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HW07-solutions

# HW07-solutions - moore(jwm2685 – HW07 – gilbert...

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Unformatted text preview: moore (jwm2685) – HW07 – gilbert – (55485) 1 This print-out should have 20 questions. Multiple-choice 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 ) = 2 t 2 , y ( t ) = 8 t 3 . 1. x = y 2 / 3 2. x = y 3 / 2 3. x = y 4 / 3 4. x = 1 2 y 3 / 2 5. x = 1 2 y 2 / 3 correct 6. x = 1 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 = 1 2 y 1 / 3 , in which case x = 2 parenleftbigg 1 2 y 1 / 3 parenrightbigg 2 = 1 2 y 2 / 3 . 002 10.0 points Describe the motion of a particle with posi- tion P ( x, y ) when x = 3 sin t , y = 5 cos t as t varies in the interval 0 ≤ t ≤ 2 π . 1. Moves once clockwise along the ellipse (3 x ) 2 + (5 y ) 2 = 1 , starting and ending at (0 , 5). 2. Moves along the line x 3 + y 5 = 1 , starting at (3 , 0) and ending at (0 , 5). 3. Moves once clockwise along the ellipse x 2 9 + y 2 25 = 1 , starting and ending at (0 , 5). correct 4. Moves once counterclockwise along the ellipse x 2 9 + y 2 25 = 1 , starting and ending at (0 , 5). 5. Moves once counterclockwise along the ellipse (3 x ) 2 + (5 y ) 2 = 1 , starting and ending at (0 , 5). 6. Moves along the line x 3 + y 5 = 1 , starting at (0 , 5) and ending at (3 , 0). 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 9 + y 2 25 = 1 ; this is an ellipse centered at the origin. At t = 0, the particle is at (3 sin0 , 5 cos0), i.e. , at the point (0 , 5) on the ellipse. Now as t increases from t = 0 to t = π/ 2, x ( t ) increases from x = 0 to x = 3, while y ( t ) decreases from y = 5 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 . moore (jwm2685) – HW07 – gilbert – (55485) 2 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 finally 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 once around the ellipse x 2 9 + y 2 25 = 1 , starting and ending at (0 , 5). 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 ) = 3 − t 2 , y ( t ) = 2 t − t 3 , where the arrows indicate the direction of increasing t ? 1. x y cor- rect 2. x y 3. x y 4. x y 5. x y 6. x y Explanation: moore (jwm2685) – HW07 – gilbert – (55485) 3 All the graphs are symmetric either about the y-axis or the x-axis. Let’s check which it is for the graph of ( x ( t ) , y ( t )) = (3 − t 2 , 2 t − t 3 ) ....
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HW07-solutions - moore(jwm2685 – HW07 – gilbert...

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