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Gilbert_Hmwk07sol - moseley(cmm3869 HW07 Gilbert(56380 This...

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moseley (cmm3869) – HW07 – Gilbert – (56380) 1 This print-out should have 12 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 ) = 1 4 t 2 , y ( t ) = 1 8 t 3 . 1. x = y 2 / 3 correct 2. x = 1 2 y 3 / 2 3. x = 1 2 y 4 / 3 4. x = 1 2 y 2 / 3 5. x = y 3 / 2 6. x = 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 4 parenleftBig 2 y 1 / 3 parenrightBig 2 = y 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 π . 1. Moves once clockwise along the ellipse (4 x ) 2 + (3 y ) 2 = 1 , starting and ending at (0 , 3). 2. Moves once counterclockwise along the ellipse x 2 16 + y 2 9 = 1 , starting 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 x 2 16 + y 2 9 = 1 , starting and ending at (0 , 3). correct 5. Moves along the line x 4 + y 3 = 1 , starting at (4 , 0) and ending at (0 , 3). 6. Moves once counterclockwise along the ellipse (4 x ) 2 + (3 y ) 2 = 1 , starting and ending at (0 , 3). 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 sin 0 , 3 cos 0), i.e. , at the point (0 , 3) on the ellipse. Now as t increases from t = 0 to t = π/ 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 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
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moseley (cmm3869) – HW07 – Gilbert – (56380) 2 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 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 ) = t 2 3 , y ( t ) = 2 t t 3 , where the arrows indicate the direction of increasing t ? 1. x y 2. x y 3. x y 4. x y 5. x y cor- rect 6. x y Explanation: All the graphs are symmetric either about the y -axis or the x -axis. Let’s check which it
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moseley (cmm3869) – HW07 – Gilbert – (56380) 3 is for the graph of ( x ( t ) , y ( t )) = ( t 2 3 , 2 t t 3 ) . Now x ( t ) = ( t ) 2 3 = t 2 3 = x ( t ) , and y ( t ) = 2( t ) ( t ) 3 = (2 t t 3 ) = y ( t ) , so ( x ( t ) , y ( t )) = ( x ( t ) , y ( t )) . Thus the graph is symmetric about the x -axis, eliminating three choices.
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