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Unformatted text preview: rhodes (ajr2283) HW07 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 Find a Cartesian equation for the curve given in parametric form by y ( t ) = 1 2 t 2 , x ( t ) = 1 8 t 3 . 1. y = 2 x 3 / 2 2. y = 2 x 4 / 3 3. y = x 2 / 3 4. y = x 4 / 3 5. y = 2 x 2 / 3 correct 6. y = x 3 / 2 Explanation: We have to eliminate the parameter t from the equations for x and y . But from the equation for x , it follows that t = 2 x 1 / 3 , in which case y = 1 2 parenleftBig 2 x 1 / 3 parenrightBig 2 = 2 x 2 / 3 . 002 10.0 points Describe the motion of a particle with posi tion P ( x, y ) when x = 4 sin t , y = 3 cos t as t varies in the interval 0 t 2 . 1. Moves once counterclockwise along the ellipse x 2 16 + y 2 9 = 1 , starting and ending at (0 , 3). 2. Moves along the line x 4 + y 3 = 1 , starting at (4 , 0) and ending at (0 , 3). 3. Moves along the line x 4 + y 3 = 1 , starting at (0 , 3) and ending at (4 , 0). 4. Moves once clockwise along the ellipse (4 x ) 2 + (3 y ) 2 = 1 , starting and ending at (0 , 3). 5. Moves once counterclockwise along the ellipse (4 x ) 2 + (3 y ) 2 = 1 , starting and ending at (0 , 3). 6. Moves once clockwise along the ellipse x 2 16 + y 2 9 = 1 , starting and ending at (0 , 3). correct Explanation: Since cos 2 t + sin 2 t = 1 for all t , the particle travels along the curve given in Cartesian form by x 2 16 + y 2 9 = 1 ; this is an ellipse centered at the origin. At t = 0, the particle is at (4 sin0 , 3 cos0), i.e. , at the point (0 , 3) on the ellipse. Now as t increases from t = 0 to t = / 2, x ( t ) increases from x = 0 to x = 4, while y ( t ) decreases from y = 3 to y = 0 ; in particular, the particle moves from a point on the positive yaxis to a point on the positive xaxis, so it is moving clockwise . In the same way, we see that as t increases from / 2 to , the particle moves to a point rhodes (ajr2283) HW07 Gilbert (56380) 2 on the negative yaxis, then to a point on the negative xaxis as t increases from to 3 / 2, until finally it returns to its starting point on the positive yaxis as t increases from 3 / 2 to 2 . Consequently, the particle moves clockwise once around the ellipse x 2 16 + y 2 9 = 1 , starting and ending at (0 , 3). keywords: motion on curve, ellipse 003 10.0 points Which one of the following could be the graph of the curve given parametrically by x ( t ) = 2 t t 3 , y ( t ) = 3 t 2 , where the arrows indicate the direction of increasing t ? 1. x y 2. x y correct 3. x y 4. x y 5. x y 6. x y Explanation: All the graphs are symmetric either about the yaxis or the xaxis. Lets check which it is for the graph of ( x ( t ) , y ( t )) = (2 t t 3 , 3 t 2 ) . rhodes (ajr2283) HW07 Gilbert (56380) 3 Now x ( t ) = 2( t ) ( t ) 3 = (2 t t 3 ) = x ( t ) , and y ( t ) = 3...
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This note was uploaded on 09/27/2010 for the course M 408 D taught by Professor Textbookanswers during the Spring '07 term at University of Texas at Austin.
 Spring '07
 TextbookAnswers
 Multivariable Calculus

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