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Unformatted text preview: Version 047 Exam02 Gilbert (56380) 1 This printout should have 12 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points Determine a Cartesian equation for the curve given in parametric form by x ( t ) = 4 ln(9 t ) , y ( t ) = t. 1. y = 1 4 e 6 /x 2. y = 1 3 e x/ 4 3. y = 1 3 e 8 /x 4. y = 1 4 e x/ 6 5. y = 1 3 e x/ 8 correct 6. y = 1 4 e x/ 3 Explanation: We have to eliminate the parameter t from the equations for x and y . Now from the equation for x it follows that t = 1 9 e x/ 4 . But then y = parenleftBig 1 9 e x/ 4 parenrightBig 1 / 2 = 1 3 e x/ 8 . 002 10.0 points Let Q, R be the points where the ray of angle intersects circles centered at the origin as shown in 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 P R Q and let P be the point of intersection of the horizontal line through Q and the vertical line through R . As varies, P traces out a curve. Write this curve in parametric form ( x ( ) , y ( )) , 2 . (Hint: find the coordinates of Q and R .) 1. (7 sec , 4 tan ) 2. (4 sec , 7 tan ) 3. (7 cos , 4 sin ) correct 4. (4 cos , 7 sin ) 5. (4 tan , 7 sec ) 6. (7 sin , 4 cos ) Explanation: From the graph it follows that the inner circle has radius 4, while the outer circle has radius 7. On the other hand, any circle of radius r centered at the origin can be written in parametric form as (cos , sin ) , 2 , Thus Q and R are the points Q (4 cos , 4 sin ) , R (7 cos , 7 sin ) . But P, Q have the same ycoordinate, and P, R have the same xcoordinate. Version 047 Exam02 Gilbert (56380) 2 Consequently, the curve traced out by P has the parametric form (7 cos , 4 sin ) for 0 2 . Eliminating , we see that P traces out the ellipse x 2 49 + y 2 16 = 1 whose relation to the circles is shown in 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 P 003 10.0 points Find an equation for the tangent line to the curve given parametrically by x ( t ) = e 2 t , y ( t ) = 3 t 2 t + 1 at the point P (1 , 1)....
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This note was uploaded on 04/12/2011 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas at Austin.
 Spring '07
 Sadler
 Multivariable Calculus

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