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exam2 - APPM 2350 EXAM 2 FALL 2007 INSTRUCTIONS Computers...

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APPM 2350 EXAM 2 FALL 2007 INSTRUCTIONS: Computers, calculators, books, and crib sheets are not permitted. Write your (1) name, (2) instructor’s name, and (3) recitation number on the front of your bluebook. Work all problems. Show your work clearly. Note that a correct answer with incorrect or no supporting work may receive no credit, while an incorrect answer with relevant work may receive partial credit. 1. (20 points) Using an appropriate technique from Calculus III, determine the extreme values of the function f ( x, y ) = x 2 + y 2 4 , assuming that xy = 1. For each point you find, be sure to state whether the function has a local maxima or minima at that location, and the value of f ( x, y ). 2. (20 points) The surface elevation, z , of a small mountain area that you have just purchased is given by the function z = - y 2 - x 4 + 4 x 2 + 54 for - 3 x 3 and - 3 y 3. (a) Where on this property would you build a hut? (Hint: perhaps a level spot that is not a summit. Clearly explain your reasoning, and show your calculations, to support your answer.) (b) A stream emerges from a spot on the surface located at P 0 (0 , 1 , 53) and flows downhill following the line of steepest descent. At the point P 0 , in which direction does the stream flow? (Give
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