A4 - -1 2 if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 6 Prove that...

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Math 237 Assignment 4 Due: Friday, Feb 13th 1. Let g : R 2 R and let f ( x,y ) = g ( y 2 , ln x y ). Find 2 f ∂x∂y . What assumptions do you need to make about g so that you can apply the chain rule? 2. Let h : R R and define g ( x,y ) = x 2 h ( x + y 2 ). Find g xx and g yy . What assumptions do you need to make about h ? 3. Let f ( u,v ) = ln( u 2 + v ) where u ( x ) = e x and v = v ( x,y ) = p x 2 + y 2 . Find f xy and f yx . You may leave your answer in terms of u , v , x and y . 4. Let f ( x,y ) = e 2 x - 3 y . a) Calculate the gradient vector and the Hessian matrix of f at (1 , 2). b) Calculate L (1 , 2) ( x,y ) and P 2 , (1 , 2) ( x,y ) of f at (2 , 3). 5. Let f ( x,y ) = e - 2 x + y . Use Taylor’s theorem to show that the error in the linear approximation L (1 , 1) ( x,y ) is at most 6 e [( x - 1) 2 + ( y
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Unformatted text preview: -1) 2 ] if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. 6. 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 ). 7. 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. 8. Consider f : R 2 → R defined by f ( x,y ) = 2 x 2 + 3 y 2 . Prove that for any ( a,b ) ∈ R 2 we have f ( x,y ) ≥ L ( a,b ) ( x,y ) for all ( x,y ) ∈ R 2 ....
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