Question 53, 59, 61,65,69,71 Chapter applications of differentiation

d 43. Investigate the family of functions f (x) In(sin x + C). What features do the members of this family have in common? How do they differ? For which values of C is f continuous on (00, u)? For which values of C does f have no graph at all? What happens as C + 00? 44. Investigate the family of functions f(x) cxe . What hap- pens to the maximum and minimum points and the inflection points as c changes? Illustrate your conclusions by graphing several members of the family. 45. Show that the equation x 101 + x51 + x 1 0 has exactly one real root. 46. Suppose that f is continuous on [0, 4], f (0) == 1, and 2 < f '(x) < 5 for all x in (0, 4). Show that 9 sf(4) 21. 47. By applying the Mean Value Theorem to the function f(x) = x 1/5 on the interval [32, 33], show that 2 < < 2.0125 48. For what values of the constants a and b is (1, 6) a point of inflection of the curve y = x3 + ax2 + bx + l? 49. Letg(x) where f is twice differentiable for all x, f '(x) > 0 for all x # 0, and f is concave downward on (00, 0) and concave upward on (0, 00). (a) At what numbers does g have an extreme value? (b) Discuss the concavity of g. 50. Find two positive integers such that the sum of the first number and four times the second number is 1000 and the product of the numbers is as large as possible. 51. Show that the shortest distance from the point (Xl, h) to the straight line Ax + By + C 0 is IAXI + BYI + cl 52. Find the point on the hyperbola xy 8 that is closest to the point (3, 0). . Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r. 54. Find the volume of the largest circular cone that can be inscribed in a sphere of radius r, 55. In AABC, D lies on AB, CD LAB, IAD! IBDI 4 CITI, and I CDI 5 cm. Where should a point P be chosen on CD so that the sum I PAI + IPBI + I PCl is a minimum? points and inflection points of f. e velocity of a wave of length L in deep water is 42, If f (x) In(2x + x sin x), use the graphs of f, f', and f" to estimate the intervals of increase and the inflection points of f on the interval (0, 15]. 2x3 + 3X2 58. 59. 60. 61. 62. 63. 64. 6 where K and C are known positive constants. What is the length of the wave that gives the minimum velocity? A metal storage tank with volume V is to be constructed in the shape of a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal? A hockey team plays in an arena with a seating capacity of 15,000 spectators. With the ticket price set at $12, average attendance at a game has been 11,000. A market survey indi- cates that for each dollar the ticket price is lowered, average attendance will increase by 1000. How should the owners of the team set the ticket price to maximize their revenue from ticket sales? A manufacturer determines that the cost of making x units of a commodity is C(x) 1800 + 25x 0.2x2 + 0.001x3 and the demand function is p(x) 48.2 0.03x. (a) Graph the cost and revenue functions and use the graphs to estimate the production level for maximum profit. (b) Use calculus to find the production level for maximum profit. (c) Estimate the production level that minimizes the average cost. se Newton's method to find the root of the equation x4 + 3x2 3x 2 0 in the interval [l, 21 correct to six decimal places. Use Newton's method to find all roots of the equation 2 3x + 1 correct to six decimal places. sin x x Use Newton's method to find the absolute maximum value of t2 correct to eight decimal the function f(t) cos t + t places. Use the guidelines in Section 4.5 to sketch the curve y = x sin 0 x < 2m Use Newton's method when necessary. 65-72 lill Find f. 66. f '(x) 8x 3 sec2x 67. f'(x) 70. f'(u) = 4 2t 3 sin t, f(0) 2 u 6x + 48x2, f (0)