a2_soln

# a2_soln - Math 237 x4 y 4 x2 + y 2 Assignment 2 Solutions...

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Math 237 Assignment 2 Solutions 1. Let f ( x, y ) = ± x 4 - y 4 x 2 + y 2 if ( x, y ) 6 = (0 , 0) 0 if ( x, y ) = (0 , 0) . Determine all points where f is continuous. Solution: Since x 2 + y 2 6 = 0 if ( x, y ) 6 = (0 , 0), we have that f is continuous for all ( x, y ) 6 = (0 , 0) by the continuity theorems. For f to be continuous at (0 , 0) we must have lim ( x,y ) (0 , 0) f ( x, y ) = 0. We try to prove this with the squeeze theorem. We have ² ² ² ² x 4 - y 4 x 2 + y 2 - 0 ² ² ² ² x 4 x 2 + y 2 + y 4 x 2 + y 2 = x 2 · x 2 x 2 + y 2 + y 2 · y 2 x 2 + y 2 x 2 ( x 2 + y 2 ) x 2 + y 2 + y 2 ( x 2 + y 2 ) x 2 + y 2 = x 2 + y 2 , since x 2 x 2 + y 2 and y 2 x 2 + y 2 . Thus, since lim ( x,y ) (0 , 0) x 2 + y 2 = 0 , we get lim ( x,y ) (0 , 0) f ( x, y ) = 0 by the Squeeze theorem. Hence, f is continuous at (0 , 0). 2. Let f ( x, y ) = ± x 3 x 2 + y 2 if ( x, y ) 6 = (0 , 0) 0 if ( x, y ) = (0 , 0) . a) Determine all points where

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## This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.

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a2_soln - Math 237 x4 y 4 x2 + y 2 Assignment 2 Solutions...

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