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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 final answer. This test should take approx. 90 min. The actual exam will have 4 problems with fewer questions. 1. (10 pts) A function z satisfies 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 satisfies 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 ) negationslash = (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 w s and w t in terms of f x and f y . In particular, show that w s = - xf x - yf y and w t = - yf x + xf y . Remark: Note that in this problem it is

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

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