SalasSV_09_04_ex_ans

SalasSV_09_04_ex_ans - ANSWERS TO ODD-NUMBERED EXERCISES 25...

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Unformatted text preview: ANSWERS TO ODD-NUMBERED EXERCISES 25. center (0, 0) transverse axis 2 vertices ( ± 1, 0) √ foci ( ± 2, 0) asymptotes y = ± x y A-67 27. center (0, 0) transverse axis 6 x y 29. center (0, 0) transverse axis 8 x y vertices ( ± 3, 0) foci ( ± 5, 0) asymptotes y = ± 4 x 3 vertices (0, ±4) foci (0, ±5) asymptotes y = ± 4 x 3 x 31. center (1, 3) transverse axis 6 vertices (4, 3) and (−2, 3) foci (6, 3) and (−4, 3) asymptotes y = ± 4 (x − 1) + 3 3 35. d ( F1 , F2 ) + k = 2(c + a) y (1,3) 33. center (1, 3) transverse axis 4 vertices (1, 5) and (1, 1) √ foci (1, 3 ± 5) asymptotes y = 2x + 1, y = −2x + 5 y (1,3) x x √ 37. 2 π 2 a4 − A2 /π a 39. (5 ± 5 21 √ 5, 0) √ √ √ √√ 41. center (0, 0), vertices (1, 1) and (−1, −1), foci ( 2, 2) and (− 2, − 2), asymptotes x = 0 and y = 0, transverse axis 2 2 √ √ 43. [2 3− ln (2 + 3)]ab 45. 3 47. 4 . 49. E1 is fatter than E2 , more like a circle 5 5 51. The ellipse tends to a line segment of length 2a. 53. x2 /9 + y2 = 1 55. 5 3 57. √ 2 59. The branches of H1 open up less quickly than the branches of H2 . 61. The hyperbola tends to a pair of parallel lines separated by the transverse axis. 63. about 0.25 mile west and 1.46 miles north of point A SECTION 9.3 1. −7. See ﬁgure to the right. 17. [1, 1 π 2 9. (0, 3) 11. (1, 0) 13. ( − 3 , 2 3 2 √ 3) 15. (0, −3) 2 1, 1 π 3 + 2nπ ], [ − 1, 3 π 2 + 2nπ ] 19. [3, π + 2nπ ], [ − 3, 2nπ ] 23. [8, 1 π + 2nπ ], [ − 8, 7 π + 2nπ ] 6 6 11 π] 6 –1, 1 π 3 polar axis – 1π √ √ 21. [2 2, 7 π + 2nπ ], [ − 2 2, 3 π + 2nπ ] 4 4 25. 2 2 r1 + r2 − 2r1 r2 cos (θ1 − θ2 ) 2 π] 3 27. (a) [ 1 , 2 1 π 3 (b) [ 1 , 5 π ] 26 (c) [ 1 , 7 π ] 26 4, 5 π 4 29. (a) [2, (b) [2, 5 π] 3 (c) [2, 31. symmetry about the x-axis 33. no symmetry about the coordinate axes; no symmetry about the origin 35. symmetry about the origin 43. θ = π/4 37. r cos θ = 2 39. r 2 sin 2θ = 1 41. r = 4 sin θ 45. r = 1 − cos θ 47. r 2 = sin 2θ √ 53. the parabola y2 = 4(x + 1) 51. the line y = 3x 57. the line y = 2x 63. (x − 59. 3x2 + 4y2 − 8x = 16, ellipse 49. the horizontal line y = 4 55. the circle x2 + y2 = 3x b2 a a2 + b2 ) + ( y − )2 = ; center: 2 2 4 61. y2 = 8x + 16, parabola √ ba d a2 + b 2 , 65. r = , radius: 22 2 2 − cos θ SECTION 9.4 1. 3. 4 5. 7. 3 –π 1 4 A-68 9. ANSWERS TO ODD-NUMBERED EXERCISES 11. 1 π 2 1 1 13. 1 15. [2, π] 3 4 π 1 1 [–2,0] 2 [–2, π ] 3 4 17. 19. 2 3 1 1 21. 1 23. θ= π 1 2 θ= π 1 5 25. 3 2 1 2 27. 29. 2 θ= π 3 31. 1 3 1 1 1 1 3 θ= π 4 3 33. yes; [1, π ] = [−1, 0] and the pair r = −1, θ = 0 satisﬁes the equation 35. yes: the pair r = 1 , θ = 1 π satisﬁes equation 2 2 √ 33 ,) 44 π 6 37. [2, π ] = [−2, 0]. The coordinates of [−2, 0] satisfy the equation r 2 = 4 cos θ , and the coordinates of [2, π ] satisfy the equation r = 3 + cos θ . 39. (0, 0), (− 1 , 1 ) 22 41. (−1, 0), (1, 0) √ 43. (0, 0), ( 1 , ± 1 3rt ) 4 4 45. (0, 0), ( ± 47. center: (b, a); radius: 5π 6 √ a2 + b 2 49. (b) The curves intersect at the pole and at [1. 175, 0. 176], [1. 86, 1. 036], [0. 90, 3. 243]. 53. (b) The curves intersect at the pole and at: r = 1 − 3 cos θ [−2, 0] [3. 800, 3. 510] [2. 412, 4. 223] [−1. 267, 0. 713] SECTION 9.5 1. 17. π/6 π /3 1 π a2 4 5π/6 51. θ = , π, 2 55. butterﬂy 57. a petal curve with 2m petals r = 2 − 5 sin θ [2, π ] [3. 800, 3. 510] [−2. 412, 1. 081] [−1. 267, 0. 713] 3. 12 a 2 5. 1 π a2 2 7. 19. 1 4 − 1 π 16 1 ([4]2 2 9. 3 π 16 + 3 8 11. 52 a 2 π /3 13. 1 (3e2π 12 − 3 − 2π 3 ) + π /2 π/3 15. 1 2π (e 4 + 1 − 2eπ ) 1 ([4 sin θ ]2 2 − [2]2 ) d θ π /3 −π/3 π /6 − [2 sec θ ]2 ) d θ + π /2 π/6 1 (1 2 21. 2 0 1 (2 sec θ )2 d θ 2 π /4 0 1 (4)2 d θ 2 √ 3 16 23. 0 1 (2 sin 3θ )2 d θ 2 25. 2 0 1 ( sin θ )2 d θ 2 π /4n 0 − sin θ )2 d θ 27. π − 8 1 ( cos 2θ )2 d θ 2 π /4n 0 29. π 6 − 31. For r = a cos 2nθ , area of one petal is a2 Total area = 35. (5/6, 0) π a2 . 2 37. 9π 2 39. cos2 2nθ d θ = π a2 . For r = a sin 2nθ , area of one petal is a2 8n sin2 2nθ d θ = π a2 . 8n √ 4π +2 3 3 41. (a) Substitute x = r cos θ , y = r sin θ into the equation and solve for r . (c) 8 − 2π SECTION 9.6 1. 4x = ( y − 1)2 3. y = 4x2 + 1, x ≥ 0 5. 9x2 + 4y2 = 36 7. 1 + x2 = y2 9. y = 2 − x2 , −1 ≤ x ≤ 1 11. 2y − 6 = x, −4 ≤ x ≤ 4 ...
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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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