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final265su2004

# final265su2004 - 4 Find and classify the extrema of f x y =...

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Math 265 Final Exam, Summer 2004. NAME 8 Problems, 80 Minutes Instructions: Answer each question completely. Show all work. No credit for mere answers with no work shown. Show the steps of calculations. State the reasons that justify conclusions. 1. Find an equation for the plane perpendicular to the curve x = 3 t , y = 1 4 t 2 , z = 1 6 t 3 , at the point where t = 2. 2. Parametric equations for a point P moving in a plane are x ( t ) = 3 cos t , y ( t ) = 2 sin t . (a) Graph the path of P . (b) Calculate v ( t ) and a ( t ) and graph these vectors at (0 , 2).

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3. The temperature T at a point ( x, y, z ) in a ball centered at the origin is given by T ( x, y, z ) = 360 p x 2 + y 2 + z 2 . (a) Find a vector in the direction of greatest increase of temperature from (1 , 2 , 2). (b) Does the vector in (a) point toward the origin?

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Unformatted text preview: 4. Find and classify the extrema of f ( x, y ) = xy-2 x-y . 5. Find the area of the part of the surface z = x 2 + y 2 that is cut o± by the plane z = 4. 6. Reverse the order of integration in the iterated integral Z 2-2 Z √ 4-y 2 f ( x, y ) dx dy. 7. Calculate the work done by the force F ( x, y, z ) = ( yz + 1) i + ( xz + 1) j + ( xy + 1) k along the path consisting of line segments from (0 , , 0) to (2 , , 0) to (2 , 3 , 0) to (2 , 3 , 4). [ hint : Is F a conservative vector Feld?] 8. Calculate I C ( x 2 + 4 xy ) dx + (2 x 2 + 3 y ) dy around the ellipse C given by x = 4 cos t , y = 3 sin t , 0 ≤ t ≤ 2 π ....
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