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SalasSV_07_05_ex_ans

# SalasSV_07_05_ex_ans - ANSWERS TO ODD-NUMBERED EXERCISES 61...

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Unformatted text preview: 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 A-58 39. 1 e ANSWERS TO ODD-NUMBERED EXERCISES 41. f (x) = px ln f (x) = x ln p f (x ) = ln p f (x ) f (x) = px ln p 43. (x + 1)x x + ln (x + 1) x+1 45. (ln x)ln x 1 + ln (ln x) x 47. xsin x cos x ln x + sin x x 49. ( sin x)cos x cos2 x − sin x ln ( sin x) sin x 51. x2 x 2x + 2x (ln x)(ln 2) x y In 53. y x 55. 2x 57. log 3 x x y 2x x log 2 x x 59. 1 4 ln 2 61. 2 63. 45 ln 10 65. 1 1 + 3 ln 2 67. approx. 16. 999999; 5(ln17)/(ln 5) = (eln 5 )(ln17)/(ln 5) = eln 17 = 17 69. (b) the x-coordinates of the points of intersection are: x ∼ −1. 198, x = 3 and x ∼ 3. 408. = = (c) for the interval [ − 1. 198, 3], A ∼ 5. 5376; for the interval [3, 3. 408], A ∼ 0. 1373 = = SECTION 7.6 1. (a) \$411.06 (b) \$612.77 (c) \$859.14 3. about 5 1 % : (ln 3)/20 ∼ 0. 0549 = 2 5. 16, 000 9. (a) e0.35 (b) k = liters 17. 5 4 5/2 5 7. (a) P (t ) = 10, 000 et ln 2 = 10, 000(2)t (b) P (26) = 10, 000(2)26 , P (52) = 10, 000(2)52 13. in the year 2112 15. 200 4 t /5 5 ln 2 15 ∼ 2. 86 gms = 11. P (20) ∼ 317. 1 million; P (11) ∼ 284. 4 million = = 19. 100[1 − 1 1/n 2 % 21. 80.7%, 3240 yrs 23. (a) x1 (t ) = 106 t , x2 (t ) = et − 1 d d [x1 (t ) − x2 (t )] = [106 t − (et − 1)] = 106 − et . (b) dt dt This derivative is zero at t = 6 ln 10 ∼ 13. 8. After that the derivative is negative = ∼ (c) x2 (15) < e15 = (e3 )5 = 205 = 25 (105 ) = 3. 2(106 ) < 15(106 ) = x1 (15) x2 (18) = e18 − 1 = (e3 )6 − 1 ∼ 206 − 1 = 64(106 ) − 1 > 18(106 ) = x1 (18) = ∼ 64(106 ) − 1 − 18(106 ) ∼ 46(106 ) x2 (18) − x1 (18) = = (d) If by time t1 EXP has passed LIN, then t1 > 6 ln 10. For all t ≥ t1 the speed of EXP is greater than the speed of LIN: for t ≥ t1 > 6 ln 10, v2 (t ) = et > 106 = v1 (t ). 25. (a) 15( 2 )1/2 ∼ 12. 25 lb/in.2 = 3 31. 176/ln 2 ∼ 254 ft = (b) 15( 2 )3/2 ∼ 8. 16 lb/in.2 = 3 35. f (t ) = Cet 2 /2 27. 6. 4% 29. (a) \$18,589.35 (b) \$20,339.99 (c) \$22,933.27 33. 11,400 years 37. f (t ) = Cesin t SECTION 7.7 1. (a) 0 (b) − π 3 13. 3. (a) √ 2 2π 3 √ (b) 3π 4 5. (a) 1 2 (b) π 4 7. (a) does not exist 2 sin−1 x 17. √ 1 − x2 (b) does not exist 9. (a) 3 2 (b) − 7 25 11. 1 x 2 + 2x + 2 1 x[1 + (ln x)2 ] x 4x4 − 1 r √ |r | 1 − r 2 2x 15. √ + sin−1 2x 1 − 4x2 27. 2x sec−1 1 x −√ x 2 19. x − (1 + x2 ) tan−1 x x2 (1 + x2 ) 1 21. (1 + 4x 2 ) 1 √ tan−1 2x c−x c+x 23. 25. − 1 − x2 29. cos [ sec−1 (ln x)] · x|ln x| (ln x)2 − 1 31. 33. Set au = x + b, a du = dx. dx a2 − (x + b)2 = a du = √ a2 − a 2 u 2 √ du 1 − u2 = sin−1 u + C = sin−1 x+b a +C ...
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