MATHEMATICS 571 Answers to Final Examination Review Problems 1. a) 2a2 + 4ab + 2b2 − 5a− 5b b) 4x2 2. a) x ≥ 4 or x ≤ −1 b) x 6= 0, 1,−1 3. 3 units to the left and 4 units down 4. a) f ◦ g(x) = 1 x− 1 , x ≥ 0, x 6= 1; g ◦ f(x) = 1√ x2 − 1 , x > 1, or x < −1 b) f ◦ g(x) = x 2 (x− 1)2 + 1, x 6= 1; g ◦ f(x) = x2 + 1 x2 , x 6= 0 5. a) (0, 0),no symmetry; b) no intercepts; origin c) (2, 0), (−2, 0), (0, 2); y-axis 6. a) DNE b) 3 7. a) − 1 b) − 12 c) 4 d) −∞ e) +∞ f) DNE g) 1 h) − 1 i) DNE j) 0 k) 1/3 l) DNE m) 0 n) 1 o) 1/3 p) 4 q) √ 3/2 r) − sinx 8. a) 15 b) 9/2 9. a) (−∞,∞) b) (−∞,−3] and (−3,∞) 10. a) x = 1, y = 1 b) x = −2, y = 1 c) y = 1, y = −1, x = 0 d) x = 2, y = 1 11. f(x) = x2 − 1 x− 1 , a = 1. f(1) DNE, but limx→1 f(x) = 2 12. − 112 13. 49 14. 73 24 15. −8 3 16. y = 3x 17. (2, 6) 1 10/00 rev. 18. y + 3 = (22/21)(x− 2) 19. − √ 3/9 20. 3 √ 2/8 21. − 3 √ 4/12 22. 1/2 23. − 8/3 24. 3 √ 3/4 25. (4 + 3 √ 2)/(3 + 2 √ 2) = √ 2 26. − √ 3 27. − 9/2 28. [3/4,∞) 29. (−3/2, 1] 30. (−4, 0), (3,∞) 31. (−∞,−2) and (0, 2) 32. 3, −4 33. None 34. At x = 2 √ 2− 3 35. At x = 0 36. a) 0 and − 16 36. b) 16 and 0 37. 192 ft, 6 sec, −128 ft/sec 38. 64 ft/sec, 288 ft 39. 5000 ft2 40. 80 41. 5 ft on each side 42. Overland for 5/8 mile 43. 2 in, 128 in3 44. 33 √ 5 mph 45. 1 ft/sec 46. 39 ft/min 47. 0.0702, 0.07 48. ± 0.06 ft3 49. 7 + 1/14 50. 0.04M 51. 15 meters 52. a) Yes, the function f(x) = x5 + 2x2 − 10x + 5 is continuous on [1, 2] and f(1) < 0 < f(2). b) Yes, the functionf(x) = x−√x is continuous on [4, 9] and f(4) < 5 < f(9). c) Yes, the function f(x) = x− cosx is continuous on [0, pi] and f(0) < 0 < f(pi). d) No, the function f(x) = x− tanx is not continuous at pi 2 . 53. f(x) = 2 3 x3 + 3 2 x2 − 8x + 47 6 2 10/00 rev. 54. f(x) = 1 3 x3 + 3 2 x2 + 2x 55. a) 4 b) 3 c) f(t) is increasing in [1, 4] and decreasing in [0, 1] and [4, 5] d) (2, 2) 56. f ′, the derivative of the function shown, is increasing on [0, 2] and is decreasing in [2, 5], as can be seen by noting the concavity of the function, which indicates the sign of the derivative of f ′(t). 57. a) When time is 12 seconds. b) 16 meters c) 4 meters d) 12 meters e) Forward; foot on the gas f) Backward; foot on the brake 58. a) 1 30 (3x2 + 4)5 + C b) 1 16 (2x4 + 4x)4 + C c) 195 64 d) − 3 x − 3 x2 − 2 3x3 + C e) 38 3 f) 1 3 (3x2 − 2x + 1)3/2 + C 3 10/00 rev. g) −1 3(x + 5)3 + C h) −1 12(10x2 + 2x + 40)3 + C i) 1 9 (6x3 + 1)3/2 + C j) 2 5 (2− x)5/2 − 4 3 (2− x)3/2 + C k) − 3 4 (3 + z−1)4/3 + C l) sin(secx) + C m) − 3 4 ( cos x 3 )4 + C n) 1 2 (tanx)2 + C 59. a) x2 − 8 b) 2x(x4 + 6)5/2 60. 4 15 61. 9 2 62. 48 5 63. 64pi 64. 96pi 5 65. a) 8pi b) 256pi 15 66. 100pi 3 67. 250 3 68. pi 240 69. 5 2 70. 2 pi 71. a) 28125pi b) 21093.75pi 72. a) 840 b) 952.5 c) 862.5 73. 1L, 2K, 3B, 4A, 5C, 6I, 7D, 8E, 9G, 10P, 11J, 12M, 13O, 14Q, 15R, 16H 4 10/00 rev. m1571ans.dvi