exam2 - Version 146 – Exam02 – gilbert – (55485) 1...

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Unformatted text preview: Version 146 – Exam02 – gilbert – (55485) 1 This print-out should have 14 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Describe the motion of a particle with posi- tion P ( x, y ) when x = 2 sin t , y = 3 cos t as t varies in the interval 0 ≤ t ≤ 2 π . 1. Moves once counterclockwise along the ellipse (2 x ) 2 + (3 y ) 2 = 1 , starting and ending at (0 , 3). 2. Moves along the line x 2 + y 3 = 1 , starting at (2 , 0) and ending at (0 , 3). 3. Moves once clockwise along the ellipse x 2 4 + y 2 9 = 1 , starting and ending at (0 , 3). correct 4. Moves once clockwise along the ellipse (2 x ) 2 + (3 y ) 2 = 1 , starting and ending at (0 , 3). 5. Moves once counterclockwise along the ellipse x 2 4 + y 2 9 = 1 , starting and ending at (0 , 3). 6. Moves along the line x 2 + y 3 = 1 , starting at (0 , 3) and ending at (2 , 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 4 + y 2 9 = 1 ; this is an ellipse centered at the origin. At t = 0, the particle is at (2 sin0 , 3 cos0), 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 = 2, 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 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 4 + y 2 9 = 1 , starting and ending at (0 , 3). keywords: motion on curve, ellipse 002 10.0 points Which, if any, of the following pairs of vectors are perpendicular? I. i + 5 j − 2 k , 3 i − 2 j − 4 k , II. ( 3 , 2 ) , ( 4 , − 6 ) . 1. II only correct 2. I only 3. both of them 4. neither of them Version 146 – Exam02 – gilbert – (55485) 2 Explanation: I Since the dot product ( i + 5 j − 2 k ) · (3 i − 2 j − 4 k ) = 1 , the vectors are not perpendicular. II Since the dot product ( 3 , 2 )·( 4 , − 6 ) = (3)(4) + (2)( − 6) = 0 , the vectors are perpendicular....
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This note was uploaded on 05/02/2011 for the course M 408 D taught by Professor Textbookanswers during the Spring '07 term at University of Texas.

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exam2 - Version 146 – Exam02 – gilbert – (55485) 1...

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