Assignment 6

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

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Unformatted text preview: Math222 Assignment 6 due Wednesday Nov. 7, 2007 1. The function z = Z(x, y) is defined implicitly by the equation x2 + y 2 + 4z 2 + z 4 = 64 z (a) Find x , (4, 4, 2). z y , 2z x2 , and then evaluate these at the point P = (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 +4y 2 )/2 3. Let z be defined implicitly by z 3 + 4z = 2x2 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 z(1, 2) = 2. 4. Let f (x, y) = 4x2 + 2y 3 - 3xy 2 - 24y + 40. (a) Find and classify all the critical points of the function f (x, y), (b) Find the absolute maximum and minimum values of f (x, y) in the region {0 x 2, 0 y 1}. 5. An open rectangular box has volume 128 cubic centimeters. Among all such boxes, what are the dimensions of the box with minimum surface area? 6. Using the method of Lagrange Multipliers, find the maximum and minimum values of the function 12x2 + 8xy + 12y 2 + z 2 if the point (x, y, z) is constrained to lie on the sphere x2 + y 2 + z 2 = 8 ...
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This note was uploaded on 04/29/2008 for the course PHYS 230 taught by Professor Harris during the Fall '07 term at McGill.

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