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# a3 - =(0 0 4 Invent a function f R 2 → R such that f x(1...

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Math 237 Assignment 3 Due: Friday, Oct 9th 1. Use the linear approximation to approximate arctan(0 . 99(1 . 01) 3 ). 2. Let f ( x, y, z ) = xyz x + y + z . Use the linear approximation to approximate f ( - 1 . 04 , - 1 . 98 , 3 . 97). 3. For each of the following functions f : R 2 R , determine if f is differentiable at (0 , 0). a) f ( x, y ) = x 4 - y 4 x 2 + y 2 , if ( x, y ) = (0 , 0) 0 , if ( x, y ) = (0 , 0) . b) f ( x, y ) = y 3 x 2 + y 2 , if ( x, y ) = (0 , 0) 0 , if ( x, y ) = (0 , 0) . c) f ( x, y ) = xy x 2 + y 2 + 1 , if ( x, y ) = (0 , 0) 1 , if (
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Unformatted text preview: ) = (0 , 0) . 4. Invent a function f : R 2 → R such that f x (1 , 1) and f y (1 , 1) both exist, but f is not diﬀerentiable at (1 , 1). Prove your function satisﬁes the conditions. 5. Prove that if f satisﬁes | f ( x, y ) | ≤ x 2 + y 2 for all ( x, y ) ∈ R 2 , then f is diﬀerentiable at (0 , 0)....
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