# a3 - 0 4 Consider the theorem ”If f is diﬀerentiable at...

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Math 237 Assignment 3 Due: Friday, Oct 8th 1. Let f ( x, y ) = xy x 2 + y 2 . a) Find the equation of the tangent plane of f at (1 , 2 , 2 / 5). b) Approximate f (0 . 9 , 2 . 1). 2. For each of the following functions f : R 2 R , determine if f is diﬀerentiable at (0 , 0). a) f ( x, y ) = ± x 3 + y 4 x 2 + y 2 if ( x, y ) 6 = (0 , 0) 0 if ( x, y ) = (0 , 0) . b) f ( x, y ) = ± x 4 + y 4 x 2 + y 2 + 1 if ( x, y ) 6 = (0 , 0) 1 if ( x, y ) = (0 , 0) . c) f ( x, y ) = ± x | y | x 2 + y 2 if ( x, y ) 6 = (0 , 0) 0 if ( x, y ) = (0 , 0) . 3. Let f ( x, y ) = ± y 3 x 2 + y 2 if ( x, y ) 6 = (0 , 0) 0 , if ( x, y ) = (0 , 0) . Prove that f is not diﬀerentiable at (0
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Unformatted text preview: , 0). 4. Consider the theorem, ”If f is diﬀerentiable at ( a, b ), the f is continuous at ( a, b ).” Prove that the converse is false by ﬁnding a function f which is continuous at (1 , 0), but not diﬀerentiable at (1 , 0). Prove that your function satisﬁes the conditions....
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