subscribers to an online fash 37. foll magazine. Due to competition from an 8-month-old competing magazine 38. The constant function g (x) = 5e" is an anti number C(t) of subscribers is expected to decrease at the rate of In Problems 39-42, could the three graphs in ed C' (t) = -600+1/3 derivatives of the same function? Explain. 40. subscribers per month, where t is the time in months since the competing mags 39. Zine began publication. How long will it take until the number of subscribers to the online fashion magazine drops to 46,000? Exercises 5.1 w Skills Warm-up Exercises 42 . In Problems 1-8, write each function as a sum of terms of the 17 . x 3 dx 18 . * 4 dx 41. form ax", where a is a constant. (If necessary, review Section A.6). 1. f (x ) = 2. f (x ) = 19 . /10 x 3 / 2 dx 20 . 8 x 1 /3 dx 3 . f (x ) = = 3x - 2 x + 5x - 1 * 5 4. f (x ) = 21 . 13dz 22 . 5 . f (x ) = Vx + 5 Vx 6. f ( x ) = Vx- - In Problems 43-54, find each indefin Vx 23 . 16e " du 24. ( 5 e" du differentiation.) 7 . f ( x ) = Vx ( 4 + x - 3x2) 8. f (x ) = Vx(1 - 5x + x3 ), A In Problems 9-24, find each indefinite integral. Check by 25. Is F ( x) = (x + 1) (x + 2) an antiderivative of 43. (5x ( 1 - x ) dx differentiating. f(x) = 2x + 3? Explain. 000.TS 26. Is F(x) = (2x + 5) (x - 6) an antiderivative of 45. du 9. / 7 dx 10 . 10 dx f(x) = 4x - 7? Explain. Vu 27. Is F(x) = 1 + x In x an antiderivative of f(x) = 1 + Ix! Explain. 47 . dx 4x3 11. / 8x dx 14 x dx 28. Is F(x) = x In x - x + e an antiderivative of f(x) = Inx! Explain. ( 2 x + 1 ) 3 49. 4 + u du u 13 . 9x 2 dx 14. 15 x 2 dx 29. Is F(x) = - an antiderivative of f(x) = (2 + 1)? 3 Explain. fixed cost (3.x - 2) 4 51. ( 5 ex + 4 ) dz 16. / x8 dx 30. Is F(x) = an antiderivative of f(x) = (3x - 2)? 15. / x5 dx Explain. 431. Is F (x) = et/s an antiderivative of f(x) = et? Explain. SECTION 5.1 Antiderivatives and Indefinite Integrals 333 32. Is F(x) = (e - 10) (e* + 10) an antiderivative of f (x ) = 2021? Explain. 53. / (32 2 - 2 ) dx 54. / ( 48 + 2) dx B In Problems 35-38, discuss the validity of each statement. If the statement is always true, explain why. If not, give a In Problems 55-62, find the particular antiderivative of each counterexample. derivative that satisfies the given condition. 33. The constant function f(x) = T is an antiderivative of the 55. C' (x) = 9x2 - 20x; C( 10) = 2,500 constant function k (x) = 0. 56. R' (x) = 500 - 0.4x; R(0) = 0 34. The constant function k (x) = 0 is an antiderivative of the 57, dx 10 constant function f(x) = dt Vix ( 1 ) = 25 58. OR dt 3 ; R (1 ) = 50 35. If'n is an integer, then x" + 1 / (n + 1) is an antiderivative of x". 59. f' (x ) = 4x-2 - 3x 1 + 2;5(1 ) = 5 36. The constant function k(x) = 0 is an antiderivative of itself. 60. f' (x) = x1 - 2x 2 + 1;f(1) =5 dy ore days. 37. The function h(x) = Se is an antiderivative of itself. 61. dt = 6e' - 7; (0) = 0 62. - dy = 3 - 2e'; y(0) = 2 38. The constant function g (x) = 5e" is an antiderivative of itself. 63. Find the equation of the curve that passes through (2, 3) if online fash- agazine, the In Problems 39-42, could the three graphs in each figure be anti- its slope is given by derivatives of the same function? Explain. 23 = 4x - 3 40. dx 39. for each x. 64. Find the equation of the curve that passes through (1, 3) if ing maga- its slope is given by cribers to dx = 12x2 - 12x for each x. In Problems 65-70, find each indefinite integral. 41. 42. 65. 2x - * dx 7 3 66. x x dx 67. (x - 2x dx 68. 1 2 dx 69. (xex - 2x dx 7 2 70 . [1 - xe dx In Problems 71-74, find the derivative or indefinite integral as In Problems 43-54, find each indefinite integral. (Check by indicated. lifferentiation.) 71. ( 6.x 2 - 7x + 2 ) dx 72 . @ ( 4x 2 - 3x + 5 ) doc 13 . / 5x ( 1 - * ) dx 44. x2 ( 1 + * 3 ) dx of d [ 2 - Int at 73 . It / 3+ + 2 74 . " ( @ 2 - 7 + 1 ) dt ve of 45 . du 46. dt 75. Use differentiation to justify the formula = 1 + Inx? 000,05 0 47, dx 6 dm 48. int It 4x 3 m - ff ( x) = In x? provided that n * -1. 19. " du 50 . 1 - y 76. Use differentiation to justify the formula u 3 y - dj = (2x + 1) 2? ( edx = ex + c 51. / ( 5 03 + 4 ) do o nhitutor - 52. ) = (3x - 2) 3