" = 0.007m085 71. Make a rough sketch of the graph of the derivative of the Calculate d A /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), one 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 dT where & is a constant. Estimate d P/dT for T = 303. 313, 323. 333. 343 using the data and the symmetric difference approximation (A) (B) dP P P(T + 10) - P(T - 10) dT FIGURE 17 20 73. Sketch the graph of f(.x) = x (x]. Then show that f'(0jess 74. Determine the values of x at which the function in Figure It 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 A? 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. 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 18-uploads.s3-us Thursday, Sept 15) 1 / 1 - 100% + . Use the limit definition of derivative and the trig identity sin(A+ B) = sin A cos B+cos A sin B to compute the derivative of f(x) = sin(x) at x = 3. . Find the values of t where the graphs of the functions y = 13 and y = }-2t have perpendicular tangents