EVEN09 - CHAPTER 9 Conics, Parametric Equations, and Polar...

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Unformatted text preview: CHAPTER 9 Conics, Parametric Equations, and Polar Coordinates Section 9.1 Section 9.2 Section 9.3 Section 9.4 Section 9.5 Section 9.6 Conics and Calculus . . . . . . . . . . . . . . . . . . . . 424 Plane Curves and Parametric Equations . . . . . . . . . . 434 Parametric Equations and Calculus . . . . . . . . . . . . 439 Polar Coordinates and Polar Graphs . . . . . . . . . . . . 444 Area and Arc Length in Polar Coordinates . . . . . . . . .452 Polar Equations of Conics and Kepler’s Laws . . . . . . . 458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 Review Exercises Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 CHAPTER 9 Conics, Parametric Equations, and Polar Coordinates Section 9.1 Conics and Calculus Solutions to Even-Numbered Exercises x 16 Center: 2, 1 Ellipse Matches (b) 2 2 2. x2 p 8y 2>0 4. y 4 1 2 1 Vertex: 0, 0 Opens upward Matches graph (a). x2 9 y2 9 6. 1 8. x 9 2 2 y2 4 1 Circle radius 3. Matches (g) Hyperbola Center: 2, 0 Horizontal transverse axis. Matches (d) 10. x2 8y x2 0 4 2y −8 −4 −4 y 12. x 1 2 8y x 2 1 2 0 4 2y 2 y Vertex: 0, 0 Focus: 0, Directrix: y 2 2 (0, 0) x 4 8 Vertex: 1, Focus: 1, Directrix: y 2 4 0 −8 −4 −4 x 4 8 −8 − 12 (1, − 2) −8 − 12 14. y 2 6y y2 8x 6y y 25 9 3 3 3 0 2 0 8x 4 25 2 y 8 4 − 20 − 16 − 12 −8 −4 x 16. y 2 9 4y y2 8x 4y y 12 4 2 2 0 8x 4 12 2 y 4 4 2x 2x Vertex: Focus: 2, 4, Vertex: 2, Focus: 0, Directrix: x 2 2 4 −6 −4 −2 −4 2 Directrix: x x 4 6 (− 2, − 3) −8 − 12 (2, − 2) −6 −8 424 S ection 9.1 18. y 6y 6y x x 10 4 4 2 2 Conics and Calculus 425 1 6 x2 4 4 2 2 8x 10 6 1 6 x2 8x 16 10 20. x2 2x x2 8y 2x x 1 1 3 1 9 1 2 0 8y 4 9 1 1 2 −8 10 x x 6y 4 3 2 2y 5 3 Vertex: 1, 5 3 4 y Focus: 1, Directrix: y Vertex: 4, 5 3 Focus: 4, 1 6 Directrix: y 19 6 −2 − 10 10 −4 22. x2 2x x 8y 1 2 4 0 2y 2 24. Vertex: 0, 2 y y 2 15 2 2 42 x 0 8 or p 0 8x 4y 4 26. x 2 y 4x y 4 0 x 2 2 4x x2 28. From Example 2: 4p Vertex: 4, 0 x x2 8x 2 2 4 2 8y 0 0 8y y 4 14 2 16 30. 5x2 x2 14 a2 7y 2 y2 10 14, b2 70 6 y 32. x 1 a2 2 y 1 −5 −4 −3 −2 −1 −1 x 1 10, c2 4 −6 −2 4 2 x 2 −4 −6 4 6 1, b2 2, 12 ,c 4 4 3 4 (− 2, − 4) −2 −3 −4 Center: Foci: Vertices: e Center: 0, 0 Foci: ± 2, 0 Vertices: ± e 34. 16x2 16 x2 2 14 25y2 4x 4 14, 0 14 7 64x 25 150y y2 6y 279 0 3 , 2± 2 1, 3 2 4, 4 3, 4 −5 0 279 10 64 225 y 1 x 5 a2, , b2 8 Center: 2, Foci: 2± 22 ,c 5 3 2 58 a2 2 y 3 25 9 40 2 1 −1 −1 1 2 3 4 −2 −3 −4 x b2 (2, −3) 3 10 , 20 10 , 4 3 3 Vertices: e c a 3 5 2± 426 36. Chapter 9 36x2 36 x2 4 x 3 9y 2 4 9 Conics, Parametric Equations, and Polar Coordinates 48x 9 y2 36y 4y 43 4 9 0 43 16 36 x 23 14 3 4 2 y 1 2 2 1 a2 1, b2 12 ,c 4 2 ,2 3 Center: Foci: Vertices: Solve for y: 9 y2 3 2 ,2 ± 3 2 2 ,3 , 3 2 ,1 3 3 4y y 4 2 2 36x2 36x2 48x 48x 9 43 7 36 −4 −1 2 y 2± 1 3 36x2 48x 7 (Graph each of these separately.) 38. 50x 2x2 2 y2 25y 2 12 x 5 4.8x 120x 36 25 y 5, c2 6.4y 160y 25 y 2 16 5 10 5 2 3.12 78 32 y 5 1 0 0 256 25 78 72 256 250 50 x2 x a2 65 5 10, b2 2 Center: Foci: Vertices: 6 16 , 55 6 16 , ± 55 5 10 10.24 y 3.2 2 7 6 16 , ± 55 6.4y Solve for y: y2 2x2 7.12 3.2 ± 4.8x 4x 7.12 2x2 3.12 10.24 −7 −1 5 y 40. Vertices: 0, 2 , 4, 2 1 2 Horizontal major axis Eccentricity: Center: 2, 2 a x 4 2, c 2 2 4x 2x2 (Graph each of these separately.) 42 Foci: 0, ± 5 Major axis length: 14 Vertical major axis Center: 0, 0 1⇒b y 3 2 2 3 1 c x2 24 5, a y2 49 7⇒b 1 24 S ection 9.1 x2 25 a y Conics and Calculus 427 44. Center: 1, 2 Vertical major axis Points on ellipse: 1, 6 , 3, 2 From the sketch, we can see that h 1, k 2, a 4, b 2 x 4 1 2 46. y2 9 5, b 1 3, c a2 b2 34 y 10 8 6 4 2 Center: 0, 0 (1, 6) 6 Vertices: ± 5, 0 Foci: ± (3, 2) y 16 2 2 1. (1, 2) −4 −2 −2 x 4 34, 0 3 ±x 5 −8 Asymptotes: y − 4 −2 −4 −6 −8 −10 x 4 8 10 48. y a 1 122 2 x 52 5, c 4 2 1 a2 b2 13 50. y 2 y 2 9x2 9 x2 2 36x 4x x 4 2, c 72 4 2 2 0 72 1 36 36 12, b Center: 4, 1 Vertices: 4, 11 , 4, 13 Foci: 4, 14 , 4, 12 12 Asymptotes: y 1± x 5 y 20 y 36 a 4 6, b a2 b2 2 10 Center: 2, 0 Vertices: 2, 6 , 2, Foci: 2, 2 10 , 2, Asymptotes: y y 6 2 10 2 ±3 x 5 x −5 1 2 6 7 8 5 x 4 −5 5 − 20 −2 −1 52. 9 x2 6x 9 9x y 4 y2 3 2 2 2y 4y x 1 1 2 2 78 1 1 81 4 1 54. 9 x2 9x 2 6x y2 9 54x y2 10y 10y 55 25 0 55 1 81 25 1 14 3 19 x y 3 a 1 ,b 2 1 ,c 3 3, 1 13 6 2 3 19 2 y 1 5 2 1 Center: Vertices: Foci: a 1 ,b 3 1, c 3, 5 10 3 10 3, 1 , 2 3, 13 3 2 −3 −1 1 x −1 Center: Vertices: Foci: 1 3, 1 ± 6 1 3 ± ,5 3 3± 10 ,5 3 −8 0 2 Asymptotes: y 1± 3 x 2 3 Solve for y: y2 10y y 25 5 2 9x2 9x2 5± 54x 54x 9x2 55 80 54x 25 y 80 (Graph each curve separately.) 428 56. Chapter 9 3y 2 3 y2 4y 4 y 1 a 1, b 3, c Conics, Parametric Equations, and Polar Coordinates x2 x2 2 2 6x 6x x 3 2 12y 9 3 2 0 0 1 6 58. Vertices: 0, ± 3 12 9 3 Asymptotes: y ± 3x Vertical transverse axis a 3 a b ±3 Slopes of asymptotes: ± Thus, b 1. Therefore, x2 1 1. 10 Center: 3, 2 Vertices: 3, 1 , 3, 3 Foci: 3, 0 , 3, 4 Solve for y: 3 y2 4y y 4 2 2 −4 −4 y 9 2 x2 x 2 6x 6x 3 x2 12 12 6x 3 12 y 2± (Graph each curve separately.) 60. Vertices: 2, ± 3 Foci: 2, ± 5 Vertical transverse axis Center: 2, 0 a 3, c 5, b2 x 16 c2 2 2 62. Center: 0, 0 Vertex: 3, 0 Focus: 5, 0 a2 1. Therefore, 16 Horizontal transverse axis a 3, c 5, b2 x 9 2 y2 Therefore, 9 c2 y2 16 1. a2 16 64. Focus: 10, 0 Asymptotes: y ±x 66. (a) 3 4 y2 4 y At x x2 2 4x 2y 1, y 2 2x y 2x2 4, 2yy 4x 0, Horizontal transverse axis Center: 0, 0 since asymptotes intersect at the origin. c 10 3 ± and b 4 a2 3 a 4 b2 At 4, 6 : y At 4, 64 and 36. 6: y 6 6 b Slopes of asymptotes: ± a c2 a2 b2 100 4: y ± 6, y ±2 4 6 4 x 3 4 x 3 ± 4 3 3y 3y 2 2 0 0 4 or 4x 4 or 4x Solving these equations, we have Therefore, the equation is x2 64 y2 36 1. (b) From part (a) we know that the slopes of the normal lines must be 3 4. At 4, 6 : y At 4, 6: y 6 6 3 x 4 3 x 4 x 4 or 3x 4 or 3x 4y 4y 36 36 0 0 68. 4x2 A y2 4, C 4x 1 3 0 70. 25x2 A 10x 25, C 200y 0 119 0 72. y2 A 4y 1 5 0 0, C AC < 0 Hyperbola Parabola Parabola S ection 9.1 74. 2x2 A 2x2 y2 2, C 2xy 3y 3y 0 y2 2xy 76. 9x2 A 4y2 9, C 9x2 54x 54x 16y Conics and Calculus 81 61 36 0 4 y2 4y 429 4 1, AC > 0 4, AC > 0 Ellipse Ellipse c ,c a 78. (a) An ellipse is the set of all points x, y , the sum of whose distance from two distinct fixed points (foci) is constant. (b) x a2 h 2 80. e a2 b2 0<e<1 For e 1 For e 0, the ellipse is nearly circular. 1, the ellipse is elongated. y b2 k 2 1 or x b2 h 2 y a2 k 2 82. Assume that the vertex is at the origin. (a) x2 82 1600 3 x2 4py 4p p 4 1600 y 3 6400 y 3 3 100 84. (a) Without loss of generality, place the coordinate system so that the equation of the parabola is x2 4py and, hence, y 1 x. 2p Therefore, for distinct tangent lines, the slopes are unequal and the lines intersect. (b) x2 2x 128 3 ± 6.53 meters. 4x 4 4 4y dy dx dy dx 0 0 1 x 2 1 (b) The deflection is 1 cm when y 2 ⇒x 100 y 5 100 4 100 ± At 0, 0 , the slope is 1: y x. At 6, 3 , the slope is 2: y 2x 9. Solving for x, x 3 ( 0, 100 ) 3 ( 8, 100 ) 2x 9 3 3. 9 ( 3 −8, 100 ) 2 100 1 100 3x x −8 −4 x 4 8 y Point of intersection: 3, 86. The focus of x2 d x 8y 0 2 3 4 2 y is 0, 2 . The distance from a point on the parabola, x, x2 8 , and the focus, 0, 2 , is x2 8 2 2. y 4 3 Since d is minimized when d 2 is minimized, it is sufficient to minimize the function fx fx fx x3 16 x2 2x x2 8 2 x2 8 2 2. 2 x 4 x3 16 x. x 2 = 8y (0, 2) 1 −3 −2 −1 −2 1 () x 2 3 x, x 8 2 0 implies that x x x 16 2 1 0⇒x 0. This is a minimum by the First Derivative Test. Hence, the closest point to the focus is the vertex, 0, 0 . 430 Chapter 9 Conics, Parametric Equations, and Polar Coordinates 12 y 4 1 y 2 1 1 0 88. (a) C (b) 32 0 0.0853t2 0.2917t 263.3559 90. x x 1 x 4 2 0 0 18 y2 4 y2 dy 4 y2 4 ln y 20 5 5.916 1 2 4 dC (c) dt y 5 4 3 2 1 0.1706t 0.2971 s 4 0 y2 dy 4 1 y4 4 1 4 20 4 25 x 1 2 3 4 5 4 y2 0 4 ln 4 ln 2 4 ln 2 −1 −1 The consumption of fruits is increasing at a rate of 0.1706 pounds/year. h 92. x2 y y S 20y x 20 x 10 r 2 94. A 2 0 4py dy h 4p 0 y1 2 dy 23 y 3 2 h 2 0 4p x 0 2 1 x 10 x2 2 r dx r 2 0 x 100 10 r2 32 x2 dx 8 3 ph3 10 2 100 3 32 0 15 100 1000 96. (a) At the vertices we notice that the string is horizontal and has a length of 2a. (b) The thumbtacks are located at the foci and the length of string is the constant sum of the distances from the foci. Focus Vertex Focus Vertex 98. e 0.0167 c c a c 149,570, 000 2,497,819 c c 147,072,181 km 152,067,819 km Least distance: a Greatest distance: a 100. e A A P P 4000 4000 0.9367 119 119 4000 4000 102. x2 a2 x2 a2 a2 x2 a2 a2 x2 a2 a2 y2 y2 b2 1 1 1 1 122,000 122,000 121,881 130,119 y2 b2 a2 c2 a2 y2 e2 a2 1 As e → 0, 1 x2 a2 y2 a2 e2 → 1 and we have 1 or the circle x2 y2 a2. S ection 9.1 x2 4.5 y2 2.5 Conics and Calculus 431 104. 2 2 1 4.5 9 ± 5 2 y x2 x V V 90 1 2.5 2 y2 2.5 y 2 5 ft x 2 3 ft 9 ft Area of bottom Length 4.5 2.5 2 144 5 36x 8yy 0.5 Area of top Length 9 5 y2 2.5 2 16 1 y 2 24y 36 8y 16 0 y 2 dy (Recall: Area of ellipse is ab.) y 2.5 0.5 2.5 2 2.5 2 arcsin 90 0 72 0.5 6 5 2.5 2 arcsin 1 5 318.5 ft3 106. 9x2 4y 2 18x 36 24y 24 y y 0 0 18x 36 18x 36 8y 24 3. y 0 when x 2, y At x At y 3, x 2. y undefined when y 0 or 6. 2, 0 , 2, 6 4, 3 2 4 3 x 16 2 4 2 Endpoints of major axis: 0 or 4. Endpoints of minor axis: 0, 3 , Note: Equation of ellipse is x y 9 3 2 1 4 108. (a) A 4 0 3 4 16 x2 dx x2 16 arcsin 9 8 x 4 4 12 0 (b) Disk: V y y 1 4 2 0 9 16 16 x2 x2 dx 16x 13 x 3 4 48 0 3 4 16 3x 4 16 x2 1 x2 7x2 9x2 16 16 x2 4 y 3 4 2 S 22 0 16 16 16 x2 9x2 dx 2 16 16 x 256 arcsin 7x 16 4 0 4 0 3 4 16 x2 256 4 16 7x2 dx x2 7 4 3 4 138.93 4 256 0 7x2 dx 3 87 7x 256 3 48 7 87 256 arcsin —CONTINUED— 432 Chapter 9 Conics, Parametric Equations, and Polar Coordinates 108. —CONTINUED— 4 (c) Shell: V 4 0 x 3 4 y2 16 x2 dx 3 1 2 2 16 3 4 x2 32 0 64 x x 2 4 3 9 4y 39 1 y2 16y 2 9 9 y2 99 y2 16y 2 dy 2 99 y 1 3 x 4 3 81 S 22 0 3 9 y2 7y 2 dy 4 0 4 9 16 927 8 3 7 12 97 x2 a2 2x a2 y2 b2 2yy b2 y At P, y b2 a2 3 7y 81 7y 2 81 ln 12 7y 81 ln 9 81 7y 2 0 81 ln 3 7 168.53 110. (a) 1 0 xb2 ya2 x0 y0 m. y0 b2x0 a2y0 b2x0 a2y0 a2b2 x0y0c2 b2 y0c b2x0 a2y0 b2x0 a2y0 (b) Slope of line through c, 0 and x0, y0 : m1 y0 x0 y0 x0 c c Slope of line through c, 0 and x0, y0 : m2 (c) tan m2 m 1 m2m x0 1 c y0 x0 b2x0c a2y0c arctan y0 c a2y02 b2x0 x0 c a2y0 x0 c b2x0 y0 b2 a2 x0c y0c x0c a2 b2 y0c a2y02 b2x02 x0y0 a2 b2 arctan b2 y0c b2x0c a2y0c tan m1 m 1 m1m a2y02 a2x0y0 arctan x0 1 b2x02 a2cy0 c y0 x0 b2x0c b2x0 y0 c a2y02 b2x0 x0 c a2y0 x0 c b2x0y0 b2 a2 x0c y0c x0c a2 b2 y0c a2b2 b2x0c x0y0 a2 b2 a2cy0 b2 y0c , the tangent line to an ellipse at a point P makes equal angles with the lines through P and the foci. Since S ection 9.1 c a x a x a (b) x 4 7 Conics and Calculus 3, 0 and 433 3, 3 112. (a) e a2 b2 ⇒ ea a h 2 2 2 2 a2 b2. Hence, 114. The transverse axis is vertical since are the foci. Center: c 3 , 2a 2 3, 3 2 c2 a2 5 4 y b 2 k 2 1 1. 1 h 2 2 y a1 2 k2 e2 2, b2 2 y 41 32 e2 Therefore, the equation is y 32 1 2 x 3 54 2 1. −3 −1 9 (c) As e approaches 0, the ellipse approaches a circle. 116. Center: 0, 0 Horizontal transverse axis Foci: ± c, 0 Vertices: ± a, 0 The difference of the distances from any point on the hyperbola is constant. At a vertex, this constant difference is a c c a 2a. Now, for any point x, y on the hyperbola, the difference of the distances between x, y and the two foci must also be 2a. x c 2 y 0 2 x c 2 y c c 4xc xc 2 2 0 2 2a 2a 4a2 4a a x x c x 4a c 2 x x y2 y2 4a2 a2 a4 a2y2 y 2 c x 2 2 y2 c y2 2 y2 x c 2 y2 y2 c2 y2 ( x, y ) y x2c2 x2 c2 x2 a2 Since a2 b2 c2, we have x2 a2 2a2cx a2 c2 a2 x2 a2 1 c2 2cx a2 (c, 0) (− c, 0) (− a, 0) (a, 0) x a2 1. y 2 b2 118. c b 150, 2a 150 x2 932 2 0.001 186,000 , a 93 2 2 93, 120. 2x a2 x2 a2 y2 b2 2yy b2 1 0 or y b2x0 x a2y0 b2x0x b2x0x b2x 0 x 1 b2x a2y x0 b2x02 a2y0 y a2y 0 y 13,851 1 y 13,851 When y x2 x 75, we have 932 1 752 13,851 a2y0y b2x02 y y0 a2y02 a2y02 a2b2 110.3 miles. x0x a2 y0 y b2 434 122. Chapter 9 Conics, Parametric Equations, and Polar Coordinates Ax2 A x2 Cy2 D x A C y2 D 2A C 2 Dx C y2 E y C y Ey F E y C 0 F F (Assume A 0 and C 0; see (b) below) A x2 D x A D2 4A 2 x E2 4C2 E 2C A 2 D2 4A E2 4C R R AC (b) If C 0, we have Ax If A D 2A 2 (a) If A x C, we have D 2A 2 y E 2C 2 R A F Ey D2 . 4A which is the standard equation of a circle. (c) If AC > 0, we have x D 2A R A 2 0, we have Cy E 2C 2 F Dx E2 . 4C y E 2C R C 2 These are the equations of parabolas. 1 (d) If AC < 0, we have x D 2A R A 2 which is the equation of an ellipse. y E 2C R C 2 ±1 which is the equation of a hyperbola. 124. True 126. False. The y4 term should be y2. 128. True Section 9.2 2. x 4 cos2 Plane Curves and Parametric Equations y 2 sin 2≤y≤2 (c) 2 4 2 2 0 4 0 4 2 2 2 −1 5 3 0≤x≤4 (a) x y (b) 3 2 1 y 0 2 0 2 (d) −3 x 4 y2 4 x 4 y2 4 x cos2 sin2 1 4 y2, 2≤y≤2 x 1 −2 2 3 5 (e) The graph would be oriented in the opposite direction. S ection 9.2 4. x y y 2y y Plane Curves and Parametric Equations 2t2 t4 x 2 2 435 3 2 2 3x 2t 3t 3 3 2 13 0 x 6. x y y 1 1 x2 4 1, x ≥ 0 For t < 0, the orientation is right to left. For t > 0, the orientation is left to right. y 6 4 2 x 2 4 6 8 6 5 4 3 2 1 −1 −1 x 1 2 3 4 5 6 8. x t2 t, y t2 t Subtracting the second equation from the first, we have x y y x 4 2t or t y 2 x 2 y y t x y 2 6 2 1 0 2 0 0 0 1 2 0 2 4 y x 2 6 2 3 2 Since the discriminant is B2 4AC 2 2 41 1 0, −1 −1 x 2 3 4 the graph is a rotated parabola. t, t ≥ 0 t x 4, y 3 2 1 −3 −2 −1 x −1 −2 −3 −2 10. x y y 4 12. x y 1 t 1 1 x y 1 1 t 1 1 implies t t 1 1 1 x 1 14. x y x y 5 4 3 2 1 x 2 t t y 1 2 2 1 y 3 3 3 x≥0 x y 1 2 3 x 1 2 3 4 5 −3 16. x y y e t, x > 0 e2t x 2 y 3 2 18. x y sec2 tan2 sec2 tan2 1 1 4 3 2 1 y 1 1 1 x2 1, x > 0 −3 −2 −1 1 x −1 −2 −3 1 3 y x x≥0 x 1 2 3 4 436 Chapter 9 Conics, Parametric Equations, and Polar Coordinates 20. x y x 2 2 2 cos 6 sin cos2 sin2 1 22. x y y 1 x2 y cos 2 sin 2 4 sin cos sin2 ± 4x 3 24. x y x y 4 1 1 2 2 2 4 1 2 cos 2 sin y 6 x2 4 2 4 cos2 4 sin2 4 y2 36 y 1 ellipse 1 x2 x 3 4 2 y 4 2 − 6 −4 4 6 x −2 −1 2 7 −2 −3 −5 26. x y x2 y2 sec tan −3 2 28. 3 x y x2 3 3 cos3 sin3 cos2 sin2 −2 1.5 2 sec2 tan2 −2 y2 − 1.5 30. x y t y ln 2t t2 ex 2 e 2x r 1 2x e 4 −2 3 32. x y 3 e2t et x −1 4 y2 3 −1 −1 y>0 y x, x > 0 34. By eliminating the parameters in (a) – (d), we get x2 domains. These curves are all smooth. (a) x 2 cos , y y 3 y2 4. They differ from each other in orientation and in restricted (b) x 4t 2 t 1 2 4 1 t2 y y 1 t 0 2 sin x ≥ 0, x y 1 −3 −1 x 1 3 2 1 x 1 −1 −2 3 −3 −1 (c) x x≥0 y t y y≥0 4 t (d) x 4 2<x≤0 e2t y et y>0 y 3 3 2 1 1 x 1 2 3 −3 −2 −1 x S ection 9.2 Plane Curves and Parametric Equations 437 36. The orientations are reversed. The graphs are the same. They are both smooth. 38. The set of points x, y corresponding to the rectangular equation of a set of parametric equations does not show the orientation of the curve nor any restriction on the domain of the original parametric equations. 40. x y cos sin cos2 x h 2 h k x r y r x r cos r sin h k h r2 2 42. x y x a y b k k b2 2 h k sec tan 1 a sec b tan h sin2 y k 2 y r2 k 2 1 x a2 h 2 y r2 46. From Exercise 40 we have x y 1 3 3 cos 3 sin . 48. From Exercise 41 we have a 5, c x y 4 2 3⇒b 5 cos 4 sin . 4 44. From Exercise 39 we have x y 1 4 4t 6t. Solution not unique Solution not unique Center: 4, 2 Solution not unique 2 x 1 50. From Exercise 42 we have a 1, c x y 2⇒b 3 tan sec . 3 52. y 54. y x2 Example x x t, y t, y 2 t 1 2 t 1 Example x x t, t 3, y y t2 t6 Center: 0, 0 Solution not unique The transverse axis is vertical, therefore, x and y are interchanged. 56. x y 1 sin cos 6 58. x y 2 2 4 sin 4 cos 9 60. x y 2 2 4 sin cos −6 −2 6 −9 −3 9 − 5 0 Not smooth at x 2n 1 Smooth everywhere 438 Chapter 9 3t 1 3t 2 1 t3 t3 Conics, Parametric Equations, and Polar Coordinates 62. x y 2 −3 3 64. Each point x, y in the plane is determined by the plane curve x f t , y g t . For each t, plot x, y . As t increases, the curve is traced out in a specific direction called the orientation of the curve. −2 Smooth everywhere 66. (a) Matches (ii) because 68. x y cos3 2 sin 2 1 ≤ x ≤ 0 and 1 ≤ y ≤ 2. (b) Matches (i) because x 70. x y cot 4 sin cos y 2 2 1 for all y. Matches (a) Matches (c) 72. Let the circle of radius 1 be centered at C. A is the point of tangency on the line OC. OA 2, AC 1, OC 3. P x, y is the point on the curve being traced out as the angle changes AB AP . AB 2 and AP ⇒ 2 . Form the right triangle CDP. The angle and 2 OCE DCP x y Hence, x 74. False. Let x 76. (a) x y t OE EC Ex CD 3 cos 2 3 sin 3 sin cos 3 , y t. Then x 2 cos 3 3 sin 2 sin 3 3 2 . cos 3 y 3 C 2 1 A α D P = ( x, y ) x θ 1 2 3 sin sin 3 . 3 cos sin 3 BE x 2 t 2 and y t t y 2 and y is not a function of x. v0 cos h x v0 cos v0 sin 16t 2 h h h v0 sin tan tan , and 16 2 v02 80. 0 −5 250 ⇒y y x v0 cos x 16 x v0 cos 2 16 sec2 2 x v02 16 sec2 2 x v02 (c) 80 (b) y h 5 x 0.005x 2 1⇒ 16 sec2 v02 32 0.005 4 x 5, tan 0.005 v02 4 6400 ⇒ v0 (d) Maximum height: y Range: 204.88 55 at x 100 Hence, x y 80 cos 45 t 5 80 sin 45 t 16t 2. Section 9.3 Parametric Equations and Calculus 439 Section 9.3 2. dy dx dy dt dx dt t, y dy dx d 2y dx2 3t Parametric Equations and Calculus 1 1 3t 2 1 6t 6 when t 1. 3t2 3 3 4. dy dx dy d dx d t2 dy dx d 2y dx2 3t 2 2t 3 1 2e 2e 2, y 2t 2 1 e 4 3 2 1 4e3 2 6. x 8. x 3 12t t 12t 3 2 when t 3 3 0. 4 3 4 when t 9 0. 6 concave upwards 2 2 2t 2t 3 2t 2 concave downward 10. x cos , y dy dx d 2y dx2 3 cos sin 3 csc2 sin 3 sin 3 cot 3 sin3 dy is undefined when dx d 2y is undefined when dx2 0. 0. 12. x dy dx t, y t 1 14. x dy dx 2 when t 2. 1 t 1 d dx2 2y sin , y 1 sin cos 1 1 cos 0 when cos cos cos 2 12t 1 12t t t 1 t . sin2 d 2y dx2 t 12t 1 13 t12 t 12t 1 when t 2. cos 1 1 1 cos 2 1 2 1 when 4 . concave downward concave downward 16. x 2 dy dx At 3 cos , y 2 cos 3 sin 3 2 cot 3 0, and 1 dy , and 2 dx 5 7 dy , and 6 dx 23 . 3 (c) 0. (b) At t dx dt dy dx dy is undefined. dx 2 sin 18. x (a) −3 t 1, y 4 1 t 1, t 1 1, 3 , 5 Tangent line: x At 2, 5 , Tangent line: y At 4 33 ,2 , 2 −4 1, x, y 1, dy dt 1, 0, 2 , and dy dx 2 y 1 1x x 2 0 1. At 0, 2 , y Tangent line: y 2 3x 3y 43 2 3 23 x 3 0 −4 4 33 2 (d) −3 4 5 440 Chapter 9 Conics, Parametric Equations, and Polar Coordinates 3 4 (b) At 6 20. x (a) 4 cos , y 4 3 sin , 3 , x, y 4 dx dt 2 2, 4 43 , , and 2 2 dy dt 3 2 dy , 2 dx 3 4 −6 −4 (d) 43 , , 2 2 y 3 2 y 3 x 4 3 x 4 4 2 32 −6 (c) dy dx 3 . At 4 6 −4 22. x t2 t, y t3 3t 1 or t 1 crosses itself at the point x, y 2. 2, 1 . At this point, t dy dx At t At t 2, 3t2 2t 1, dy dx 9 3 21 3 1 0 and y 3 and y 1. Tangent Line 1 3x 2 or y 3x 5. Tangent Line dy dt 24. x 2,y cos dy d 2 sin 0 when 0, ± , ± 2 , . . . . Horizontal tangents: Points: 4n , 0 , 2 2n 1 , 4 where n is an integer. Points shown: 0, 0 , 2 , 4 , 4 , 0 Vertical tangents: dx d 3t dy dt 2t 3 0 when t 3 . 2 2 0; none 26. x t 1, y t2 28. x t2 t 2, y t3 dy dt 3t 3t 2 3 0 when t ± 1. Horizontal tangents: Point: 1 , 2 9 4 Horizontal tangents: Points: 2, 2 , 4, 2 dx dt 2t 1 0 when t 1 . 2 Vertical tangents: dx dt Vertical tangents: 1 0; none Point: 7 , 4 11 8 30. x cos , y 2 sin 2 dy d 4 cos 2 2 , 2 dx d 1, 0 2, 0 when 2 ,2 , 2 0 when 357 ,,,. 4444 2 , 2 0, . 2 Horizontal tangents: Points: 2 ,2 , 2 Vertical tangents: Points: 1, 0 , sin S ection 9.3 32. x 4 cos2 , y 2 sin dy d 2 cos 0 when 2, exclude them. 0 when 3 ,. 22 34. x Parametric Equations and Calculus cos dy d sin 0 when x 441 cos2 , y Horizontal tangents: Since dx d 0 at Horizontal tangents: Since dx d 0, . 2 and 3 dx d 0 at these values, exclude them. dx d 2 cos sin 3 ,. 22 Exclude 0, . 0 when Vertical tangents: 8 cos sin 0, . Vertical tangents: Point: 4, 0 Point: 0, 0 36. x dx dt s t2 2t, 0 1, y dy dt 4t2 1 4t3 12t2, 3, dx dt 2 1≤t≤0 dy dt 0 2 38. x 4t2 144t4 36t2 dt 2 arcsin t, y 1 1 12 ln 1 t 2, 0 ≤ t ≤ t 1 2 1 2 t2 dx dt s dy , t 2 dt dx dt 1 1t ln 2t 1 2t 2 1 t2 2 144t4 dt 1 232 0 1 2t 1 1 1 54 373 0 dy dt dt dt 12 1 36t 54 4.149 12 0 1 t2 2 1 1 12 0 1 1 t2 dt 0 1 1 ln 2 3 t5 10 2 1 ln 3 2 dx d 0.549 40. x t, y S 1 1 dx , 6t 3 dt 1 t4 2 t4 2 1 2t 4 1, 1 2t 4 2 2 dy dt dt t4 2 1 2t 4 42. x a cos , y 2 a sin , a2 sin2 a sin , a2 cos2 2 dy d a cos S 4 0 2 d 2 1 2 1 4a dt 0 d 4a 0 2a t4 2 1 dt 2t 4 1 6t 3 2 1 t5 10 779 240 dx d 44. x dy d cos sin 2 sin , y sin cos , cos S 0 2 2 cos2 22 2 sin2 d 2 2 d 0 2 0 442 Chapter 9 4t 1 t3 Conics, Parametric Equations, and Polar Coordinates 4t 2 1 4 xy 4 46. x ,y t3 (b) dy dt 1 t 3 8t 4t 2 3t2 32 1t 0 when t 43 2 43 4 , 3 3 0 or t 3 (a) x 3 y3 4t 2 t3 1 t3 2 6 2. −6 Points: 0, 0 , −4 1.6799, 2.1165 1 (c) s 2 0 1 41 1 t8 4t 6 2t 3 t3 2 2 4t 2 t 3 1 t3 2 4t 2 2 1 dt 1 dt 2 0 1 16 t3 4 t8 4t 6 4t 5 4t 3 4t 2 1 dt 8 0 4t 5 4t 3 1 t3 2 6.557 48. x dx d 3 cos , y 3 sin , 2 4 sin dy d 4 cos −6 4 6 s 0 9 sin2 16 cos2 d 22.1 −4 50. x t, y 2 4 2 2t, 4 dx dt 2t t2 1 1, dy dt 4 dt 2 52. x dx dt 13 t ,y 3 t2, S dy dt t 1 2 1, 1 ≤ t ≤ 2, y-axis (a) S 0 2 2 5 4t 2 8 0 5 2 2 1 13 4 tt 3 2 2 1 dt 2 9 x4 1 32 1 (b) S 2 0 t1 4 dt 5 t2 0 4 5 9 173 23 23.48 54. x a cos , y 2 b sin , b sin dx d a2 sin2 1 1 a sin , dy d b cos (a) S 4 0 2 b2 cos2 d a2 a2 b2 cos2 d 4ab e 2 0 2 4 0 ab sin 2ab e e cos e sin 0 1 e2 e2 cos2 arcsin e d e2 cos2 arcsin e cos a2 b2 a 2 b2 2 ab e e1 2 b2 e a2 b2 a 2 a2b arcsin a2 b2 c : eccentricity a ab arcsin e e —CONTINUED— S ection 9.3 54. —CONTINUED— 2 Parametric Equations and Calculus 443 (b) S 4 0 a cos a2 sin2 b2 cos2 d 2 4 0 a cos c sin c b2 b2 c2 b2 c2 sin2 c2 sin2 b2 ln c a2 b d 4a c 2 c cos 0 b2 c 2 sin2 c 2 sin2 2 d 2a c 2a c 2 a2 b2 ln c sin b2 b2 c2 b2 b2 ln b 0 2 ab2 a ln 2 2 a b 2 a2 b2 1 ln e 1 e e 56. (a) 0 (b) 4 58. One possible answer is the graph given by x t, y t. 4 3 2 1 −4 −3 −2 y x 1 2 3 4 −2 −3 −4 b 60. (a) S (b) S 2 a gt b dx dt dx dt 2 dy dt dy dt 2 dt 2 2 2 a ft dt 62. Let y be a continuous function of x on a ≤ x ≤ b. a, f t 2 b. Suppose that x f t , y g t , and f t1 Then using integration by substitution, dx f t dt and b t2 y dx a t1 g t f t dt. 64. x A 4 t, y 0 t, 1 24 dx dt t dt 1 24 2 t ,0≤t≤4 u2 du t dt and 1 2 1 4 0 4 0 t 4 4 0 1 u4 2 t t dt 4 u2 u2. 4 arcsin u 2 2 0 Let u x y x, y 1 1 2 4 0 t, then du 4 tt t 4 2 124 1 24 t t dt dt 4 4 0 1 23 t 23 1 4 0 2 4 8 3 t 0 1 24 t 4 t dt 28 3 4 t 4 8 3 88 , 33 dx d 2 66. x cos , y 0 3 sin , 3 sin 2 0 sin sin d 18 cos cos3 3 0 V 2 18 sin3 2 d 12 2 444 Chapter 9 Conics, Parametric Equations, and Polar Coordinates dx d 2 csc2 68. x 2 cot , y 0 2 sin2 , 2 sin2 2 csc2 0 0 A 2 2 d 8 2 d 8 2 4 70. 3 8 a2 is area of asteroid (b). x2 72. 2 a2 is area of deltoid (c). 74. 2 ab is area of teardrop (e). 76. (a) y 12 ln 12 144 x 144 x2 (b) x 60 12 sech t ,y 12 t 12 tanh t ,0≤t 12 0 < x ≤ 12 60 0 0 0 0 12 12 Same as the graph in (a), but has the advantage of showing the position of the object and any given time t. 1 sech2 t 12 sech t 12 tan t 12 t0 sinh t 12 t sinh 0 x 12 y t0 sinh t 12 sech 0 12 y 24 (c) dy dx (0, y0) 16 12 Tangent line: y t 12 tanh 0 12 t0 x 12 8 4 ( x, y ) y-intercept: 0, t0 Distance between 0, t0 and x, y : d d 12 sech t0 12 2 x 2 4 6 10 12 12 tanh t0 12 2 12 12 for any t ≥ 0. 78. False. Both dx dt and dy dt are zero when t 0. By eliminating the parameter, we have y x 2 3 which does not have a horizontal tangent at the origin. Section 9.4 2. 2, 7 4 2 cos 2 sin 7 4 7 4 2, π Polar Coordinates and Polar Graphs 4. x y x, y 0, 7 6 0 cos 0 sin 7 6 7 6 0, 0 π 2 6. x 0 0 y 3, 1.57 3 cos 3 sin 1.57 1.57 0.0024, 3 π 2 0.0024 3 x y x, y 2 2 2 x, y (− 0.0024, 3) 2 (− 2, 2) (0, 0) 0 1 2 0 1 1 2 0 Section 9.4 11 6 Polar Coordinates and Polar Graphs 445 8. r, x, y 2, 10. r, x, y 8.25, 1.3 2.2069, 7.9494 y 8 12. x, y r tan ±5 0, 5 1.7321, 1 y undefined 3 3 , , , 5, 22 2 y 2 6 4 (2.2069, 7.9494) 5, (−1.7321, 1) 1 x 1 2 2 2 −2 −1 x −2 −4 −6 −8 1 2 −2 −1 −1 −2 −3 −4 −5 x 1 2 3 −2 −1 −1 −2 (0, −5) 14. x, y r tan ± 4, 16 2 4 0.464 2 4 1 2 ±2 y 5 −1 −2 −3 −4 x 1 2 3 4 5 (4, −2) 2 5, 0.464 , 2 5, 2.678 −5 16. x, y r, 3 2, 3 2 6, 0.785 18. x, y r, 0, 5, 5 1.571 20. (a) Moving horizontally, the x-coordinate changes. Moving vertically, the y-coordinate changes. (b) Both r and values change. (c) In polar mode, horizontal (or vertical) changes result in changes in both r and . 22. x2 r2 rr y2 2ax 0 0 0 r 2a cos 0 π 2 24. r cos x 10 10 π 2 2ar cos 2a cos r a 2a 10 sec 0 2 4 6 8 12 26. r cos r sin xy 4 4 28. r2 2 x2 y2 2 9 x2 y2 0 0 0 9 r 2 cos2 r2 r2 r 2 sin2 9 cos 2 r2 r2 4 sec csc 8 csc 2 9 cos 2 π 2 π 2 0 2 4 1 2 0 446 Chapter 9 Conics, Parametric Equations, and Polar Coordinates 5 6 tan y x 2 30. r r2 x2 y2 4 4 y 2 32. r r2 x2 x2 5x x 25 4 5 2 2 5 cos 5r cos 5x 25 4 5 2 34. y2 y2 y2 tan 5 6 3 3 3 x 3 y 1 x 1 −1 y −1 y 4 3 2 1 −2 −1 −2 −3 −4 x 1 2 3 4 6 −2 −1 −1 −2 1 2 x 1 2 36. r sin r 2 csc 2 38. r 0≤ −10 51 <2 2 sin 40. r 0≤ 4 3 cos <2 6 y y 2 y 2 0 3 10 −4 10 3 −18 −6 1 −1 x 1 2 42. r 4 2 3 sin ≤2 44. r 0≤ 3 sin 5 2 46. r2 1 . Traced out once on 0 ≤ 3 <4 4 Graph as r1 6 1 , r2 1 . . −6 It is traced out once on 0, 1.5 −3 −1 3 −4 −2 2 − 1.5 S ection 9.4 Polar Coordinates and Polar Graphs 447 48. (a) The rectangular coordinates of r1, r2 cos 2, r2 sin 2 . d2 x2 x1 2 1 are r1 cos 1, r1 sin 1 . The rectangular coordinates of r2, 2 are y2 r1 cos y1 1 2 2 r2 cos r22 cos2 r2 r1 d (b) If 1 2, 2 2 2 2 2 r2 sin 1 2 r1 sin r12 cos2 1 1 1 1 2 2r1r2 cos sin2 2 1 cos r1 2 2 r22 sin2 2 2 2r1r2 sin 1 1 sin 1 2 r12 sin2 2 1 cos2 r2 2 cos2 2 sin2 2 r1r2 cos cos 2 sin sin 2r1r2 cos r12 r22 2r1r2 cos 1 2 the points lie on the same line passing through the origin. In this case, r12 r1 r22 r2 2 d 2r1r2 cos 0 r1 1 r2 2 (c) If 1 2 90 , then cos 0 and d r12 r22, the Pythagorean Theorem! 1 (d) Many answers are possible. For example, consider the two points r1, d 1 22 2 1 2 cos 0 2 2 1, 0 and r2, 2 2, 2. 5 2 2 Using r1, 1 1, and r2, 2, 5 2 ,d 1 2 2 1 2 cos 5 2 5. You always obtain the same distance. 50. 10, 7 , 3, 6 102 32 7 2 10 3 cos 6 109 30 3 7.6 52. 4, 2.5 , 12, 1 d 42 160 122 2 4 12 cos 2.5 12.3 1 d 96 cos 1.5 109 60 cos 6 54. r dy dx 21 sin 2 cos sin 2 cos cos dy dx 1. 2 cos 2 sin 1 1 sin sin 56. (a), (b) r 3 2 cos 4 −8 4 At 2, 0 , At 3, 7 dy , is undefined. 6 dx 0. −4 r, 1, 0 ⇒ x, y 1 1, 0 dy 3 At 4, , 2 dx Tangent line: x (c) At 0, dy does not exist (vertical tangent). dx 448 Chapter 9 4 6 Conics, Parametric Equations, and Polar Coordinates 60. r dy d 8 58. (a), (b) r a sin a sin cos 2a sin cos 0, 2 , , 3 2 a cos2 , a1 2 sin2 0 a cos sin 0 −8 −6 at r, 4, 4 ⇒ x, y 22 y 1x x 2 2, 2 2 22 42 sin dx d ± a sin2 1 , 2 Tangent line: y 4 357 , , 444 2 (c) At dy , 4 dx 1. Horizontal: 0, 0 , a, Vertical: a2 a 23 ,, , 24 2 4 62. r dy d a sin cos2 a sin cos3 2a sin cos3 2a sin cos 0, tan2 1, 2a ,, 44 2 2 64. r 2a sin2 sin3 cos2 3 44 , 2a 3 , , 0, 0 4 4 68. cos sin2 0 cos a cos3 sin −2 3 cos 2 sec 2 4 −2 Horizontal tangents: 2.133, ± 0.4352 Horizontal: 66. r 2 cos 3 r r2 x2 y2 y2 3 2 3 ,0 2 3 cos 3r cos 3x π 2 −3 3 x −2 3 2 2 0 9 4 1 2 4 Horizontal tangents: 1.894, 0.776 , 1.755, 2.594 , 1.998, 1.442 Circle: r Center: Tangent at pole: 2 π 2 70. r 31 cos Cardioid Symmetric to polar axis since r is a function of cos . 0 1 0 r 0 3 3 2 2 3 2 3 9 2 6 S ection 9.4 72. r sin 5 74. r Polar Coordinates and Polar Graphs 449 3 cos 2 Rose curve with five petals Symmetric to 2 Rose curve with four petals Symmetric to the polar axis, Relative extrema: 3, 0 , 3579 ,,,,. 10 10 10 10 10 Tangents at the pole: 3, 2 2 , and pole , 3, 3 2 Relative extrema occur when dr d 5 cos 5 0 at , 3, 3 , 44 Tangents at the pole: π 2 234 0, , , , 5555 7 5 and given the same tangents. 4 4 π 2 0 1 0 2 76. r 2 78. r 4 π 2 1 sin 80. r 5 4 sin Circle radius: 2 x2 y2 Cardioid π 2 Limaçon Symmetric to 2 2 0 1 1 2 π 2 6 7 0 5 6 3 2 1 0 r 9 0 2 4 82. 2r sin 3r cos 2y Line π 2 r 6 2 sin 6 6 3 cos 84. r 1 Hyperbolic spiral 3 4 4 3 1 5 4 4 5 3 2 2 3 3x 4 r 4 2 2 π 2 0 1 0 1 450 86. r2 Chapter 9 Conics, Parametric Equations, and Polar Coordinates 4 sin π 2 Lemniscate Symmetric to the polar axis, Relative extrema: ± 2, 2 5 6 ± 2 , and pole 0 2 0 r 0 ± 6 2 2 ±2 2 0 Tangent at the pole: 88. Since r 2 csc 2 0 90. r 1 , sin 2. r⇒ as r r sin 2 r y →± 2 cos 2 sec Strophoid r⇒ as ⇒ ⇒ 2 2 2 2 cos2 1 sec the graphs has symmetry with respect to Furthermore, r⇒ r⇒ Also, r ry r as as 2 2y 2y y 1 ⇒0 ⇒ 1 sin r . as y ⇒ 1. . 2 cos 2 sec 4 cos2 2 2 2 r cos x lim 2 2 4 cos2 2 4 cos2 2 x = −2 Thus, r ⇒ ± 4 −3 3 −2 −4 4 y=1 −2 92. x x2 r cos , y y2 r 2, tan r sin y x 94. Slope of tangent line to graph of r dy dx If f pole. f f cos sin f f sin . cos f at r, is 0 and f 0, then is tangent at the 96. r 4 cos 2 98. r Line 2 sec Rose curve Matches (b) Matches (d) S ection 9.4 100. r (a) 61 cos 0, r 9 Polar Coordinates and Polar Graphs 451 61 cos (c) r 15 2 61 61 61 cos cos cos sin 15 2 2 sin sin 2 −9 −9 (b) 4 ,r 12 61 cos 4 − 12 −3 12 −9 15 The graph of r angle 2. cos is rotated through the 61 cos is rotated through the −6 The graph of r angle 4. 61 102. (a) sin 2 sin cos cos 2 cos sin 2 (b) sin sin cos sin r f sin f sin cos sin r f sin f cos sin cos cos r f sin 2 (c) sin 3 2 3 2 cos sin 3 2 3 2 f cos 104. r 2 sin 2 4 sin 4 sin cos 6 2 (a) r cos 6 (b) r 4 sin 2 2 cos 2 4 sin cos −3 3 −3 3 −2 −2 (c) r 4 sin 2 2 3 cos 2 3 (d) r 4 sin 2 cos 4 sin cos −3 −3 3 −2 −2 3 452 Chapter 9 Conics, Parametric Equations, and Polar Coordinates π 2 106. By Theorem 9.11, the slope of the tangent line through A and P is f cos f sin f sin f cos Radial line ψ P = (r, θ) Polar curve r = f (θ) This is equal to tan tan tan 1 tan tan sin cos cos tan . sin tan θ Tangent line A 0 Equating the expressions and cross-multiplying, you obtain f cos f cos2 f cos sin tan f sin cos sin tan f sin2 f cos2 tan sin2 tan sin f sin2 f tan f f cos tan f sin f cos f sin cos f cos2 tan f sin cos f sin cos tan cos2 sin2 r . dr d r dr d 6 , tan 3 2 4 sin 2 8 cos 2 sin 2 cos 3 3 3 . 2 40.89 108. tan At r dr d 3 , tan 4 arctan 2 3 1 cos 3 sin 1 2 2 2 5 110. tan 22 2 2 59.64 2 . At 1.041 arctan 4 0.7137 −8 2 −6 6 −5 −4 112. tan r dr d 6 5 undefined ⇒ 0 2 . −9 9 −6 114. True 116. True Section 9.5 2. (a) r 3 cos Area and Arc Length in Polar Coordinates π 2 (b) A 2 9 1 2 2 2 3 cos 0 2 d cos2 0 2 d cos 2 2 0 0 2 4 9 2 9 2 1 0 d 9 4 A 3 2 2 9 4 sin 2 2 S ection 9.5 Area and Arc Length in Polar Coordinates 453 4. A 2 36 1 2 4 4 6 sin 2 0 4 2 d 36 0 sin2 2 d 6. A 2 1 2 1 2 10 cos 5 0 2 d 10 0 1 0 cos 4 d 2 sin 4 4 4 0 1 sin 10 10 20 18 18 4 9 2 8. A 2 1 2 3 2 2 1 0 sin 2 d 2 0 10. A 3 4 8 2 1 2 2 2 4 arcsin 2 3 6 sin 48 sin 48 sin 2 d 36 sin 36 1 2 2 cos 1 sin 2 4 16 arcsin 2 3 2 d cos 2 2 d 16 arcsin 2 3 2 34 2 −8 48 cos 8 9 sin 2 arcsin 2 3 1.7635 −12 12. Four times the area in Exercise 11, A 2 1 2 2 4 2 3 3 . More specifically, we see that the area inside the outer loop is 4 6 21 6 2 sin 2 d 16 sin 16 sin2 d 8 6 3. 6 The area inside the inner loop is 2 1 2 3 7 2 21 6 2 sin 2 d 4 6 3. 63 4 63 4 12 3. −4 −1 4 Thus, the area between the loops is 8 14. r r 31 31 sin sin 16. r r 2 cos 3 cos Solving simultaneously, 31 sin 2 sin 31 0 0, . Replacing r by r and by in the first equation 3 1 sin , sin 1, and solving, 3 1 sin 2. Both curves pass through the pole, 0, 3 2 , and 0, 2 , respectively. Points of intersection: 3, 0 , 3, , 0, 0 sin Solving simultaneously, 2 3 cos cos cos 1 2 5 , . 33 Both curves pass through the pole, (0, arccos 2 3), and 0, 2 , respectively. Points of intersection: 1 15 ,,, , 0, 0 23 23 454 Chapter 9 Conics, Parametric Equations, and Polar Coordinates 18. r r 1 cos 20. r 3 cos 1 2 5 ,. 33 4 2 π 2 3 cos Solving simultaneously, 1 cos cos Line of slope 1 passing through the pole and a circle of radius 2 centered at the pole. Points of intersection: 2, , and 0, 2, 4 , 2, 4 0 1 3 Both curves pass through the pole, 0, respectively. Points of intersection: 3 35 ,,, , 0, 0 23 23 π 2 22. r r 3 sin Points of intersection: 17 2 0 1 2 2 csc 3 , arcsin , 17 2 3 17 2 , 3 , 17 2 3 arcsin 3.56, 0.596 , 3.56, 2.545 The graph of r 3 sin is a limaçon symmetric to 2, and the graph of r 2 csc is the horizontal line y 2. Therefore, there are two points of intersection. Solving simultaneously, 3 sin2 3 sin sin sin 2 2 csc 0 3± 2 arcsin 17 17 2 3 0.596. 24. r r 31 1 cos r= 6 cos 31 cos cos is a cardioid with polar axis symmetry. The graph of − 10 6 1 − cos θ 5 The graph of r r 61 5 is a parabola with focus at the pole, vertex 3, , and polar axis symmetry. Therefore, there are two points of intersection. Solving simultaneously, 31 1 cos cos cos 2 r = 3(1 − cos θ ) −5 1 2 1± 6 cos 2 2. 2 4.243, 1.998 , 3 2, 2 arccos 1 2 4.243, 4.285 arccos 1 Points of intersection: 3 2, arccos 1 S ection 9.5 Area and Arc Length in Polar Coordinates 2 455 26. r r 4 sin 21 sin 28. A 4 1 2 91 0 2 sin sin 2 2 d 9 3 2 7 18 Points of intersection: 0, 0 , 4, 0 1 d 8 2 (from Exercise 14) −7 The graphs reach the pole at different times ( values). 6 7 r = 4 sin θ −7 −6 −2 6 r = 2 (1 + sin θ) 30. r 5 3 sin 4 and 5 4 4 and r 5 5 4. 5 3 sin 2d 3 cos intersect at 32. A 2 1 2 2 2 3 sin 6 2 d 4 sin 1 2 d 2 2 2 6 sin 2 d A 1 2 2 59 2 4 cos 2 6 5 4 4 30 cos 30 30 2 8 9 sin 2 4 2 2 9 4 4 2 sin 2 2 2 9 4 −4 4 cos 6 33 59 5 24 59 2 59 24 30 4 50.251 −4 −12 12 −8 34. Area Area of r 2a cos Area of sector twice area between r 2a cos and the lines 3 A , a2 2 a2 3 2 a2 3 2 π 2 36. r tan A a cos , r 1, 2 a2 0 a sin 4 2 . a2 2 3 2a 2 1 2 2 1 3 2 1 2 2 a cos 0 4 2 d 2a cos 3 2 d 1 cos 2 d 2 sin 2 2 1 2 4 0 cos 2 2 3 d 2a 2 2a 2 π θ=3 sin 2 2 2 3 12 a 2 12 a 2 4 12 a 4 π 2 a2 3 3 4 2 a2 6 3 3a 2 12 a 8 r = a sin θ a a 2a 0 0 a π θ =−3 r = a cos θ 456 Chapter 9 Conics, Parametric Equations, and Polar Coordinates A2 and A3 6 38. By symmetry, A1 A1 A2 a2 2 a2 2 A3 A5 A4 1 2 6 A4. 2 2a cos 3 a 2 d 4 1 2 4 2a cos 6 2 2a sin 2 d 4 cos2 3 6 1d 2a2 6 4 6 cos 2 d a2 22 3 a2 1 3 2 a2 1 π 2 sin 2 3 a 2 sin 2 a2 4 4 1 a2 22 2 1 2 π θ= 3 A2 15 a2 26 5 a2 12 5 a2 12 2a 2 r = 2a sin θ 2a π θ= 4 A7 2a sin 5 6 2 d 5π θ= 6 a π θ= 6 A3 1 5 6 cos 2 d A5 A6 A4 a A1 2a 0 a2 2 6 sin 2 5 6 2 5 a2 12 1 2 a2 a2 4 a2 a2 d 3 3 2 a2 12 3 2 r=2 r = 2a cos θ π θ = −3 A6 2 1 2 2a sin 0 6 d d a2 12 2 6 4 6 2a 2 0 1 sin 2 cos 2 6 0 a2 2 A7 2 a2 6 3 3 2 a2 12 a2 5 12 3 2 1 2 4 4 2a sin 6 2 a 2 d 4 4 sin2 A6 A7 A4 1d a2 a2 sin 2 6 a2 12 1 3 2 [Note: A1 area of circle of radius a] 40. r cos r sec 1 2 cos , 2 cos2 2 r 2 cos2 r2 2x 2 2 < < y 2 1 x x2 y2 x y2 x 1 y2 1 2 4 1 x2 x2 x2 1 1 4 1 2 x2 x 2 x y2 −1 y2 x3 x x A 2 sec 0 2 cos 4 cos2 2 d 4 4 sec2 0 4 d 0 sec2 4 21 cos 2 d tan 2 sin 2 0 2 2 Section 9.5 42. r r s 2 2 Area and Arc Length in Polar Coordinates 81 cos ,0 ≤ ≤2 457 2a cos 2a sin 2 44. r r 2a cos 2 8 sin 2 0 2a sin 2 2 2 d s 81 1 0 cos 2 cos 1 2 8 sin cos2 2 d d 2a d 2 2 2a 16 16 2 sin2 cos d cos sin d 1 1 cos cos d 0 16 2 0 1 16 2 0 1 cos cos 0 32 2 1 64 46. r sec , 0 ≤ 3 ≤ 3 48. r e ,0 ≤ ≤ 10 50. r 2 sin 2 cos 2 ,0≤ ≤ −3 −2 4 − 25 5 4 −3 −5 −2 Length 52. r r S 2 a cos 1.73 exact 3 Length 31.31 Length 7.78 a sin 2 a cos 0 2 cos d 2 0 a2 cos 2 a2 sin2 1 cos 2 d d 2 a2 0 cos2 sin 2 2 a2 0 2a2 a2 2 54. r r S a1 cos a sin 2 0 a1 2 2 a2 0 cos 1 sin cos a2 1 32 cos sin d 2 a2 sin2 d 1 2 a2 0 sin 52 0 1 cos 32 a2 5 2 2 cos d 4 2 a2 5 cos 56. r r S 1 2 0 58. The curves might intersect for different values of : See page 696. sin 2 1d 42.32 458 Chapter 9 Conics, Parametric Equations, and Polar Coordinates 60. (a) S (b) S 2 2 f f sin cos f f 2 f f 2 d d 2 2 62. r 8 cos , 0 ≤ 1 2 r2 d 0 ≤ 1 2 r2 0.4 12.14 64 cos2 0 (a) A d 32 0 1 cos 2 d 2 16 sin 2 2 16 0 (Area circle (b) A 0.2 6.32 42 0.6 17.06 16 ) 0.8 20.80 1.0 23.27 1.2 24.60 1.4 25.08 1 (c), (d) For 4 of area 4 12.57 : 0.42 25.13 : 1.57 37.70 : 2.73 2 For For 1 2 3 4 of area 8 of area 12 (e) No, it does not depend on the radius. 64. False. f 0 and g intersection. sin 2 have only one point of Section 9.6 2. r 2e e cos 1, r 0.5, r 1.5, r 1 Polar Equations of Conics and Kepler’s Laws 4. r 2 , parabola cos 1 1 4 1 1 2e e sin 1, r 0.5, r 1.5, r 9 (a) e (b) e (c) e (a) e 2 , ellipse cos 6 , hyperbola 3 cos (b) e (c) e 1 1 1 2 , parabola sin 1 0.5 sin 3 1.5 sin 2 2 2 , ellipse sin 6 , hyperbola 3 sin 1 0.5 cos 3 1.5 cos 2 2 e = 1.5 −9 e=1 3 −9 e = 1.5 e=1 9 −3 e = 0.5 −4 e = 0.5 6. r 1 4 0.4 cos (b) r 1 4 0.4 cos (a) Because e 0.4 < 1, the conic is an ellipse with vertical directrix to the left of the pole. (c) − 10 7 9 10 −8 8 The ellipse is shifted to the left. The vertical directrix is to the right of the pole 4 r . 1 0.4 sin The ellipse has a horizontal directrix below the pole. −7 −5 8. Ellipse; Matches (f) 10. Parabola; Matches (e) 12. Hyperbola; Matches (d) Section 9.6 6 cos 1 Ellipse since e Vertices: π 2 0 4 8 Polar Equations of Conics and Kepler’s Laws 1 3 5 sin 459 14. r 1 16. r 5 5 3 sin 1 3 <1 5 18. r 3 r 3 1 2 cos 6 2 cos 6 Parabola since e Vertex: 3, 0 π 2 5 53 ,,, 82 22 2 2 3 cos 2 <1 3 6 , 5 Ellipse since e Vertices: 6, 0 , π 2 0 1 2 0 1 2 3 4 5 20. r 3 6 7 sin 1 2 7 3 sin 7 > 1. 3 22. r 1 4 2 cos 2>1 4, Hyperbola since e Vertices: π 2 Hyperbola since e Vertices: π 2 3 33 ,,, 52 22 4 ,0 , 3 0 2 2 0 24. −6 4 Hyperbola 6 26. −6 4 Hyperbola 6 −4 −4 28. r 1 6 cos 3 1 6 cos 30. r 6 3 7 sin 2 3 3 6 . 7 sin 3. Rotate graph of r Rotate the graph of r Clockwise through angle of 2 4 counterclockwise through the angle . 3 6 −6 − 14 10 6 −4 − 10 460 Chapter 9 Conics, Parametric Equations, and Polar Coordinates 2 1 sin 6 32. Change to 6 :r 34. Parabola e r 1, y 1 1, d ed e sin 1 1 1 sin 36. Ellipse e r 3 ,y 4 1 1 4 42. Ellipse 3 Vertices: 2, , 4, 2 2 e r 1 ,d 3 1 1 3 8 ed e sin 83 1 3 sin 8 sin 4 sin ed e sin 23 4 3 4 sin 6 3 sin 2, d 2 38. Hyperbola e r 3 ,x 2 1 1 2 1, d ed e cos 32 3 2 cos 3 3 cos 44. Hyperbola 1 40. Parabola Vertex: 5, e r 1, d 1 10 ed e cos 1 10 cos Vertices: 2, 0 , 10, 0 e r 3 ,d 2 1 1 2 10 3 ed e cos 5 3 2 cos 10 3 cos 46. r 1 is a parabola with horizontal directrix above the pole. (b) Parabola with horizontal directrix below pole. (d) Parabola (b) rotated counterclockwise x2 a2 x 2b 2 b 2r 2 cos 2 y2 b2 y 2a 2 4. (a) Parabola with vertical directrix to left pole. (c) Parabola with vertical directrix to right of pole. x2 a2 x 2b 2 b 2 r 2 cos 2 r 2 b 2 cos 2 r2 a2 r2 a2 1 b2 a2 1 cos 2 y2 b2 y 2a 2 48. (a) 1 a 2b 2 a 2b 2 a 2b 2 a 2b 2 a2 a 2b 2 c 2 cos 2 (b) 1 a 2b 2 a 2b 2 a 2b2 a 2b 2 1 b2 c 2 a 2 cos 2 a 2r 2 sin 2 a2 1 cos 2 b2 a2r 2 sin2 cos 2 b2 a2 r 2 b 2 cos 2 r2 r2 a2 cos 2 a 2 a2 a 2b 2 c 2 cos 2 b2 e cos 2 2 a 2b 2 a 2 cos 2 1 1 b2 c a 2 cos 2 b2 e 2 cos 2 R eview Exercises for Chapter 9 5 4 3 2 461 50. a r2 4, c 5, b 3, e 52. a r2 2, b 1, c 1 3 4 cos 2 3, e 1 9 25 16 cos 2 1 54. A 2 1 2 2 2 3 2 2 sin 2 2 d 4 2 3 1 2 sin 2 d 3.37 56. (a) r 1 ed e cos 0, r c a ea a a1 e. (b) The perihelion distance is a When ,r 1 1 e2 e c a c a a1 a a1 ea e. ea e. a1 e. When Therefore, The aphelion distance is a a1 e e e2 e2 a . e cos 60. a r 1.422792505 109 1 0.0543 cos e e 1.3495139 1.5044861 b cos br cos bx 109 km 109 km 36.0 1 1 10 6 mi, e e2 a e cos ed 1 ed ed. e When 0, r 1 1 e2 a e a1 e. a1 e1 a1 Thus, r 1 1 58. a e r 1.427 0.0543 1 1 109 km 0.206 e2 a e cos 34.472 10 6 1 0.206 cos e e 28.582 43.416 10 6 mi 10 6 mi Perihelion distance: a 1 Aphelion distance: a 1 Perihelion distance: a 1 Aphelion distance: a 1 62. r r2 x2 x2 y2 bx y2 ay a sin a r sin ay 0 represents a circle. Review Exercises for Chapter 9 2. Matches (b) - hyperbola 4. Matches (c) - hyperbola 6. y 2 12y y2 8x 12y y 20 36 6 2 0 y 8x 20 2 36 16 12 42 x Parabola Vertex: 2, 6 −4 x 8 12 462 8. 4x 2 Chapter 9 y2 4x x Ellipse Center: 2, 0 Vertices: 2, ± 1 16x 4 2 Conics, Parametric Equations, and Polar Coordinates 15 y2 y2 1 1 −1 0 15 16 1 y 4 x2 (2, 0) x 1 2 3 2 14 −2 10. 4 x2 4x 2 x x Hyperbola Center: Vertices: 1 ,1 2 4y 2 1 4 4x 4 y2 8y 2y y 2 11 1 1 2 0 4 y 11 1 1 4 3 12 2 2 1 −2 −1 −2 x 1 3 4 1 ± 2 2, 1 1± x 1 2 Asymptotes: y 12. Vertex: 4, 2 Focus: 4, 0 Parabola opens downward p x x2 4 2 2 4 8y 2y 0 2 8x 14. Center: 0, 0 Solution points: 1, 2 , 2, 0 Substituting the values of the coordinates of the given points into x2 b2 y2 a2 1, 16. Foci: 0, ± 8 ± 4x Asymptotes: y Center: 0, 0 Vertical transverse axis c y 8 a x b c2 4x asymptote → a a2 64 4b 2 we obtain the system 1 b2 4 a2 1, 4 b 2 1. 4b ⇒ 17b2 64 b2 Solving the system, we have a2 16 and b 2 3 4, x2 4 3y 2 16 1. ⇒ b2 y2 1024 17 64 ⇒ a2 17 x2 64 17 1 1024 17 18. x2 4 y2 25 1, a 5, b 2, c 21, e 21 5 By Example 5 of Section 9.1, 2 C 20 0 1 21 2 sin d 25 23.01. R eview Exercises for Chapter 9 12 x 200 200y 4 50 y y 1 y 2 463 20. y (a) x 2 x2 (b) y 12 x 200 1 x 100 1 100 Focus: 0, 50 x2 10,000 x 1 x2 dx 10,000 38,294.49 S 2 0 a 22. (a) A 4 0 b a 2 a2 b x2 dx a2 2 b b2 b2 y2 4b 1 a2 y2 dy x a2 2 a2 b2 x2 b a 2 arcsin x a a ab 0 (b) Disk: V b2 0 y2 dy 0 b 2 a2 2 by b2 13 y 3 b 0 42 ab 3 S 4 0 a b b b4 a2 b2 y2 dy b b2 y2 2a cy b 4 b 2c b b2 b 4a b2 b4 0 c 2y 2 d y c 2y 2 b 4 ln cy b4 c 2y 2 0 2a 2 b c b2 b 2c 2 a2 a c2 b 4 ln cb 2 c2 b 4 ln b 2 e e 2 b2 2 ax a2 13 x 3 a 0 ab2 ca ln c e b2 2 a a2 b a a4 0 2 a2 2 b2 a2 a b2 1 ln e 1 a2 0 (c) Disk: V 2 0 a x2 dx x2 dx 4 ab2 3 S 22 0 a2 x2 a4 a2 b2 x2 a a2 x2 2b cx a 4 a 2c c a 3 x 3 x 9 Ellipse y 7 6 5 4 3 2 1 −2 −1 −2 −3 dx cx a2 a 0 4b a2 a c 2x 2 d x c 2x 2 a 4 arcsin ab 2 a c a2 a 2c 24. x t t x 4, y t2 x 4 2 c2 a 4 arcsin 2 b2 2 ab arcsin e e 2 2 26. x 3 cos , y 2 5 sin 1 28. x x 5 5 sin3 , y 23 5 cos3 1 4⇒y 3 3 y 5 y 2 2 25 Parabola y 7 6 5 4 3 2 1 −1 x 1 2 3 4 5 6 7 y 5 3 y 6 4 2 23 2 2 1 x2 y2 3 52 3 −6 −4 −4 −6 x 2 4 6 x 12345678 464 Chapter 9 2 2 Conics, Parametric Equations, and Polar Coordinates 2 2 30. x x h 5 y y k 3 r2 2 2 32. a 4 Let 4, c y2 16 5, b2 sec 2 and c2 x2 9 a2 9, y2 16 x2 9 1 tan 2 . 4 sec . Then x 34. x y (a) a x y a a b cos t b sin t 2, b cos t sin t y=0 −2 ≤ x ≤ 2 −3 3 −6 6 3 tan and y b cos b sin a b a b b b t t (b) a 3, b 2 cos t 2 sin t 4 1 cos t sin t 2 1 cos 2t sin 2t (c) a x y 4, b 3 cos t 3 sin t 1 cos 3t sin 3t 4 2 cos t 0 x y −6 6 −2 −4 −4 (d) a x y 10, b 9 cos t 9 sin t 1 cos 9t sin 9t 10 (e) a x y 15 3, b cos t sin t 2 2 cos 2 sin 4 (f) a t 2 x y 4, b cos t sin t 3 3 cos 3 sin 4 t 3 t 2 t 3 − 15 − 10 −6 6 −6 6 −4 −4 36. x t u r cos sin r sin cos r sin 38. x y t t2 dy dx 4 r cos y v w r cos (a) r sin y 2t 1 2t 0 when t 0. Point of horizontal tangency: 4, 0 (b) t y θ w v u rθ ( x, y ) x x x y 6 5 4 3 2 1 4 4 2 (c) r θ t x 1 2 3 4 5 6 R eview Exercises for Chapter 9 1 t t2 dy dx 1 x 1 x2 y 4 3 465 40. x y (a) 42. x y 2t 1 t2 2t 3 t 0 2t 1 t2 1 2t t2 1 t2 t x 2 1 x y dy (a) dx 2t 2 t 2 2 2 2t 2 No horizontal tangents (b) t y (c) 0 when t 1. 1 Point of horizontal tangency: 1, (b) t y (c) 2 1 4 3x 12 2 2x 12 x 1 2 1 −2 −2 −1 x 1 2 x 2 4 44. x y (a) 6 cos 6 sin dy dx 6 cos 6 sin cot 0 when 3 . 22 , 6 46. x y (a) et e dy dx t e et ln x e y t 1 e 2t 1 x2 Points of horizontal tangency: 0, 6 , 0, (b) (c) 4 2 −4 −2 −2 −4 x 2 4 No horizontal tangents (b) t y 1 ,x > 0 x x 6 2 y 6 2 1 y ln x e ln 1 x (c) 3 2 1 x 1 2 3 48. x y 2 2 sin cos 8 50. x y dx d 6 cos 6 sin 6 sin 6 cos 36 sin 2 0 (a), (c) −8 8 dy d s −4 36 cos 2 d 6 0 6 (b) At dy dt dx , 6d 1.134, 2 dy dx 3 , 2 (one-half circumference of circle) 0.5, and 0.441 466 Chapter 9 1, 3 1 arctan r, 2 Conics, Parametric Equations, and Polar Coordinates y 52. x, y r 32 3 10 1.89 108.43 10, 5.03 −3 (−1, 3) 2 1 −2 −1 −1 −2 −3 x 1 2 3 10, 1.89 , 54. r r2 x2 y 2 10 100 100 56. 2r 2± x2 r cos y2 4 x2 3x 2 4 3 cos 1 2 cos 3 4 3 2 sin 4y 2 3 4 tan y x y 1 1 x 2x r 2 1 1 cos x y2 1 1 x 0 1 2 58. r 4 sec 60. r cos x 3 sin 3y 8 8 62. x 2 r2 y2 4x 0 0 64. x 2 y 2 arctan y x 2 a2 2 4r cos r r2 a2 4 cos π 2 66. Line 12 68. r 3 csc , r sin 3, y 3 π 2 Horizontal line 0 1 2 0 1 2 3 4 70. r 3 4 cos Limaçon Symmetric to polar axis π 2 2 0 0 r 1 3 1 2 3 2 3 5 7 R eview Exercises for Chapter 9 72. r 2 2 467 Spiral Symmetric to π 2 0 2 4 8 0 r 0 4 5 2 3 4 3 2 2 5 4 5 2 3 2 3 74. r cos 5 π 2 Rose curve with five petals Symmetric to polar axis Relative extrema: 1, 0 , Tangents at the pole: 2 , 1, , 5 5 3 79 , ,, , 10 10 2 10 10 1, 1, 3 4 , 1, 5 5 0 1 76. r 2 cos 2 Lemniscate Symmetric to the polar axis Relative extrema: ± 1, 0 Tangents at the pole: 3 , 44 r 0 ±1 ± π 2 6 2 2 4 0 0 1 2 78. r 2 sin cos 2 2 −1 0.75 80. r 4 sec cos −1 3 Bifolium Symmetric to 1 − 0.25 Semicubical parabola Symmetric to the polar axis r⇒ r⇒ as as ⇒ ⇒ 2 2 5 −3 82. r 2 4 sin 2 dr d dr d 8 cos 2 4 cos 2 r 0, 2 (b) dy dx r cos r sin cos 2 cos 2 sin cos 4 cos 2 sin r 4 cos 2 cos r sin 2 cos sin 2 sin (a) 2r Tangents at the pole: (c) −3 2 Horizontal tangents: dy dx tan 0 when cos 2 tan 2 , sin sin 2 cos 0, 2 3, 3 0, , 0, 0 , ± 3 3 0: Vertical tangents when cos 2 cos −2 sin 2 sin 2 3, 6 tan 2 tan 1, 0, , 0, 0 , ± 6 468 Chapter 9 Conics, Parametric Equations, and Polar Coordinates 1, 0 84. False. There are an infinite number of polar coordinate representations of a point. For example, the point x, y 1, 0 , 1, 2 , 1, , etc. has polar representations r, 86. r a sin , r a cos 2, 4 and 0, 0 . For r 2 sin cos . cos 2 0. For r cos 2 . 2 sin cos 2, a cos , a sin , The points of intersection are a m1 At a m2 At a 88. r A 51 1 2 2 dy dx 2, dy dx 2, a cos sin a cos 2 a sin cos a sin 2 4 , m1 is undefined and at 0, 0 , m1 a sin 2 a sin cos 4 , m2 sin 3 2 4 a cos 2 a cos sin 0 and at 0, 0 , m2 is undefined. Therefore, the graphs are orthogonal at a 90. r 4 sin 3 3 1 2 3 4 and 0, 0 . 2 51 sin 2 d 117.81 75 A 2 4 sin 3 0 2 d 12.57 4 8 4 −8 −6 − 12 −4 6 92. r 9 3, r 2 r2 18 sin 2 18 sin 2 −6 4 6 sin 2 1 2 −4 12 A 2 1 2 12 18 sin 2 d 0 1 2 5 12 9d 12 1 2 2 18 sin 2 d 5 12 1.2058 9.4248 1.2058 11.84 dr d 94. r A e ,0 ≤ 1 2 e 0 2 ≤ d 10 96. r 133.62 s a cos 2 , 4 2a sin 2 4a 2 sin 2 2 d 4) 2 8 0 4 a 2 cos 2 2 1 0 8a − 25 5 3 sin 2 2 d (Simpson’s Rule: n 2 1.5811 4 1.8870 a 1 6 9.69a 4 1.1997 −5 ...
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This note was uploaded on 05/18/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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