SalasSV_11_SkillReview_ans

SalasSV_11_SkillReview_ans - A-112 21 ANSWERS TO...

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Unformatted text preview: A-112 21. ANSWERS TO ODD-NUMBERED EXERCISES 23. 9 9 2 2 1 1 + g 2 (x ) + C 2 25. 1 8 27. 4 − 64/3 4 29. 19 3 31. 9 2 33. 1 9 3 35. 3 4 37. (a) −1 1 2 + x − x2 dx = 39. (a) 1 4 2 2(x − 1) dx + 2(x − 1) − x + 5 dx (b) −2 2 − y + y dy + 43. 2 2 − y dy 45. (a) −1 2 π r3 3 (b) −2 y + 4 − 1 y2 dy = 18 2 47. 8 + π2 2π y= 64 21 41. 1 1 + x2 2x 1 − 1 + x4 1 + x2 57. (a) (b) f (x) = − sin x √ 59. 3 3 13 2 49. 4 π 15 51. 6 π 7 53. π − 1 π 2 4 65. x = y = 16 ; 35 55. 2π (b) 16 π 35 43 r 3 61. x = 16 , 15 63. x = 0, y = 1 π 8 the two volumes are equal: V = 67. inches 69. 1100 ft-lbs ANSWERS, SKILL MASTERY REVIEW THREE (p. 512) 1. f −1 (x) = (x − 2)3 7. f (x) = 3. f is not one-to-one 1 2 5. f (x) = −ex < 0 on (−∞, ∞); ( f −1 ) (1 + ex )2 11. ex − e3x (1 + e2x )2 13. 1 2 = −4 3x2 + 3x ln 3 x 3 + 3x 3 − 4 ln |x + 1| + C x 23. − 2 cos3 x + C 3 33. π 12 √ 4 + x2 > 0 on (−∞, ∞); ( f −1 ) (0) = x tanh x − ln ( cosh x) x2 17. 2 3 9. 24 [ln x]2 x 15. ( cosh x)1/x 25. 35. 1 2 19. 2[x cosh x − sinh x] + C 1 2 21. 4 ln |x| + − 1 e −2 4 41. 1 2 [x ln |x + 1| − x + ln |x + 1|] + C √ 37. − 27. ln | sec x| − sin2 x + C 39. 1 6 29. 3 4 31. − 1 [ln ( cos x)]2 + C 2 ln 2 + √ 3− 1 4 x 2x 2x − +C ln 2 (ln 2)2 1 2 a2 − x 2 − sin−1 x 45. 1 π 2 1 2 x +C a tan6 x + C 49. x = ln 3 43. sin−1 x − 1 2 √ √ x 1 − x2 − 1 − x2 + C 47. a2 ln 2 √ 6−3 3 3 ln 3 , y= π 2π 51. (b) A = 2 (c) x = 1 π , y = 5 π 2 8 53. [1 − ln 4 + e2 (1 + ln 4)] 55. (a) f increases on (−∞, −1] and [0, 1]; f decreases on [ − 1, 0] and [1, ∞]; (b) absolute max: f (−1) = f (1) = e−1 ; absolute min: f (0) = 0 √ √ √ (c) concave up on −∞, − 1 5 + 17 , − 1 5 − 17, 1 5 − 17 , and 2 2 2 √ √ and 1 5 − 17, 1 5 + 17 2 2 (d) 1 2 5+ √ 17, ∞ ; concave down on − 1 5 + 2 √ 17, − 1 5 − 2 √ 17 0.2 –4 –2 2 4 57. T = − ln 2 ∼ = 3. 1 years ln 0. 8 59. (a) 6.94 grams (b) 18,935 years ago ANSWERS, SKILL MASTERY REVIEW FOUR (p. 704) 1. 2 1 –4 –3 –2 –1 –1 –2 –2 –1 –1 –2 –3 3. 2 1 1 2 6π 9 π 4 ANSWERS TO ODD-NUMBERED EXERCISES 5. 1 − 1 π 4 √ 7. x + y = 2 2 9. parabola 11. ln 3 − 1 2 A-113 13. 992 π 3 15. (a) min speed at t = 1 , max speed at t = 1 4 (b) 1 23. converges to 0 41. 4 43. 2 25. diverges 45. divergent 17. unbounded, increasing 19. unbounded, increasing from a3 on 35. a − a ln a 21. converges to 1 2 e 27. converges to ln 8 47. absolutely convergent 57. divergent 29. 5 31. − 1 2 33. 0 37. 39. diverges 49. conditionally convergent 59. convergent ∞ 51. absolutely convergent 11 ,) 55 53. conditionally convergent ∞ 55. convergent (−1)k −1 k x 2k − 1 61. [ − 63. (−∞, ∞) ∞ 65. (−9, 9) (−1)k +1 (x − 1)k , R = 1 k 67. k =0 2 k 2 k +1 x k! ∞ 69. k =1 71. 1 − 1 x − 1 x2 − 3 9 53 x 81 73. e2 k =0 (−1)k 2k (x + 1)k , R = ∞ k! 75. k =1 ANSWERS, SKILL MASTERY REVIEW FIVE (p. 940) √ 1. −9 i − 5 j − 2 k 5. cos θ = 3. 90 √2 714 1 7. ± √293 (6 i − j + 16 k ) 9. (a) x = 5 + 4t , y = 6 − 7t , z = −3 + 5t (b) x = 5 − 10t , y = 6 − 5t , z = −3 + t (c) x − 3y − 5z − 2 = 0 11. (a) no 19. (b) yes 13. 2x + 3y − 4z − 19 = 0 15. 2x − 2y − z − 6 = 0 21. v(t ) = 2 i + a (t ) = − 17. f (t ) = 2e2t i + 2t 2 − 2t 2 j; f (t ) = 4e2t i + 2 j t2 + 1 (t + 1)2 f (t ) = 2 cosh 2t i + (te−t − e−t ) j + sinh t k ; f (t ) = 4 sinh 2t i + (2e−t − te−t ) j + cosh t k 1 1 j − 2t k ; ||v(t )|| = 2t + ; t t 23. f (t ) = 13 t 3 +1 i+ 1 2t e 2 +t− 7 2 j+ 1 [2t 3 + 1]3/2 + 8 3 k 1 j − 2k t2 25. 2 –1 –2 –3 10 15 27. 29. r π 3 √ √ = − 3 i − j + k ; x = − 1 − 3t , y = 2 √ 3 2 − t, z = π 3 +t 1 31. T(t ) = − √2 sin t i − 1 N(t ) = − √2 cos t i − 1 √ 2 1 √ 2 sin t j − cos t k cos t j + sin t k 33. 38 3 35. √ 2 sinh 1 37. (a) 1 2(1 + e−2t )3/2 (b) 1 |t |(1 + t 2 )3/2 39. κ = 1, aT = 0, aN = 1 43. circles 3 2 1 –3 –2 –1 –1 x –2 –3 1 2 3 (3, 0, 0) 41. (a) dom ( f ) = {(x, y) : x2 − y2 > 1}; range ( f ) = (−∞, ∞) (b) dom ( f ) = {(x, y, z ) : x2 + y2 < z }; range ( f ) = [0, ∞) 45. plane z (0, 0, 2) (0, 6, 0) y 47. ∂z ∂z = x2 y2 cos (xy2 ) + 2x sin (xy2 ); = 2x3 y cos (xy2 ) ∂x ∂y fxx = 4yz 3 + y2 z 2 exyz ; fzx = 12xyz 2 + yexyz + xy2 zexyz fyz = 6x z + xe 22 xyz 49. gx = x y z ; gy = 2 ; gz = 2 x2 + y2 + z 2 x + y2 + z 2 x + y2 + z 2 53. (a) ∇ f = (4x − 4y) i + (−4x + 3y2 ) j (b) ∇ f = y3 − x2 y x3 − xy2 i+ 2 j ( x 2 + y 2 )2 (x + y2 )2 51. + x yze 2 xyz 55. 2 √ 5 √ 57. e 5 59. √ 1 1 tangent plane: − √2 (x − 1) + √2 (y + 1) − (z − 2) = 0; √ 1 1 normal line: x = 1 − √2 t , y = −1 + √2 t , z = 2 − t ...
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