Final Review S10

# 6 pts gremlins when they come into contact with water

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5. (6 pts) Gremlins, when they come into contact with water, begin to reproduce at an unrestricted rate (that is, at a rate directly proportional to their current population). If there was one gremlin before they started reproducing ( t = 0) and an hour later ( t = 1), there were 100 gremlins, how many gremlins will there be after two hours? 6. (6 pts) Find f x , f y , and f xy for the function below. f ( x, y ) = xe xy .

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Final Exam, Page 6 of 15 May 8, 2010 7. (9 pts) Find and classify all extrema for the function f ( x, y ) = x 3 + y 3 - 6 xy
Final Exam, Page 7 of 15 May 8, 2010 8. (8 pts) An open rectangular box is constructed out of materials that cost \$1 per square foot for the sides and \$2 per square foot for the bottom. Assume that one side is 4 ft long, and the volume of the box is 36 ft 3 . Use Lagrange multipliers to find the dimensions of the box that can be constructed at minimum cost.

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Final Exam, Page 8 of 15 May 8, 2010 9. (9 pts) Evaluate ZZ R 4 x sin( y 2 ) dA where R is the region bounded by y = x 2 , x 0 , y = π .
Final Exam, Page 9 of 15 May 8, 2010 10. (7 pts) Verify that y = tan( x

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