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Assignment 6 - Math222 Assignment 6 due Wednesday Nov 7...

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Math222 Assignment 6 due Wednesday Nov. 7, 2007 1. The function z = Z ( x, y ) is defined implicitly by the equation x 2 + y 2 + 4 z 2 + z 4 = 64 (a) Find ∂z ∂x , ∂z ∂y , 2 z ∂x 2 , and then evaluate these at the point P = (4 , 4 , 2). (b) Find the equation of the tangent plane to the surface defined above at the point P . (c) Find Z at P and find the direction of maximum increase of the function Z . 2. Find the critical points indicating (with your justification) which are local maxima, local minima, and which are saddle points of the function. z = xy e - ( x 2 +4 y 2 ) / 2 3. Let z be defined implicitly by z 3 + 4 z = 2 x 2 y + 12. Taking x and y as independent variables, find all the first and second partials of z and evaluate these at ( x, y ) = (1 , 2) given that
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