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 " " # # 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 ±± ² ² indicating cusps and local minima at both x 0 and x 1. œœ 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. ÄÄ 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 ±² ÈÈ on ( ) and a local maximum at x , a local minimum at ß Ê œ 17 33 17 33 17 33 16 16 16 ± È x . The derivative approaches as x 1, and approaches as x 0 , œ± _ Ä Ä _ Ä 17 33 16 ² È indicating a cusp and local minimum at x 0 and a vertical tangent at x 1.
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Chapter 4 Practice Exercises 281 43. y 1 44. y 2 œœ ± ² x 1 4 2x 10 x 3x 3 x 5x 5 ± ²² ±± 45. y x 46. y x 1 ± œ œ ² ± x 1 x 1 xx x x # # ±" ² ± "
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282 Chapter 4 Applications of Derivatives 47. y 48. y x œœ ± ² x 2 x 1 xx x x $# % ## ±" ² " # 49. y 1 50. y 1 ² ± 4 x4 x 3x 3 x 4x 4 # # ²" ²² 51. lim Ä" x x x # ±$ ²% # ±$ " & 52. xa x a xb x b aa bb ²" ²" " " 53. x Ä 1 tan x tan x ! 1 1 54. Ä ! Ä ! tan x sec x xs i n x c o s x ± "±" # "" œ # 55. 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 sin x cos x # # # # # ab a b ab ab œœœ #† ### # # ± # sec x #! ± # " # " 56. Ä ! Ä ! sin mx m cos mx sin nx n cos nx n m 57. lim sec x cos x lim lim x ÄÎ# 11 1 ($ œ œ œ cos x sin x cos x sin x $ $ ( ( ( $ 58. x sec x Ä ! Ä ! È œ ! È x cos x ! " 59. csc x cot x x Ä ! Ä ! Ä ! ²œ œ œ œ ! ! " cos x sin x sin x cos x
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Chapter 4 Practice Exercises 283 60. lim x x xx x x x Ä ! Ä ! Ä ! Ä ! Ä ! ˆ‰ Š‹ ab "" " ± " " ## x x x x %% % % # # ± œ œ" ± ± œ œ " _ œ _ 61. x x x x x x x x ÄÄ __ ²² ± œ ± † ÈÈ " ² ± ² ²"² ± È È È È x x x x x œ _ x Ä " ± x x x È È Notice that x x for x so this is equivalent to œ³ ! È # œœ œ œ " # " " " " # x x x x ÉÉ É É ± "± ± ± "² # " 62. x x x Ä Ä Ä _ _ _ ±œ œ œ œ x x x $$ $# # # $ % ±" ²" % ²" ± #' " # x x "# # œ ! "# " #% # 63. (a) Maximize f(x) x 36 x x (36 x) where 0 x 36 œ± ± ± Ÿ Ÿ È È "Î# "Î# f (x) x (36 x) ( 1) derivative fails to exist at 0 and 36; f(0) 6, Êœ ±± ± œ Ê œ ± w ±"Î# ±"Î# ±² È È È È 36 x x x 36 x and f(36) 6 the numbers are 0 and 36 œÊ (b) Maximize g(x) x 36 x x (36 x) where 0 x 36 œ² ± ± Ÿ Ÿ È È "Î# "Î# g (x) x (36 x) ( 1) critical points at 0, 18 and 36; g(0) 6, ²± ± 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 Ê È È 64. (a) Maximize f(x) x (20 x) 20x x where 0 x 20 f (x) 10x x œ ± Ÿ Ÿ Ê È "Î# $Î# w ±"Î# "Î# # 3 0 x 0 and x are critical points; f(0) f(20) 0 and f 20 Ê œ œ œ œ œ ± 20 3x 20 20 20 20 x 33 3 3 ± # È ˆ É the numbers are and .
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This note was uploaded on 09/20/2010 for the course MATHEMATIC 09991051 taught by Professor Dr.maenshadeed during the Fall '10 term at Norwegian Univ. of Science & Technology.

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ISM_T11_C04_I - 280 Chapter 4 Applications of Derivatives...

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