746 CHAPTER 10 Analytic Geometry In Problems 7–26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. 7. x t t y t t t 3 2, 1; 0 4 ( ) ( ) = + = + ≤ ≤ 8. x t t y t t t 3, 2 4; 0 2 ( ) ( ) = − = + ≤ ≤ 9. ( ) ( ) = + = ≥ x t t y t t t 2, ; 0 10. x t t y t t t 2 , 4 ; 0 ( ) ( ) = = ≥ 11. x t t y t t t 4, 4; 2 2 ( ) ( ) = + = − −∞< <∞ 12. x t t y t t t 4, 4; 0 ( ) ( ) = + = − ≥ 13. x t t y t t t 3 , 1; 2 ( ) ( ) = = + −∞< <∞ 14. x t t y t t t 2 4, 4 ;2 ( ) ( ) = − = −∞< <∞ 15. x t e y t e t 2 , 1 ; 0 t t ( ) ( ) = = + ≥ 16. x t e y t e t , ; 0 t t ( ) ( ) = = ≥ − 17. x t t y t t t , ; 0 3 2 ( ) ( ) = = ≥ 18. x t t y t t t 1, ; 0 3 2 ( ) ( ) = + = ≥ 19. x t t y t t t 2cos , 3sin ; 0 2π ( ) ( ) = = ≤ ≤ 20. x t t y t t t 2cos , 3sin ; 0 π ( ) ( ) = = ≤ ≤ 21. x t t y t t t 2cos , 3sin ; 0 π ( ) ( ) = = − ≤ ≤ 22. x t t y t t t 2cos , sin ; 0 2 π ( ) ( ) = = ≤ ≤ 23. π ( ) ( ) = = ≤ ≤ x t t y t t t sec , tan ; 0 4 24. x t t y t t t csc , cot ; 4 2 π π ( ) ( ) = = ≤ ≤ 25. x t t y t t t sin , cos ; 0 2 2 2 π ( ) ( ) = = ≤ ≤ 26. x t t y t t t , ln ; 0 2 ( ) ( ) = = > Skill Building In Problems 27–34, find two different pairs of parametric equations for each rectangular equation. 27. y x4 1 = − 28. y x8 3 = − + 29. y x 1 2 = + 30. y x2 1 2 = − + 31. y x3 = 32. y x 1 4 = + 33. x y3/2 = 34. x y = In Problems 35–38, find parametric equations that define the plane curve shown. 35. x 2 4 6 (7, 5) (2, 0) y 4 2 6 36. x 1 3 21 22 2 (21, 2) (3, 22) y 2 21 22 23 1 37. x 1 3 21 22 23 2 y 2 21 22 1 38. x 2 22 (0, 24) y (0, 4) 2 22 In Problems 39–42, find parametric equations for an object that moves along the ellipse x y 4 9 1 2 2 + = with the motion described. 39. The motion begins at 2, 0 , ( ) is clockwise, and requires 2 seconds for a complete revolution. 40. The motion begins at 0, 3 , ( ) is counterclockwise, and requires 1 second for a complete revolution. 41. The motion begins at 0, 3 , ( ) is clockwise, and requires 1 second for a complete revolution. 42. The motion begins at 2, 0 , ( ) is counterclockwise, and requires 3 seconds for a complete revolution. In Problems 43 and 44, parametric equations of four plane curves are given. Graph each of them, indicating the orientation. 43. C x t t y t t t C x t t y t t t C x t e y t e t C x t t y t t t : , ; 4 4 : cos , 1 sin ; 0 : , ; 0 ln4 : , ; 0 16 t t 1 2 2 2 3 2 4 π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = − ≤ ≤ = = − ≤ ≤ = = ≤ ≤ = = ≤ ≤ 44. C x t t y t t t C x t t y t t t C x t t y t t t C x t t y t t t : , 1 ; 1 1 : sin , cos ; 0 2 : cos , sin ; 0 2 : 1 , ; 1 1 1 2 2 3 4 2 π π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = − − ≤ ≤ = = ≤ ≤ = = ≤ ≤ = − = − ≤ ≤ In Problems 45–48, use a graphing utility to graph the plane curve defined by the given parametric equations. 45. x t t t y t t t t sin , cos , 0 ( ) ( ) = = > 46. x t t t y t t sin cos , sin cos ( ) = + = − 47. x t t t y t t t 4 sin 2 sin 2 4 cos 2cos 2 ( ) ( ) ( ) ( ) = − = − 48. x t t t y t t t 4 sin 2 sin 2 4cos 2 cos 2 ( ) ( ) ( ) ( ) = + = + 4. Multiple Choice If a circle rolls along a horizontal line without slipping, a fixed point P on the circle will trace out a curve called a n( ) . (a) cycloid (b) epitrochoid (c) hyptrochoid (d) pendulum 5. True or False Parametric equations defining a curve are unique. 6. True or False Plane curves defined using parametric equations have an orientation.
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