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Math+22+-+Final+Review+-+Winter+11

# Math+22+-+Final+Review+-+Winter+11 - Math 22 Final Review...

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Math 22 - Final Review - Winter 11 1.) Find and classify the critical points of each function. (a.) f ( x, y ) = x 2 - xy + y 2 + 9 x - 6 y + 10 (b.) f ( x, y ) = 2 x 3 + xy 2 + 5 x 2 + y 2 (c.) f ( x, y ) = 3 xy - x 2 y - xy 2 (d.) f ( x, y ) = x 4 + y 4 - 4 xy + 2 (e.) f ( x, y ) = x 2 y + xy 2 + x + y + 1 2.) Find the absolute maximum and minimum values of f on the set D . Give all points where these extreme values occur. (a.) f ( x, y ) = 3 + xy - x - 2 y , D is the triangular region with vertices (1 , 0),(5 , 0), and (1 , 4) (b.) f ( x, y ) = x 2 + y 2 + x 2 y + 4, D = { ( x, y ) R 2 : | x | ≤ 1 , | y | ≤ 1 } 3.) A cardboard box without a lid is to have a volume of 32 , 000 cubic centimeters. Find the dimensions of the box that minimize the amount of cardboard used. 4.) Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). Give all points where these extreme values occur. (a.) f ( x, y ) = x 2 y ; x 2 + 2 y 2 = 6 (b.) f ( x, y, z ) = 2 x + 6 y + 10 z ; x 2 + y 2 + z 2 = 35 (c.) f ( x, y, z ) = x + 2 y ; x + y + z = 1, y 2 + z 2 = 4 5.) Evaluate. (a.) R 1 0 R 1 0 xy p x 2 + y 2 dy dx (b.) R R D x 1 + xy dA , D = [0 , 1] × [0 , 1] (c.) R R D x cos( y ) dA , D is bounded by y = 0, y = x 2 , and x = 1 (d.) R R D y 3 dA , D is the triangular region with vertices (0 , 2),(1 , 1), and (3 ,

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Math+22+-+Final+Review+-+Winter+11 - Math 22 Final Review...

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