23 Sample Exam 2B - Sample Second Exam from 2005 1...

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Sample Second Exam from 2005 1. Calculate the following for the function f ( x, y ) = x ln xy . (a) f x ( x, y ) (b) f y ( x, y ) (c) f xy ( x, y ) 2. Let f be the function f ( x, y ) = 5 - 2 xy . Find the rate of change of f at the point (2 , - 1) in the direction of < 4 , - 3 > . Find the maximal rate of increase of f at the point (2 , - 1) and the direction in which it occurs. 3. Find the tangent plane to the level surface 2 xy + 3 xz 2 = 4 at the point (1 , - 4 , - 2). 4. Find the local maximum and local minimum values and saddle points of - x 2 y + x 2 + 2 y 3 + 3 y 2 . 5. Use the method of Lagrange multipliers to find the maximum and minimum values of f ( x, y ) = x 2 + 2 x - y 2 on 2 x 2 + y 2 = 2. 6. Find the volume of the solid under z = x 2 + y 2 and above the triangular region bounded
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Unformatted text preview: by the lines x = 0, y = 0, x + y = 1. 7. Use polar coordinates to find the surface area of the part of the surface z = x 2 + y 2 above the region in the first quadrant bounded by x 2 + y 2 = 1. 8. Evaluate the triple integral R R R E xdV , where E lies under the plane z = 1 + x + 2 y and above the region in the xy-plane bounded by y = √ x , y = 0, x = 1. NOTE: These problems do not cover all the material that you will be held responsible for on Exam 2. You should look at examples from 15.5, 15.6, and 15.8 as well....
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