A-44 Answers to Selected Exercises (c) no 91. r sin u = k 92. r = k sin u 93. r = k csc u 75. x + y = 2 y 2 x 2 0 r = x + y = 2 cos U + sin U 2 1808 08 908 2708 3 r2 = 9 sin 2U lemniscate 1808 08 908 2708 2 r2 = 4 cos 2U lemniscate 59. 61. 0 P r = U 3P 2 P 2 1808 08 908 2708 4 r = 2 sin U tan U 63. 65. y 2 x –2 –2 0 r = 1 + sin U 2 1 4 y = – x2 + 1 y x r = 2 sin U x2 + (y – 1)2 = 1 2 0 1 67. x2 + 1y - 122 = 1 69. y = -1 4x 2 + 1 y x 0 r = 2 sec U x = 2 2 0 y x (–1, –1) r = –2 cos U – 2 sin U (x + 1)2 + (y + 1)2 = 2 √2 71. 1x + 122 + 1y + 122 = 2 73. x = 2 –60 –40 40 60 Earth is closest to the sun. –2.4 –1.6 1.6 2.4 89. (a) (b) –20 –20 20 20 r = 1.5u, −4p ≤ u ≤ 4p –15 –15 15 15 r = u, 0 ≤ u ≤ 4p 81. 83. 85. A2, p 6 B , A2, 5p 6 B , 10, 02 87. Q 4 + 22 2 , p 4R , Q 4 - 22 2 , 5p 4 R 77. r = 2 2 cos u + sin u 79. (a) 1r, -u2 (b) 1r, p - u2 or 1-r, -u2 (c) 1r, p + u2 or 1-r, u2 95. r cos u = k 96. r = k cos u 97. r = k sec u x y 0 3 r = 3 csc U y = 3 y = 3 94. x y 0 3 r = 3 sec U x = 3 x = 3 98. 8.8 Exercises 1. 110, -32 3. A1 2 , 23 B 5. C 7. A x y x = √t y = 3t – 4 for t in [0, 4] (0, –4) (2, 8) 0 2 8 (b) y = 3x2 - 4, for x in 30, 24 x y x = t + 2 y = t2 for t in [–1, 1] 0 1 1 2 3 (b) y = x2 - 4x + 4, for x in 31, 34 9. (a) 11. (a) y x 0 x = 2 sin t y = 2 cos t for t in [0, 2P] 2 2 (b) x2 + y2 = 4, for x in 3-2, 24 x y x = t3 + 1 y = t3 – 1 for t in (–`, `) 0 2 –2 4 (b) y = x - 2, for x in 1-∞, ∞2 13. (a) 15. (a) y x (1, 1) 0 1 1 5 x = sin t y = csc t for t in (0, P) (b) y = 1 x , for x in 10, 14 y x x = 3 tan t y = 2 sec t –3 0 3 for t in (– , ) P 2 P 2 (b) y = 231 + x 2 9 , for x in 1-∞, ∞2 17. (a) 19. (a) y x 0 1 3 2 x = 2 + sin t y = 1 + cos t for t in [0, 2P] (b) 1x - 222 + 1y - 122 = 1, for x in 31, 34 y x 4 –4 4 0 x = t y = √t2 + 2 for t in (–`, `) (b) y = 2x2 + 2, for x in 1-∞, ∞2 21. (a) 23. (a) y x x = t + 2 y = t – 4 for t in (–`, `) 0 6 –6 (b) y = x - 6, for x in 1-∞, ∞2 y x 0 2 2 –2 –2 x = t + 2 y = for t 3 –2 t + 2 1 (b) y = 1 x , for x in 1-∞, 02 ´10, ∞2 25. (a) 27. (a)
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