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Unformatted text preview: zakaria (mmz255) HW07 gilbert (55485) 1 This printout should have 20 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 x ( t ) = 8 t 2 , y ( t ) = 8 t 3 . 1. x = 4 y 2 / 3 2. x = 2 y 3 / 2 3. x = 2 y 4 / 3 4. x = 2 y 2 / 3 correct 5. x = 4 y 4 / 3 6. x = 4 y 3 / 2 Explanation: We have to eliminate the parameter t from the equations for x and y . But from the equation for y , it follows that t = 1 2 y 1 / 3 , in which case x = 8 parenleftbigg 1 2 y 1 / 3 parenrightbigg 2 = 2 y 2 / 3 . 002 10.0 points Describe the motion of a particle with posi tion P ( x, y ) when x = 5 sin t , y = 4 cos t as t varies in the interval 0 t 2 . 1. Moves along the line x 5 + y 4 = 1 , starting at (0 , 4) and ending at (5 , 0). 2. Moves once clockwise along the ellipse x 2 25 + y 2 16 = 1 , starting and ending at (0 , 4). correct 3. Moves once clockwise along the ellipse (5 x ) 2 + (4 y ) 2 = 1 , starting and ending at (0 , 4). 4. Moves along the line x 5 + y 4 = 1 , starting at (5 , 0) and ending at (0 , 4). 5. Moves once counterclockwise along the ellipse x 2 25 + y 2 16 = 1 , starting and ending at (0 , 4). 6. Moves once counterclockwise along the ellipse (5 x ) 2 + (4 y ) 2 = 1 , starting and ending at (0 , 4). 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 25 + y 2 16 = 1 ; this is an ellipse centered at the origin. At t = 0, the particle is at (5 sin0 , 4 cos0), i.e. , at the point (0 , 4) on the ellipse. Now as t increases from t = 0 to t = / 2, x ( t ) increases from x = 0 to x = 5, while y ( t ) decreases from y = 4 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 zakaria (mmz255) HW07 gilbert (55485) 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 25 + y 2 16 = 1 , starting and ending at (0 , 4). 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 ) = t 3 2 t , y ( t ) = t 2 3 , where the arrows indicate the direction of increasing t ? 1. x y correct 2. x y 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 )) = ( t 3 2 t, t 2 3) . zakaria (mmz255) HW07 gilbert (55485) 3 Now x ( t ) = ( t ) 3 2( t ) = ( t 3 2 t ) = x ( t ) , and y (...
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This note was uploaded on 06/05/2011 for the course MATH 408 D taught by Professor Gilbert during the Spring '11 term at University of Texas at Austin.
 Spring '11
 Gilbert

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