SalasSV_07_04_ex_ans

SalasSV_07_04_ex_ans - A-56 ANSWERS TO ODD-NUMBERED...

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Unformatted text preview: A-56 ANSWERS TO ODD-NUMBERED EXERCISES (v) 2 –1 –2 –3 –4 4 6 8 10 73. (i) domain (0, ∞) (ii) increases on (0, 1]; decreases on [1, ∞) (iii) f (1) = ln 1 local and absolute max 2 (iv) concave down on (0,2.0582); concave up on (2.0582,∞); point of inflection (2.0582, −0.9338) (approx.) 75. average slope = 1 b−a b a 1 1 dx = ln (b/a) x b−a 77. x-intercept: 1; absolute min at x = e−2 ; absolute max at x = 10 79. x-intercepts: 1,23.1407; absolute max at x ∼ 4. 8105, absolute min at x = 100 = 81. (a) v(t ) = 2 + 2t − t 2 + 3 ln (t + 1) (c) max velocity at t ∼ 1. 5811; min velocity at t = 0 = 83. (b) x-coordinates of points of intersection: x = 1, 3. 30278 ∼ (c) A = 2. 34042 85. (a) f (x) = ln x (b) f (x) = x ln x (c) f (x) = x2 ln x 87. (a) f (x) = 1 − 2 ln x −5 + 6 ln x ; f (x ) = x3 x4 1/2 5/6 (b) f (1) = 0; f (e ) = 0; f (e ) = 0 (c) f (x) > 0 on (1, ∞); f (x) < 0 on (0, 1); f (x) > 0 on (0, e1/2 ); f (x) < 0 on (e 1/2 f (x) > 0 on (e5/6 , ∞); f (x) < 0 on (0, e5/6 ) , ∞); SECTION 7.4 dy 1. = − 2 e −2 x dx 11. 1√ dy = ex dx 2 dy dy 1 dy dy 2−1 = 2x e x 5. = ex + ln x 7. = −(x−1 + x−2 )e−x 9. = 1 ( e x − e −x ) 2 dx dx x dx dx √ ln x 1 dy dy 2e x dy dy 2 2 +√ = 4x e x (e x + 1) 15. = x 2 ex 17. =x 19. = 4x 3 13. x dx dx dx (e + 1)2 dx x 3. 23. f (x) = −e−2x (2 cos x + sin x) √ 37. 2 e x + 1 + C 47. 2 − 1 e 49. ln 3 2 21. f (x) = 2 cos (e2x )e2x 33. 12 x 2 25. 39. 51. 1 4 1 2x e 2 +C 27. 1 kx e +C k 41. esin x + C 29. 1 x2 e 2 +C 31. −e1/x + C +C 45. 35. −8e−x/2 + C 1 (1 6 ln (2 e2x + 3) + C + 1 2 43. e − 1 − π −6 ) 1 e 2 53. (a) f (n) (x) = an eax (b) f (n) (x) = (−1)n an e−ax 1 1 55. at ± √ , √ e 2 57. (a) f is an even function; symmetric with respect to the y-axis. (b) f increases on (−∞, 0]; f decreases on [0, ∞). (c) f (0) = 1 is a local and absolute maximum. √ √ √ √ √ (d) the √ graph is concave up on (−∞, −1/ 2) ∪ (1/ 2, ∞); the graph is concave down on (− 1/ 2, 1/ 2); points of inflection at (−1/ 2, e−1/2 ) and −1/2 (1/ 2, e ) (e) the x-axis (f) 1 1 2 59. (a) π 1 − 1 1 e 2 (b) 0 π e−2x dx ANSWERS TO ODD-NUMBERED EXERCISES 61. 1 (3 2 A-57 e4 + 1) 63. e2 − e − 2 (e) horizontal asymptote y=1 y 65. (a) domain (−∞, 0) ∪ (0, ∞) (b) increases on (−∞, 0), decreases on (0, ∞) (c) no extreme values (d) concave up on (−∞, 0) and on (0, ∞) y=1 x vertical asymptote x = 0 67. (a) domain (0, ∞) (b) f increases on (e−1/2 , ∞); f decreases on (0, e−1/2 ). (c) f (e−1/2 ) = −1/2e is a local and absolute minimum. (d) the graph is concave down on (0, e−3/2 ); the graph is concave up on (e−3/2 , ∞); point of inflection at (e−3/2 , −3/2e3 ) (e) 0.5 0.5 1 1.5 69. (a) lim f (x) = 0 for all k > 0 x→0+ (b) e−1/k 71. (a) 1 ± ,e a (b) 1 (e − 2) a (c) 1 + 2a2 e a3 e 73. for x > (n + 1)!, ex > 1 + x + · · · + 77. xn+1 x n +1 x > = xn > xn (n + 1)! (n + 1)! (n + 1)! 75. (a) y y f 4 g 1 –2 2 x 1 0 1 g x (b) x = −1. 9646; x = 1. 0580 (c) 6. 4240 79. (a) x ∼ 1. 14477, 1. 83788 = (b) x = ln (| nπ |), n = ±1, ±2, . . . 81. (b) x ∼ 1. 3098 = ∼ (c) f (1. 3098) = −0. 26987; g (1. 3098) ∼ 0. 76348 = (d) no 83. (a) x− ln |ex − 1| + C (b) − 1 e−5x + e−4x − 2e−3x + 2e−2x − e−x + C 5 (c) etan x + C SECTION 7.5 1. 6 3. − 1 6 5. 0 7. 3 9. logp xy = 13. 0 23. g (x) = ln xy ln x + ln y ln x ln y = = + = logp x + logp y ln p lnp ln p ln p 15. 2 17. t1 < ln a < t2 25. f (x) = − 3 −x +C ln 3 19. f (x) = 2(ln 3)32x sec2 ( log5 x) x ln 5 33. log5 |x| + C 35. 3 (ln x)2 + C ln 4 37. 1 e ln 3 11. logp xy = ln xy ln x =y = y logp x ln p ln p ln 3 5x lnx 23 x 21. f (x) = 5 ln 2 + 1 1 · 2 ln 3 x log3 x 31. 14 x 4 27. F (x) = ln 2(2−x − 2x ) sin (2x + 2−x ) 29. 3x +C ln 3 ...
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