SalasSV_09_06_ex

SalasSV_09_06_ex - 9.6 CURVES GIVEN PARAMETRICALLY 557...

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EXERCISES 9.6 Express the curve by an equation in x and y . 1. x ( t ) = t 2 , y ( t ) = 2 t + 1. 2. x ( t ) = 3 t 1, y ( t ) = 5 2 t . 3. x ( t ) = t 2 , y ( t ) = 4 t 4 + 1. 4. x ( t ) = 2 t 1, y ( t ) = 8 t 3 5. 5. x ( t ) = 2 cos t , y ( t ) = 3 sin t . 6. x ( t ) = sec 2 t , y ( t ) = 2 + tan t . 7. x ( t ) = tan t , y ( t ) = sec t . 8. x ( t ) = 2 sin t , y ( t ) = cos t . 9. x ( t ) = sin t , y ( t ) = 1 + cos 2 t . 10. x ( t ) = e t , y ( t ) = 4 e 2 t . 11. x ( t ) = 4 sin t , y ( t ) = 3 + 2 sin t . 12. x ( t ) = csc t , y ( t ) = cot t . Express the curve by an equation in x and y ; then sketch the curve. 13. x ( t ) = e 2 t , y ( t ) = e 2 t 1, t 0. 14. x ( t ) = 3 cos t , y ( t ) = 2 cos t ,0 t π . 15. x ( t ) = sin t , y ( t ) = csc t < t 1 4 π . 16. x ( t ) = 1 / t , y ( t ) = 1 / t 2 < t < 3. 17. x ( t ) = 3 + 2 t , y ( t ) = 5 4 t , 1 t 2. 18. x ( t ) = sec t , y ( t ) = tan t t 1 4 π . 19. x ( t ) = sin π t , y ( t ) = 2 t t 4. 20. x ( t ) = 2 sin t , y ( t ) = cos t t 1 2 π . 21. x ( t ) = cot t , y ( t ) = csc t , 1 4 π t < 1 2 π . 22. ( Important ) Parametrize: (a) the curve y = f ( x ), x [ a , b ]; (b) the polar curve r = f ( θ ), θ [ α , β ].

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558 ± CHAPTER 9 THE CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONS 23. A particle with position given by the equations x ( t ) = sin 2 π t , y ( t ) = cos 2 π t , t [0, 1]. starts at the point (0, 1) and traverses the unit circle x 2 + y 2 = 1 once in a clockwise manner. Write equations of the form x ( t ) = f ( t ), y ( t ) = g ( t ), t [0, 1]. so that the particle (a) begins at (0,1) and traverses the circle once in a counterclockwise manner; (b) begins at (0,1) and traverses the circle twice in a clockwise manner; (c) traverses the quarter circle from (1,0) to (0,1); (d) traverses the three-quarter circle from (1,0) to (0,1). 24. A particle with position given by the equations x ( t ) = 3 cos 2 π t , y ( t ) = 4 sin 2 π t , t [0, 1]. starts at the point (3, 0) and traverses the ellipse 16 x 2 + 9 y 2 = 144 once in a counterclockwise manner. Write equations of the form x ( t ) = f ( t ), y ( t ) = g ( t ), t [0, 1], so that the particle (a) begins at (3, 0) and traverses the ellipse once in a clockwise manner; (b) begins at (0, 4) and traverses the ellipse once in a clockwise manner; (c) begins at ( 3, 0) and traverses the ellipse twice in a counterclockwise manner; (d) traverses the upper half of the ellipse from (3, 0) to (0, 3). 25. Find a parametrization x = x ( t ), y = y ( t ), t ( 1, 1), for the horizontal line y = 2. 26. Find a parametrization x ( t ) = sin f ( t ), y ( t ) = cos f ( t ), t (0, 1), which traces out the unit circle in±nitely often.
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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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SalasSV_09_06_ex - 9.6 CURVES GIVEN PARAMETRICALLY 557...

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