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Answer 43-46 only.

5.2 The Indefinite Integral 331 41. Use a graphing utility to generate some representative in- 58. In each part, use a CAS to solve the initial-value problem. tegral curves of the function f(x) = 5x* - sec x over the (a) dx = x' cos 3x, y(1/2) = -1 interval (-7/2. x/2). 42. Use a graphing utility to generate some representative inte- dy gral curves of the function f(x) = (x - 1)/x over the in- ( dx (4 + x2)3/2 . y(0) = -2 terval (0, 5). 59. (a) Use a graphing utility to generate a slope field for the dif- ferential equation dy/dx = x in the region -5 5 x 5 5 43-46 Solve the initial-value problems. and -5 S y s 5 43. (a) N = Vx. y(1) = 2 (b) Graph some representative integral curves of the func- dx tion f(x) = x. (c) Find an equation for the integral curve that passes ( b ) NI - di " = sin / + 1, y () = through the point (2, 1). dy x+1 (c) VX . y(1) = 0 60. (a) Use a graphing utility to generate a slope field for dx the differential equation dy/dx = e'/2 in the region dy 1 -1 5 x $ 4 and -1 S y s 4. 44. (a) dx (2r)3 . )'(1 ) =0 (b) Graph some representative integral curves of the func- tion f(x) = ex/2. ( b ) di = sec2 1 - sint, y () = 1 (c) Find an equation for the integral curve that passes (c) dy = xVx, y(0) = 0 through the point (0, 1). dx 45. (2) - dy -= 4e', y(0) = 1 (b) = = 2. y(-1) =5 51-64 The given slope field figure corresponds to one of the differential equations below. Identify the differential equation 46. (a) dy 3 that matches the figure, and sketch solution curves through the =0 di highlighted points. ( b ) dy x2 - 1 dy = 2 (b ) dy =-X dx x2 + 1 ' y'() = (a) dx dx dy 47-50 A particle moves along an s-axis with position function (C) 7x = x - 4 (d) =ex/3 s = s(r) and velocity function u(() = s'(1). Use the given in- 61. 62. formation to find s(1). 47. v(1) = 321; s(0) = 20 48. u(1) = cost; s(0) = 2 49. v(1) = 3 7; s(4) = 1 50. v(1) = 3e'; s(1) = 0 51. Find the general form of a function whose second derivative -2 / is x. [Hint: Solve the equation f"(x) = vx for f(x) by integrating both sides twice.] 52. Find a function f such that f"(x) = x + cosx and such that f(0) = 1 and f'(0) = 2. [Hint: Integrate both sides of 63. 64. the equation twice.] 53-57 Find an equation of the curve that satisfies the given conditions. 53. At each point (x, y) on the curve the slope is 2x + 1; the curve passes through the point (-3, 0). 54. At each point (x, y) on the curve the slope is (x + 1)?; the curve passes through the point (-2, 8). 55. At each point (x, y) on the curve the slope is - sin x; the curve passes through the point (0, 2). FOCUS ON CONCEPTS 56. At each point (x, y) on the curve the slope equals the square of the distance between the point and the y-axis; the point 65. Critique the following "proof" that an arbitrary constay (-1, 2) is on the curve. must be zero: 57. At each point (x, y) on the curve, y satisfies the condition d' y/dx' = 6x; the line y = 5 - 3.x is tangent to the curve C= 0dx = 0. 0dx =0 0dx =0 at the point where x = 1