Practice Exam 2

Practice Exam 2 - MATH 241: CALCULUS III Department of...

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MATH 241: CALCULUS III Department of Mathematics, UMCP Fall 2007 Practice Exam 2 Handed out : Tuesday, 10/23/07 WORK ON ALL PROBLEMS. Justify your answers. Cross out what is not meant to be part of your Fnal answer. This test should take approx. 90 min. The actual exam will have 4 problems with fewer questions. 1. (10 pts) A function z satisFes Laplace’s equation in the variables x and y if 2 z ∂x 2 + 2 z ∂y 2 = 0 . Show that z ( x, y ) = tan - 1 ( y/x ) + Ax + By + Cxy satisFes Laplace’s equation where A , B and C are arbitrary constants. Hint: Recall that d dx tan - 1 x = (1 + x 2 ) - 1 . 2. (10 pts) Consider the function f ( x, y ) = xy n x 4 + y 4 , for ( x, y ) n = (0 , 0) 0 , for ( x, y ) = (0 , 0) , where n is a non-negative integer ( n = 0 , 1 , 2 , . . . ). Show that f ( x, y ) is continuous at the point (0 , 0) if n 4 . 3. (10 pts) Let w = f ( x, y ) where x = e - s cos t and y = e - s sin t . (a)(4 pts) By use of the Chain Rule, compute
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Practice Exam 2 - MATH 241: CALCULUS III Department of...

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