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# A6 - MATH 222 HOMEWORK 6 DUE 1 P ROBLEMS Problem 1.1 There...

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MATH 222 HOMEWORK 6 DUE OCTOBER 31, 2011 1. P ROBLEMS Problem 1.1. There is only one point where the plane tangent to the surface z = x 2 +2 xy + 2 y 2 - 6 x + 8 y is horizontal. Find it. Problem 1.2. Find an equation of the plane tangent to the surface z = f ( x, y ) at the indi- cated point P . a.) z = x 2 + y 2 , P = (3 , 4 , 25) , b.) z = x 3 - y 3 , P = (3 , 2 , 19) , c.) z 2 = x 2 + y 2 , P = (3 , - 4 , 5) . Problem 1.3. Find the local maxima, minima, and saddle points of the given surfaces. a.) z = 2 x 2 + 8 xy + y 4 , b.) z = e 2 x - 4 y - x 2 - y 2 , c.) z = (1 + 2 x 2 ) e - x 2 - y 2 . Problem 1.4. Calculate the directional derivative of f at P in the direction of ~v . a.) f ( x, y ) = arctan( y/x ) , P = ( - 3 , 3) , ~v = (3 , 4) . b.) f ( x, y, z ) = xy + yz + zx , P = (1 , - 1 , 2) , ~v = (1
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