tut2 - 0 and f(0 0 = 0 Show that f is continuous at(0 0 4...

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MATH 237 Tutorial 2 Wednesday, May 17th, 2006 Level Curves, Limits and Continuity 1) For each of the following functions f : R 2 R sketch and discuss typical level curves, identify any exceptional level curves and sketch and identify the graph. a) f ( x, y ) = 2 xy - y 2 b) f ( x, y ) = x 2 + 4 y 2 - 9 c) f ( x, y ) = p 36 - 4 x 2 - 9 y 2 2) A function f : R 2 R is defined by f ( x, y ) = x 2 - 2 y 3 | x | + y 2 , ( x, y ) 6 = (0 , 0) . a) Derive an inequality of the form | f ( x, y ) | ≤ M ( x, y ) for all ( x, y ) 6 = (0 , 0) where the function M is continuous at (0 , 0) and M (0 , 0) = 0. b) What conclusion can you draw from (a)? c) Define f (0 , 0) so as to make f continuous at (0 , 0). 3) Let f : R 2 R be defined by f ( x, y ) = x 3 + y 4 x 2 + y 2 for ( x, y )
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Unformatted text preview: 0) and f (0 , 0) = 0 . Show that f is continuous at (0 , 0). 4) Let g : R 2 → R be given by g ( x, y ) = x 2 y 3 x 4 + y 4 for ( x, y ) 6 = (0 , 0) and g (0 , 0) = 0 . Determine whether g is continuous at (0 , 0). 5) For each function f , determine whether or not lim ( x,y ) → (0 , 0) f ( x, y ) exists. Determine f (0 , 0) so that f is continuous at (0 , 0), whenever this is possible: a) f ( x, y ) = xy 2 x 2 + 3 y 2 , ( x, y ) 6 = (0 , 0), b) f ( x, y ) = xy 2 x 2 + 2 y 2 , ( x, y ) 6 = (0 , 0)....
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