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**Unformatted text preview: **polar transformation x = r cos , y = r sin , express 2 f x 2 + 2 f y 2 in terms of r , and the partial derivatives of f with respect to r and . (7) If z = y + f ( x 2-y 2 ), where f is dierentiable, show that y z x + x z y = x . (8) Find the direction in which f ( x,y,z ) = ze xy increases most rapidly at the point (0 , 1 , 2). What is the maximum rate increase. (9) Suppose a > 0 and a i > 0 for i = 1 ,...,n . Find the extremes of the function f ( x 1 ,...,x n ) = x a 1 1 x a 2 2 x a n n under the constraint x 1 + x 2 + + x n = a where x i > 0 for i = 1 ,...,n . (10) Find the points on the surface xy 2 z 3 = 2 that are closed to the origin. 1...

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