AN8 Answers: Chapter 2 57. 4 22 0 2 Increasing: [ ] [ ] − − 2, 1 , 1, 2 Decreasing: [ ] −1, 1 Local maximum: ( ) − = f 1 4 Local minimum: ( ) = f 1 0 59. 0.5 22 20.5 2 Increasing: [ ] [ ] − − 2, 0.77 , 0.77, 2 Decreasing: [ ] −0.77, 0.77 Local maximum: ( ) − = f 0.77 0.19 Local minimum: ( ) = − f 0.77 0.19 61. 0 26 220 4 Increasing: [ ] −3.77, 1.77 Decreasing: 6, 3.77 , 1.77, 4 [ ] [ ] − − Local maximum: ( ) = − f 1.77 1.91 Local minimum: ( ) − = − f 3.77 18.89 63. 8 23 0 2 Increasing: [ ] [ ] −1.87, 0 , 0.97, 2 Decreasing: 3, 1.87 , 0, 0.97 [ ] [ ] − − Local maximum: ( ) = f 0 3 Local minima: ( ) − = f 1.87 0.95, ( ) = f 0.97 2.65 65. (a) −4 (b) −8 (c) −10 67. (a) 15 (b) −3 (c) 9 69. (a) 5 (b) = − y x5 2 71. (a) −1 (b) = − y x 73. (a) 4 (b) = − y x4 8 75. (a) Odd (b) Local maximum value is 54 at = − x 3. 77. (a) Positive (b) Increasing (c) Positive (d) Negative (e) Decreasing (f) Negative (g) 0 79. (a) Even (b) Local maximum value is 25 at = − x 2. (c) 50.4 sq. units 81. (a) 0 2500 40 0 (b) 10 riding lawn mowers (c) $239 mower 83. (a) On average, the population is increasing at a rate of 0.036 g h from 0 to 2.5 h. (b) On average, from 4.5 to 6 h, the population is increasing at a rate of 0.1 g h. (c) The average rate of change is increasing over time. 85. (a) 1 (b) 0.5 (c) 0.1 (d) 0.01 (e) 0.001 (f) 2 23 22 3 y = x y = 0.5x y = 0.1x y = 0.01x y = 0.001x y = x2 (g) They are getting closer to the tangent line at ( ) 0, 0 . (h) They are getting closer to 0. 87. (a) 2 (b) 2; 2; 2; 2 (c) = + y x2 5 (d) 10 210 210 10 (b) Domain: { } − ≤ ≤ x x 3 3 or [ ] −3, 3 ; Range: { } − ≤ ≤ y y 1 2 or [ ] −1, 2 (c) Increasing on [ ] 2, 3 ; Decreasing on [ ] −1, 1 ; Constant on [ ] − − 3, 1 and [ ] 1, 2 (d) Neither 33. (a) 0; 3 (b) −2, 2; 0, 0 35. (a) π 2 ; 1 (b) π − − 2 ; 1 37. Odd 39. Even 41. Odd 43. Neither 45. Even 47. Odd 49. Absolute maximum: ( ) = f 1 4; absolute minimum: ( ) = f 5 1; local maximum: ( ) = f 3 3; local minimum: ( ) = f 2 2 51. Absolute maximum: ( ) = f 3 4; absolute minimum: ( ) = f 1 1; local maximum: ( ) = f 3 4; local minimum: ( ) = f 1 1 53. Absolute maximum: none; absolute minimum: ( ) = f 0 0; local maximum: ( ) = f 2 3; local minima: ( ) = f 0 0 and ( ) = f 3 2 55. Absolute maximum: none; absolute minimum: none; local maximum: none; local minimum: none 103. Not necessarily. It just means ( ) ( ) > f f 5 2 . 105. 6 15 106. − + a ab b 16 8 2 2 107. ( ) = + C x x 0.80 40 108. 8.25 days 109. ( ) ( ) − + + = x y 3 2 3 2 2 2 110. ( ) ( ) ( ) − + − + x x x x x 6 2 3 1 8 10 1 2 2 5 5 3 111. ( ) − − 7 10 , 3 2 112. { } −5, 1 3 113. { } −2, 1 114. ( ) ( ) = − − = − − D x y x y y x xy x y x y 2 2 2 2 2 2 2.4 Assess Your Understanding (page 108) 4. ( ] −∞, 0 5. piecewise-defined 6. T 7. F 8. F 9. b 10. a 11. C 13. E 15. B 17. F 89. (a) + + x h 2 2 (b) 4.5; 4.1; 4.01; 4 (c) = − y x 4.01 1.01 (d) 10 25 22 3 91. (a) + − x h 4 2 3 (b) 2; 1.2; 1.02; 1 (c) = − y x 1.02 1.02 (d) 10 22 25 4 93. (a) ( ) − +x h x 1 (b) − − − − 2 3 ; 10 11 ; 100 101 ; 1 (c) = − + y x 100 101 201 101 (d) 2 22 22 4 95. − +2 19 3 99. At most one 101. Yes; the function ( ) = f x 0 is both even and odd. 19. y x 10 (4, 4) (0, 0) (24, 24) 10 21. (2, 8) (0, 0) (22, 28) y x 10 5 23. (1, 1) (21, 21) 1 2 2, y x 5 5 25. y 5 x (0, 0) 5 (1, 1) (21, 21) 27. (a) −9 (b) 4 (c) 7 29. (a) 0 (b) 4 (c) 6 (d) 26
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