a2 - Math 237 Assignment 2 Due Friday Oct 2nd 1 Find the...

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Unformatted text preview: Math 237 Assignment 2 Due: Friday, Oct 2nd 1. Find the ﬁrst and second partial derivatives of a) f (x, y ) = x3 + 3x2 y + 4y 2 . b) g (x, y ) = 2x2 + y 2 . 2. A function f : R2 → R is deﬁned by f (x, y ) = g (x − 2y ), where g : R → R. If g (1) = 3, calculate fx (3, 1) and fy (3, 1). Show that fy (x, y ) = −2fx (x, y ) in general. √ xy 3. Let f (x, y ) = 1, x2 + y 2 + 1, if (x, y ) = (0, 0) if (x, y ) = (0, 0). a) Determine if f is continuous at (0, 0). b) Find fx (0, 0) and fy (0, 0). c) Find the equation of the tangent plane for f at (0, 0). 4. Determine if f is continuous at (0, 0) where f (x, y ) = ( x− y ) 2 , |x| + |y | 0, if (x, y ) = (0, 0) if (x, y ) = (0, 0). 5. Find a function f (x, y ) such that fx (0, 0) and fy (0, 0) both exist but f (x, y ) is not continuous at (0, 0). 6. Let f (x, y ) = y ln(2x2 + y 2 ), if (x, y ) = (0, 0) Prove that fx exists for all (x, y ) ∈ R2 , 0, if (x, y ) = (0, 0). but fx is not continuous at (0, 0). ...
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This note was uploaded on 10/12/2010 for the course MATH 237 taught by Professor Wolczuk during the Fall '08 term at Waterloo.

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