SalasSv_07_03_ex_ans - ANSWERS TO ODD-NUMBERED EXERCISES so 1 n A-55 γ<1− Letting n → ∞ we have 1 2< γ< 1(b r ∼ 2 2191 = 29 1 27(a

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Unformatted text preview: ANSWERS TO ODD-NUMBERED EXERCISES so 1 . n A-55 γ <1− Letting n → ∞, we have 1 2 < γ < 1. (b) r ∼ 2. 2191 = 29. 1 27. (a) ln 3 − sin 3 ∼ 0. 96 > 0; ln 2 − sin 2 ∼ −0. 22 < 0 = = 31. (b) eπ/2 ∼ 4. 81048; e3π/2 ∼ 111. 318 = = SECTION 7.3 1. domain (0, ∞), f (x) = 1 x x4 3. domain (−1, ∞), f (x) = 4x 3 −1 (c) eπ/2+2nπ ; e3π/2+2nπ 3x 2 x3 + 1 5. domain (−∞, ∞), f (x) = x 1 + x2 7. domain all x = ±1, f (x) = 9. domain (− 1 , ∞), f (x) = 2(2x + 1) 1 + 2 ln (2x + 1) 2 13. domain (0, ∞); f (x) = 23. 1 +C 2(3 − x2 ) 1 cos (ln x) x 15. ln |x+1|+C 17. − 1 ln |3 − x2 | + C 2 27. ln | ln x| + C 11. domain (0, 1) ∪ (1, ∞), f (x) = − 19. 1 3 1 x(ln x)2 ln | sec 3x| + C 21. 1 2 ln | sec x2 + tan x2 | + C 33. 45. 1 2 2 3 25. − ln |2 + cos x| + C 29. −1 +C ln x 39. 1 31. − ln | sin x + cos x| + C 41. 1 2 √ ln |1 + x x| + C 35. x + 2 ln | sec x + tan x| + tan x + C 37. 1 ln 8 5 43. ln 4 3 ln 2 4x 5 3 + + +1 x−1 x 61. 2π ln (2 + √ 3) 51. g (x) = x4 (x − 1) 1 1 2x 4 + − −2 (x + 2)(x2 + 1) x x−1 x+2 x +1 65. (−1)n−1 (n − 1)! xn 47. The integrand is not defined at x = 2. 53. 67. 1 π 3 49. g (x) = (x2 +1)2 (x−1)5 x3 57. 15 − 8 x2 − 1 2 ln 3 55. 1 π 4 − 1 2 ln 2 ln 4 59. π ln 9 63. ln 5 ft csc x dx = csc x( csc x − cot x) dx csc x − cot x Let u = csc x − cot x, du = csc x( csc x − cot x)dx. Then csc x dx = 1 du = ln |u| + C = ln | csc x − cot x| + C . u (v) y x=4 69. (i) domain (−∞, 4) (ii) decreases throughout (iii) no extreme values (iv) concave down throughout; no pts of inflection (3, 0) x 71. (i) domain (0, ∞) (ii) decreases on (0, e−1/2 ), increases on [e−1/2 , ∞] (iii) f (e−1/2 ) = − 21e local and absolute min. (iv) concave down on (0, e−3/2 ); concave up on (e−3/2 , ∞) pt of inflection at (e−3/2 , − 3 e−3 ) 2 (v) y (e –1/2, –1/2e) x 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 ...
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This note was uploaded on 10/12/2010 for the course MATH 12345 taught by Professor Smith during the Spring '10 term at University of Houston - Downtown.

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