# HW2S - Gauthier, Joseph Homework 2 Due: Sep 6 2007, 3:00 am...

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Gauthier, Joseph – Homework 2 – Due: Sep 6 2007, 3:00 am – Inst: JEGilbert 1 This print-out should have 17 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. The due time is Central time. YES, homework 2 is due BEFORE homework 1 001 (part 1 oF 1) 10 points Determine A so that the curve y = 8 x + 21 can be written in parametric Form as x ( t ) = t - 3 , y ( t ) = At - 3 . 1. A = - 7 2. A = - 6 3. A = - 8 4. A = 7 5. A = 6 6. A = 8 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 + 3. Thus y = 8 x + 21 = A ( x + 3) - 3 . Consequently A = 8 . keywords: Cartesian equation, parametric equations, eliminate parameter 002 (part 1 oF 1) 10 points Determine a Cartesian equation For the curve given in parametric Form by x ( t ) = 2 e t , y ( t ) = 4 e - 2 t . 1. x 2 y = 16 2. x 2 y = 32 3. x 2 y = 32 4. x 2 y = 16 correct 5. x y 2 = 8 6. xy 2 = 8 Explanation: We have to eliminate the parameter t From the equations For x and y . Now From the equation For x it Follows that e t = x 2 , From which in turn it Follows that y = 4 2 x · 2 . Consequently, x 2 y = 16 . keywords: Cartesian equation, parametric equations, eliminate parameter 003 (part 1 oF 1) 10 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 π . 1. Moves along the line x 2 + y 4 = 1 , starting at (2 , 0) and ending at (0 , 4).

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Gauthier, Joseph – Homework 2 – Due: Sep 6 2007, 3:00 am – Inst: JEGilbert 2 2. Moves once counterclockwise along the ellipse x 2 4 + y 2 16 = 1 , starting 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 once counterclockwise along the circle (2 x ) 2 + (4 y ) 2 = 1 , starting and ending at (0 , 4). 5. Moves along the line x 2 + y 4 = 1 , starting at (0 , 4) and ending at (2 , 0). 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 sin 0 , 4 cos 0), i.e. , at the point (0 , 4) on the ellipse. Now as t increases from t = 0 to t = π/ 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 π/ 2 to π , the particle moves to a point on the negative y -axis, then to a point on the negative x -axis as t increases from π to 3 π/ 2, until Fnally it returns to its starting point on the positive y -axis as t increases from 3 π/ 2 to 2 π . Consequently, the particle moves clockwise
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## This note was uploaded on 03/19/2008 for the course M 408M taught by Professor Gilbert during the Fall '07 term at University of Texas.

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HW2S - Gauthier, Joseph Homework 2 Due: Sep 6 2007, 3:00 am...

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