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# ISM_T11_C04_I - 280 Chapter 4 Applications of Derivatives...

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280 Chapter 4 Applications of Derivatives 40. The values of the first derivative indicate that the curve is rising on and ( ), and falling on ( ) ˆ "ß _ _ß ! " # and . The derivative changes from positive to negative at x , indicating a local maximum there. The ˆ " " # # ß " œ slope of the curve approaches as x 0 and x 1 , and approaches as x 0 and as x 1 , _ Ä Ä _ Ä Ä indicating cusps and local minima at both x 0 and x 1. œ œ 41. The values of the first derivative indicate that the curve is always rising. The slope of the curve approaches _ as x 0 and as x 1, indicating vertical tangents at both x 0 and x 1. Ä Ä œ œ 42. The graph of the first derivative indicates that the curve is rising on and , falling Š Š ß _ 17 33 17 33 16 16 È È on ( ) and a local maximum at x , a local minimum at _ß ! ß Ê œ Š 17 33 17 33 17 33 16 16 16 È È È x . The derivative approaches as x 0 and x 1, and approaches as x 0 , œ _ Ä Ä _ Ä 17 33 16 È indicating a cusp and local minimum at x 0 and a vertical tangent at x 1. œ œ

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Chapter 4 Practice Exercises 281 43. y 1 44. y 2 œ œ œ œ x 1 4 2x 10 x 3 x 3 x 5 x 5 45. y x 46. y x 1 œ œ œ œ x 1 x x 1 x x x x " "
282 Chapter 4 Applications of Derivatives 47. y 48. y x œ œ œ œ x 2 x x 1 x x x x " " # # # 49. y 1 50. y 1 œ œ œ œ x 4 x 4 x 3 x 3 x 4 x 4 " 51. lim lim x x Ä " Ä " x x x x \$ % # \$ " " œ œ & 52. lim lim x x Ä " Ä " x ax a x bx b a a b b " " œ œ 53. lim x Ä 1 tan x tan x œ œ ! 54. lim lim x x Ä ! Ä ! tan x sec x x sin x cos x " " " # " " œ œ œ 55. lim lim lim lim x x x x Ä ! Ä ! Ä ! Ä ! sin x sin x cos x tan x x sec x x sec x x sec x tan x x sin x cos x a b a b a b a b a b a b a b a b œ œ œ # # # # # †# # # # sec x # ! #†" # a b œ œ " 56. lim lim x x Ä ! Ä ! sin mx m cos mx sin nx n cos nx n m a b a b a b a b œ œ 57. lim sec x cos x lim lim x x x Ä Î # Ä Î # Ä Î # 1 1 1 a b a b ( \$ œ œ œ cos x sin x cos x sin x a b a b a b a b \$ \$ \$ ( ( ( ( \$ 58. lim x sec x lim x x Ä ! Ä ! È œ œ œ ! È x cos x ! " 59. lim csc x cot x lim lim x x x Ä ! Ä ! Ä ! a b œ œ œ œ ! " ! " cos x sin x sin x cos x

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Chapter 4 Practice Exercises 283 60. lim lim lim x lim x lim x x x x x Ä ! Ä ! Ä ! Ä ! Ä ! ˆ Š a b a b " " " " " # # x x x x x x œ œ " œ " œ œ " † _ œ _ 61. lim x x x x lim x x x x x x Ä Ä _ _ " œ " Š Š È È È È # # # # " " È È È È x x x x x x x x lim œ _ x Ä # " " x x x x x È È Notice that x x for x so this is equivalent to œ ! È # lim lim œ œ œ œ " _ _ x x Ä Ä x x x x x x x x x x x x É É É É # " " # " " È È 62. lim lim lim lim lim x x x x x Ä Ä Ä Ä Ä _ _ _ _ _ œ œ œ œ Š x x x x x x x x x x x x x x " " " " % " " # ' "# " a b a b a ba b x x "# lim lim œ œ œ ! _ _ x x Ä Ä "# " #% # x x 63. (a) Maximize f(x) x 36 x x (36 x) where 0 x 36 œ œ Ÿ Ÿ È È "Î# "Î# f (x) x (36 x) ( 1) derivative fails to exist at 0 and 36; f(0) 6, Ê œ œ Ê œ w "Î# "Î# " " # # # È È È È 36 x x x 36 x and f(36) 6 the numbers are 0 and 36 œ Ê (b) Maximize g(x) x 36 x x (36 x) where 0 x 36 œ œ Ÿ Ÿ È È "Î# "Î# g (x) x (36 x) ( 1) critical points at 0, 18 and 36; g(0) 6, Ê œ œ Ê œ w "Î# "Î# " " # # # È È È È 36 x x x 36 x g(18) 2 18 6 2 and g(36) 6 the numbers are 18 and 18 œ œ œ Ê È È 64. (a) Maximize f(x) x (20 x) 20x x where 0 x 20 f (x) 10x x œ œ Ÿ Ÿ Ê œ È "Î# \$Î# w "Î# "Î# # 3 0 x 0 and x are critical points; f(0) f(20) 0 and f 20 œ œ
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