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calc hw 5

# calc hw 5 - lawrence(cdl678 – Homework 5 – ODELL...

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Unformatted text preview: lawrence (cdl678) – Homework 5 – ODELL – (56280) 1 This print-out should have 30 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 ) = 8 t 2 , y ( t ) = 8 t 3 . 1. x = 4 y 4 / 3 2. x = 2 y 4 / 3 3. x = 2 y 2 / 3 correct 4. x = 2 y 3 / 2 5. x = 4 y 3 / 2 6. x = 4 y 2 / 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 = 8 parenleftbigg 1 2 y 1 / 3 parenrightbigg 2 = 2 y 2 / 3 . 002 10.0 points Find a Cartesian equation for the curve given in parametric form by x ( t ) = 5 cos 2 3 t, y ( t ) = 4 sin 2 3 t. 1. 4 x + 5 y = 20 correct 2. 5 x + 4 y = 20 3. x 5- y 4 = 1 20 4. 4 x- 5 y = 20 5. x 4- y 5 = 1 20 6. x 4 + y 5 = 1 20 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 5 + y 4 = 1 . But then after simplification, the curve has Cartesian form 4 x + 5 y = 20 . 003 10.0 points Determine a Cartesian equation for the curve given in parametric form by x ( t ) = 3 ln(4 t ) , y ( t ) = √ t. 1. y = 1 3 e 4 /x 2. y = 1 3 e x/ 2 3. y = 1 2 e x/ 6 correct 4. y = 1 2 e 6 /x 5. y = 1 3 e x/ 4 6. y = 1 2 e x/ 3 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 4 e x/ 3 . lawrence (cdl678) – Homework 5 – ODELL – (56280) 2 But then y = parenleftBig 1 4 e x/ 3 parenrightBig 1 / 2 = 1 2 e x/ 6 . 004 10.0 points Describe the motion of a particle with posi- tion P ( x, y ) when x = 4 sin t, y = 2 cos t as t varies in the interval 0 ≤ t ≤ 2 π . 1. Moves once clockwise along the ellipse x 2 16 + y 2 4 = 1 , starting and ending at (0 , 2). correct 2. Moves along the line x 4 + y 2 = 1 , starting at (4 , 0) and ending at (0 , 2). 3. Moves along the line x 4 + y 2 = 1 , starting at (0 , 2) and ending at (4 , 0). 4. Moves once clockwise along the ellipse (4 x ) 2 + (2 y ) 2 = 1 , starting and ending at (0 , 2). 5. Moves once counterclockwise along the ellipse (4 x ) 2 + (2 y ) 2 = 1 , starting and ending at (0 , 2). 6. Moves once counterclockwise along the ellipse x 2 16 + y 2 4 = 1 , starting and ending at (0 , 2). 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 4 = 1 ; this is an ellipse centered at the origin. At t = 0, the particle is at (4 sin0 , 2 cos0), i.e. , at the point (0 , 2) 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 = 2 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 ....
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calc hw 5 - lawrence(cdl678 – Homework 5 – ODELL...

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