18_new - HW06 gilbert(55035 This print-out should have 17...

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HW06 gilbert (55035) 1 This print-out should have 17 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 cos 2 3 t , y ( t ) = 5 sin 2 3 t . x y 1 1. 5 + 2 = 10 2. x 5 y 2 = 1 10 3. x 2 y 5 = 1 10 4. 5 x + 2 y = 10 correct 5. 2 x + 5 y = 10 6. 5 x 2 y = 10 Explanation: We have to eliminate the parameter t from the equations for x and y . Now cos 2 θ + sin 2 θ = 1 . Thus x 2 + y 5 = 1 . But then after simplification, the curve has Cartesian form 5 x + 2 y = 10 . 002 10.0 points Determine a Cartesian equation for the curve given in parametric form by x ( t ) = 4 ln(9 t ) , y ( t ) = t . 1. y = 1 3 e X / 4 2. y = 1 4 e X / 6 3. y = 1 4 e 6/ X 4. y = 1 4 e X / 3 5. y = 1 3 e 8/ X 6. y = 1 3 e X / 8 correct Explanation: We have to eliminate the parameter t from the equations for x and y . Now from the equation for x it follows that t = 1 9 e X / 4 . But then y = 1 9 e X / 4 1/2 = 1 3 e X / 8 . 003 10.0 points Determine a Cartesian equation for the curve given in parametric form by x ( t ) = 4 e 2 T , y ( t ) = 2 e T . 1. x 2 = 8 y x 2. y 2 = 16 x 3. y 2 = 32 4. x 2 y = 8 5. xy 2 = 32 6. xy 2 = 16 correct 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 1/2 , 4

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HW06 gilbert (55035) 2 from which in turn it follows that y = 2 x 4 1/2 . Consequently, xy 2 = 16 . 004 10.0 points Describe the motion of a particle with posi- tion P ( x, y ) when x = 2 sin t , y = 5 cos t as t varies in the interval 0 t 2 π . 1. Moves once clockwise along the ellipse x 2 + y 2 = 1 , 4 25 starting and ending at (0 , 5). correct 2. Moves once counterclockwise along the ellipse (2 x ) 2 + (5 y ) 2 = 1 , starting and ending at (0 , 5). 3. Moves along the line x 2 + y 5 = 1 , starting at (0 , 5) and ending at (2 , 0). 4. Moves along the line x 2 + y 5 = 1 , starting at (2 , 0) and ending at (0 , 5). 5. Moves once clockwise along the ellipse (2 x ) 2 + (5 y ) 2 = 1 , starting and ending at (0 , 5). 6. Moves once counterclockwise along the ellipse x 2 y 2 + 25 = 1 , 4 starting and ending at (0 , 5). 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 + y 2 = 1 ; 4 25 this is an ellipse centered at the origin. At t = 0, the particle is at (2 sin 0 , 5 cos 0), 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 = 2, 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 . 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 .
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