sampletes22 - x,y,z ) in the space is T ( x,y,z ) = x + y-z...

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Math 137 Sample Test II 1. Consider the function w = f ± x 2 , y 3 x . a) Show that xw x + yw y = 2 w 2. Compute w xx + w yy where w = f ( p x 2 + y 2 ). 3A A bird is flying in R 3 . The temperature at the point ( x,y,z ) is T ( x,y,z ) = x - y + z 2 . Now the bird is at the point (1 , 2 , 1). In which direction should it fly such that the temperature (a) increases the fastest? (b) decreases the fastest? (c) does not change? 3B a) Find the tangent plane for the surface x 2 + xyz + z 2 = 4 at the point (0 , 1 , 2) b) Find the tangent plane for the surface z = x 2 + xy + y 2 at the point (0 , 0 , 1) 4. (a) Find the absolute extrema of the function f ( x,y ) = 1 x 2 + y 2 +1 over the region x 2 + xy + y 2 1. (b) Change the region in (a) to the triangle bounded by x = 0, y = 0 and x + y = 1, and redo (a). 5. A satllite is moving around the earth. Its orbit is described by the equations x 2 + 4 y 2 = 1 and z = 0 The temperature at the point (
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Unformatted text preview: x,y,z ) in the space is T ( x,y,z ) = x + y-z + 273. Find the hottest and coldest point on the satlllites orbit. 6. An ice tray for a refrigerator is made from material that cost 2 cents per in 2 . It has 12 identical rectangular compartments with square bottoms. The total volume of the ice tray, ignoring the partitions, is 12 in 3 . Find the dimension of the tray with the lowest material cost. 7. Compute the following integrals, changing the order of integration if necessary. a) R D R xydxdy where the region D is the unit disc x 2 + y 2 1. b) R D R ( x 2 + y 2 ) dxdy where the region D is bounded by the curves x = y and y = x 2 . c) Z 1 dy Z 1 y e x 3 1...
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This note was uploaded on 03/10/2010 for the course MATH 137 taught by Professor Phann during the Fall '09 term at Georgetown KY.

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