# ODDREV05 - 272 Chapter 5 Logarithmic Exponential and Other...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 272 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions 85. As k increases, the time required for the object to reach the ground increases. ex 2 x 87. y y cosh x ex 2 e e x 89. y cosh y cosh x 1 1 sinh y 1 x sinh x sinh y y y 1 cosh2 y 1 1 x2 1 91. y y sech x 2 ex 2 ex e x e 2 x ex e x e x 2 e x ex ex e e x x sech x tanh x Review Exercises for Chapter 5 1. f x ln x 3 5 4 3 2 1 x 1 2 3 4 5 y Vertical shift 3 units upward Vertical asymptote: x 0 x=0 3. ln 5 4x2 4x2 1 1 1 2x 1 2x ln 5 4x2 1 1 1 ln 2x 5 1 ln 2x 1 ln 4x2 1 5. ln 3 1 ln 4 3 x2 ln x ln 3 ln 3 4 x2 ln x ln 33 4 x x2 7. ln x x x 1 1 1 x 2 e2 e4 e4 1 53.598 9. g x gx ln x 1 2x 1 ln x 2 11. f x fx x ln x x ln x 2 1 2 ln x 1 a bx ln a x 1 b a a bx 1 x 12 13. 1 x ln x 1 ln x 2 ln x 2 ln x y dy dx 1 ln a b2 b 1 b2 a bx bx a a a ab bx bx x 2 a bx 2 15. y dy dx 1 ln a a 1 xa bx bx ln x 17. u 7x 7x 1 2 2, du dx 7dx 1 1 7 dx 7 7x 2 1 ln 7x 7 2 C R eview Exercises for Chapter 5 sin x dx 1 cos x sin x dx 1 cos x ln 1 3 4 273 19. 21. 1 x x 1 4 dx 1 1 1 dx x 4 x ln x 1 3 ln 4 cos x C 3 23. 0 sec d ln sec 1 2x 1 2x tan 0 ln 2 3 25. (a) fx y 2y 2x f 1 3 3 (b) f − 11 −1 7 10 3 3 x x y 2x 6 (c) f 1 −7 f fx 1 f 11 2x 3 6 1 2 2 1x 2 2x 6 3 3 6 x x ff 27. (a) fx y y2 x2 f 1 x 4 f 2x x x x y x2 1 1 (b) f −1 f −3 6 1 1 x −2 1, x ≥ 0 (c) f 1 fx 1 f 1 x 1 1 x2 x2 1 1 2 1 x ff x f x2 x2 1 x for x ≥ 0. 29. (a) fx y y3 x3 f 1 3 3 x x 1 1 (b) 4 f −1 f 1 1 x x y x3 1 −4 5 −2 (c) f 1 fx 1 f f fx f 6 fx 13 x 1 1 3 3 x 1 1 3 1 x x ff 31. f f f 1 1 x x3 x3 1 fx 1 x3 x 1 x 3 1 3 1 35 3 2 2 13 33. tan x 3 3 sec2 x 4 3 1 f 6 3 4 x x 1 2 1 23 2 23 3 1 3 23 f f 1 6 3 3 0.160 274 35. (a) Chapter 5 fx y ey e2y e f 1 2x Logarithmic, Exponential, and Other Transcendental Functions x (b) −3 2 ln f −1 f 3 ln x x x y e2x (c) f 1 −2 fx 1 f f ln e 2x x2 1 ln x ln x2 e2 ln e2x x e ln x ex x x x ff 39. f x fx x e2x ln 37. y e x2 y 6 4 2 x 2 −2 4 −2 41. g t gx t2et t2et 2tet tet t 2 43. y y e2x 1 2x e 2 e e 2x 2x 12 2e2x 2e 2x e2x e 2x e2x e 2x 45. g x gx x2 ex ex 2x e2x x2ex x2 ex x 47. y 1 x ln x dy dx y ln x 2y y2 dy dx dy dx dy dx 0 0 y x y x 2y ln x 2y ln x 49. Let u xe 3x2 3x2, du dx 1 e 6 6x dx. 3x2 51. 1 e 6 3x2 e4x e2x ex 1 dx e3x 1 3x e 3 e4x ex ex 3e2x 3ex e e x x dx C C 6x dx C 3 53. xe1 x2 dx 11 e 2 11 e 2 x2 x2 2x dx C 55. Let u ex ex 1 ex dx 1, du ln ex ex dx. 1 C 57. y y ex a cos 3x e x b sin 3x 3b cos 3x a 3a 8a 3a b ex a cos 3x 3b cos 3x 3b sin 3x 6b cos 3x 10b sin 3x 8a 6b 2a 3b 10a cos 3x 0 ex 3a b sin 3x a 3b cos 3x b sin 3x 3a sin 3x 3a 3a 6a 6a ex y ex 3 ex y 2y 10y e x b sin 3x b cos 3x 8b sin 3x 8b 2 R eview Exercises for Chapter 5 4 275 59. Area 0 xe x2 dx 1 e 2 4 x2 0 1 e 2 16 1 0.500 61. y 33 2 y 63. y y log2 x 4 3 2 1 1 6 5 4 3 2 x −1 1 2 34 −2 −3 −4 x 1 2 34 5 67 −4 −3 −2 −1 −2 65. f x fx 3x 3x 1 1 67. ln 3 y ln y y y y x2x 2x 2x x y 1 1 ln x 1 2 ln x 1 x 1 2 2x 2 ln x x2x 1 2x x 1 2 ln x 69. g x gx log3 1 1 21 xa ax a 1 x 1 log3 1 2 2x x 71. x 1 5x dx 11 5x 2 ln 5 1 2 C 1 x ln 3 1 1 ln 3 (b) y y ax ln a ax (c) y ln y 1 y y y y xx x ln x x y1 xx Ph P 18,000 k Ph P 35,000 dy dx dy y x2 2 x2 x x 3 dx x 3 ln x C 1 1 x 1 ln x ln x ln x 30ekh 30e18,000k ln 1 2 18,000 30e 30e 3 15 ln 2 18,000 (d) y y aa 0 73. (a) y y 75. 10,000 P Pe 0.07 10,000 e1.05 15 77. \$3499.38 h ln 2 18,000 35,000 ln 2 18,000 7.79 inches 79. P 2C 2 ln 2 t Ce0.015t Ce0.015t e0.015t 0.015t ln 2 0.015 46.21 years 81. 276 83. y Chapter 5 2xy dy dx 1 dy y ln y e x2 C1 Logarithmic, Exponential, and Other Transcendental Functions 0 2xy 2x dx x2 y Cex y2 2 C1 y dy dx x2 2xy x2 Let y x2 x2 85. (homogeneous differential equation) 2xy dy x dv dx 0 y2 dx vx, dy v2x2 v dx. v dx 2x3v dv x2v2 dx 1 v2 dx dx x ln x x 1 1 x2 0 0 2x3v dv 2x dv 2v 1 ln 1 C v2 Cx y2 1 v2 dv v2 C1 C yx 2 2x vx x dv 2x2v2 dx x2 v2x2 ln 1 Cx2 x2 x x2 y2 v2 y2 ln C or C1 87. y y y x2 y C1x C1 6C2 x 3xy C2 x3 3C2 x2 3y x2 6C2 x 6C2 x3 3x C1 3C1x 3C2 x2 9C2 x3 4C2 3C1x C1x C2 x3 0 3C2 x3 x x 2, y 2, y 0: 0 4: 4 4 2C1 C1 8C2 ⇒ C1 12C2 4C2 12C2 8C2 ⇒ C2 1 ,C 21 2 y 2x 13 x 2 3 4 y 89. f x 2 arctan x −6 −4 −2 −2 −4 x 2 R eview Exercises for Chapter 5 1 2 2 1 277 91. (a) Let arcsin 1 2 sin arcsin 1 2 cos sin 1 sin arcsin 2 (b) Let θ 1 . 2 1 2 3 sin cos arcsin 1 2 3 . 2 x 93. y y tan arcsin x 1 x2 12 1 x2 x2 12 95. y 1 x2 32 x arcsec x x x x2 1 arcsec x x2 1 1 x2 2x y 97. y y x arcsin x 2 21 2 x2 arcsin x 2 21 1 x2 x2 2x arcsin x 1 x2 arcsin x 2 2x arcsin x 1 x2 e2x, du 1 e2x e 2x arcsin x 99. Let u 2e2x dx. 1 e2x dx e4x 1 21 1 e2x 2 dx 2e2x dx 1 arctan e2x 2 103. Let u C 101. Let u x 1 x2, du x4 dx 2x dx. 1 2 x , du 2 1 2 1 1 x2 2 4 arctan k dt m k t m C 0. Thus, x2 x 2 2 16 x x2 dx x2, du 2x dx. 1 ln 16 2 x2 C 2x dx 1 arcsin x2 2 C 16 1 1 2x dx 2 16 x2 105. Let u arctan dx. 2 4 x 2 arctan x 2 dx 4 x2 107. dy A2 y2 y arcsin A Since y sin 0 when t k t m y dx 1 x arctan 4 2 2 C 109. y y 2x 2 cosh x 1 sinh 2x x 2 sinh x 2x 0, you have C y A A sin k t m 111. Let u x x4 x2, du 1 dx 2x dx. 1 2 1 x2 2 1 2x dx 1 ln x2 2 x4 1 C 278 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions Problem Solving for Chapter 5 1. tan tan 3 x 6 2 1 10 1 x 2: 6 3 Minimize fx fx 1 1 2 arctan 3 x2 3 x arctan 1 6 10 10 x 6 x 3 x2 9 36 36 118 x 10 2x 2 2 θ1 0 x θ a θ2 10 1 9 x2 1 36 10 x 2 0 6 x2 9 18 36 10 100 20x x2 x 2 x2 20x 2x2 0 20 ± 202 2 4 118 10 ± 218 a 10 1 218 2 4.7648 1.7263 1.0304 1.2793 10 fa 1.4153 or 98.9 Endpoints: a a 0: 10: Maximum is 1.7263 at a 3. f x sin ln x or 0, 218 4.7648. (a) Domain: x > 0 (b) f x 1 (f) 2 . 3 2 2k . 0 2 sin ln x ⇒ ln x e 2 5 Two values are x (c) f x 1 ,e 2 2 sin ln x ⇒ ln x e 2, −2 2k . x→0 Two values are x e3 2. lim f x seems to be e 1. e 1. 2, 1 . (This in incorrect.) 2 e 3 2, 1, 1 , (d) Since the range of the sine function is 1, 1 . parts (b) and (c) show that the range of f is (e) f x fx 1 cos ln x x 0 ⇒ cos ln x 0 ⇒ ln x x fe f1 f 10 2 (g) For the points x we have f x For the points x e 7 2, ... 2 ,e 5 2 ,e 9 2 ,... 2 e 2 k ⇒ we have f x on 1, 10 1 0 0.7440 Maximum is 1 at x e 2 That is, as x → 0 , there is an infinite number of 1, and an infinite number where points where f x fx 1. Thus lim sin ln x does not exist. x→0 4.8105 You can verifiy this by graphing f x on small intervals close to the origin. P roblem Solving for Chapter 5 5. (a) Area sector Area circle t ⇒ Area sector 2 1 base height 2 1 cosh t 2 1 cosh2 t 2 1 cosh2 t 2 1 cosh2 t 2 At 1 t 2 C. But, A 0 1 t or t 2 sinh t 1 279 t 2 x2 t 2 1 dx 1 dx 1 sinh t cosh t 1 cosh t (b) Area AOP At At x2 cosh2 t sinh2 t 1 2 0⇒C sinh2 t sinh2 t sinh2 t C 2A t . 0 Thus, A t 7. y y y b y If x ln x 1 x 1 x a 1 x a 0, c b a b 1 Tangent line 1. Thus, b c b b 1 1. 9. Let u dx Area 1 2u 4 1 x, x u 1, x u2 2u 1, 2 du. 1 dx xx 3 2 2u u u u2 du 3 2 u2 2u 1 1 du 32 2 1 du u 32 2 u 2 ln u 2 2 ln 3 0.8109 dy dt y 1.01 2 ln 2 2 ln 3 2 11. (a) y1.01 dt t C1 0.01t 1 0.01t 1 0.01t 100 (b) y 1 dy y y y kt k dt C1 kt 1 C 1 C1 1 y0 . kt 1 dy y 0.01 C 0.01 1 y0.01 y0.01 y y0 1: 1 C y0 y0 ⇒ C1 1 1 1 ⇒C y0 . 1 y0 C C Hence, y kt 1 ⇒C C100 1 t→T 1 For t → 1 , y→ y0 k Hence, y For T 1 0.01t 100 . 100, lim y . 280 Chapter 5 dy dt 1 y ln y 20 Logarithmic, Exponential, and Other Transcendental Functions 13. Since ky dy 20 , k dt kt Cekt C 20. 52. 52ek 22.35 . dS dt d 2S dt2 4 S 100 9 4 9 S dS dt 2S 50 or 20, ek 28 52 7 13 , and k ln 7 13 . 20 y When t When t Thus, y When t dS dt S dS dt 0, y 72. Therefore, C 1, y 48. Therefore, 48 52e ln 7 13 t 20. 5, y 52e5 ln 7 13 20 15. (a) k1S L L Ce L1 LC ke 1 Ce k L1 k L1 k1S L S is a solution because kt 2 (b) ln ln ln S 100 dS dt dS dt 0. S dS dt 1 kt Ce kt kt 2 C ke kt 4 100 9 0 when S C Le kt 1 Ce kt L 1 k . L 9. And, (d) 140 120 100 80 60 40 20 S L Ce L Ce Choosing S 50 50, we have: 1 1 t 100 9eln 4 9eln 4 9t 9t kt kt L Ce kt 2 ln 1 9 ln 4 9 t S , where k1 L S 100. Also, S 10 when t 0 ⇒ C 20 when t 1 ⇒ k ln 4 9 . 1 1 100 9eln 4 9t 2.7 months Particular Solution. S 100 9e 0.8109t (c) 125 1 2 3 4 t (e) Sales will decrease toward the line S 0 0 10 L. ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online