# A4f - R 1(0 0 x,y | ≤ 3 4 x 2 y 2 x ≥ 0 y ≥ 0 5 Prove...

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Math 237 Assignment 4 Due: Friday, Oct 17th 1. Let f ( u, v ) = uv 2 where u = u ( x, y ) = x sin y and v = v ( x, y ) = e xy . Find f xy . 2. Let f : R 2 R and define g ( x, y ) = f (cos y, sin x ). Find g xx and g yy . 3. Let f ( x, y ) = (1 + x ) y . a) Calculate the gradient vector and the Hessian matrix of f at (0 , 0). b) Write the linear approximation L (0 , 0) ( x, y ) and the Taylor polynomial P 2 , (0 , 0) ( x, y ) of f . 4. Let f ( x, y ) = 1 + x + 2 y . a) Find the second degree Taylor polynomial P 2 , (0 , 0) ( x, y ) of f . b) Use Taylor’s theorem to show that
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Unformatted text preview: R 1 , (0 , 0) ( x,y ) | ≤ 3 4 ( x 2 + y 2 ), x ≥ 0, y ≥ 0. 5. Prove that if all the second partial derivatives of f are continuous at a point ( a,b ) then f x , f y and f are all continuous at ( a,b ). 6. Find a function f ( x,y ) such that f xy (0 , 0) and f yx (0 , 0) both exist, but f xy (0 , 0) 6 = f yx (0 , 0). Prove you are correct....
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