Historical Problems AN23 45. (a) T x 0 6 12 18 24 60 54 48 42 36 30 The relation appears to be cubic. (b) 1.2°/h (c) − ° 0.75 h (d) ( ) = − + − + T x x x x 0.0082 0.2868 2.4176 54.5857; 3 2 ( ) ≈ ° T 17 56.09 F (e) 60 46 0 27 (f) The predicted temperature at midnight is 54.6°F. 47. (a) (b) (c) (d) As more terms are added, the values of the polynomial function get closer to the values of f . The approximations near 0 are better than those near −1 or 1. 49. (a) Vertical scale may vary. (0, 0) (2c, 0) (b, 0) x y (b) ( ) −c, 0 and ( )b 0, (c) −c and 0 (d) ( )∞ b, (e) ( ) −∞ − −b , 4 (f) Decreasing 50. { } < − > x x x 2 3 or 4 3 , or ( ) ( ) −∞ − ∪ ∞ , 2 3 4 3 , 51. = − y x 1 3 52. ( ) 7 4 , 25 8 53. 140 54. 17 4 55. [ )∞ 4, 56. 3 57. Center: ( ) −2, 1 ; radius: 4 58. even 59. 6.25 years Historical Problems (page 226) 1. ( ) ( ) ( ) ( ) ( ) − + − + − + = − + −+ − ++−+= + − + − + = = − = − + + + = x b b x b c x b d x bx b x b bx b x b cx bc d x c b x b bc d p c b q b bc d x px q 3 3 3 0 3 27 2 3 9 3 0 3 2 27 3 0 Let 3 and 2 27 3 . Then 0. 3 2 3 2 2 3 2 2 3 3 2 3 2 3 3 2. ( ) ( ) + + + + = + + + + + + = = − − − + + + + = + = − H K p H K q H HK HK K pHpKq HK p H pHpKK pHpKq H K q 0 3 3 0 Let 3 . 0, 3 3 2 2 3 3 3 3 3 3. ( ) ( ) ( ) ( ) ( ) ( ) ( ) = − = − + − = − − = − − = − + − = = − ± − − ⋅ = − ± + = − ± + = − + + HK p K p H H p H q H p H q H p qH H qH p H q q p H q q p H q q p H q q p 3 3 3 27 27 27 27 27 0 27 27 4 27 2 27 2 27 2 27 4 27 2 27 2 4 27 2 4 27 3 3 3 3 3 6 3 3 6 3 3 3 2 3 3 2 2 2 2 3 2 2 3 2 3 2 3 3 Choose the positive root for now. 4. + = − = − − = − − − + + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − − + = − − + H K q K q H K q q q p K q q p K q q p 2 4 27 2 4 27 2 4 27 3 3 3 3 3 2 3 3 2 3 2 3 3 5. = + = − + + + − − + x H K x q q p q q p 2 4 27 2 4 27 2 3 3 2 3 3 (Note that if we had used the negative root in 3, the result would have been the same.) 6. = x 3 7. = x 2 8. = x 2
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