hw1 - This print-out should have 15 questions....

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Unformatted text preview: This print-out should have 15 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Determine A so that the curve y = 6 x + 10 can be written in parametric form as x ( t ) = t- 2 , y ( t ) = At- 2 . 1. A = 5 2. A =- 5 3. A =- 7 4. A =- 6 5. A = 7 6. A = 6 correct Explanation: We have to eliminate t from the parametric equations for x and y . Now from the equation for x it follows that t = x + 2. Thus y = 6 x + 10 = A ( x + 2)- 2 . Consequently A = 6 . 002 10.0 points Find a Cartesian equation for the curve given in parametric form by x ( t ) = 1 2 t 2 , y ( t ) = 1 8 t 3 . 1. x = y 2 / 3 2. x = y 4 / 3 3. x = y 3 / 2 4. x = 2 y 2 / 3 correct 5. x = 2 y 3 / 2 6. x = 2 y 4 / 3 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 = 2 y 1 / 3 , in which case x = 1 2 parenleftBig 2 y 1 / 3 parenrightBig 2 = 2 y 2 / 3 . 003 10.0 points Describe the motion of a particle with posi- tion P ( x, y ) when x = 2 sin t, y = 4 cos t as t varies in the interval 0 t 2 . 1. Moves once counterclockwise along the ellipse x 2 4 + y 2 16 = 1 , starting and ending at (0 , 4). 2. Moves along the line x 2 + y 4 = 1 , starting at (2 , 0) and ending at (0 , 4). 3. Moves once clockwise along the ellipse x 2 4 + y 2 16 = 1 , starting and ending at (0 , 4). correct 4. Moves along the line x 2 + y 4 = 1 , starting at (0 , 4) and ending at (2 , 0). 5. Moves once counterclockwise along the circle (2 x ) 2 + (4 y ) 2 = 1 , starting and ending at (0 , 4). 6. Moves once clockwise along the circle (2 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 4 + y 2 16 = 1 ; this is an ellipse centered at the origin. At t = 0, the particle is at (2 sin0 , 4 cos 0), 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 = 2, while y ( t ) decreases from y = 4 to y = 0 ; in particular, the particle moves from a point on the positive y-axis to a point on the positive x-axis, so it is moving clockwise . In the same way, we see that as t increases from / 2 to , the particle moves to a point on the negative y-axis, then to a point on the negative x-axis as t increases from to 3 / 2, until finally it returns to its starting point on the positive y-axis as t increases from 3 / 2 to 2 . Consequently, the particle moves clockwise once around the ellipse x 2 4 + y 2 16 = 1 , starting and ending at (0 , 4)....
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This note was uploaded on 09/28/2008 for the course M 408M taught by Professor Gilbert during the Fall '07 term at University of Texas at Austin.

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hw1 - This print-out should have 15 questions....

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