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Unformatted text preview: 5. (a) Find and classify the critical points of f ( x,y ) = x 2 y-x 2-y 2-2 y as local maxima, local minima or saddle points using the test involving the second partial deriva-tives of f ( x,y ). (b) Use the Lagrange multiplier method to nd the shortest distance from the origin to the curve xy 2 = 1. 6. For each of the following double integrals (a) Z 1 Z 1 x 1 / 3 p 1-y 4 dy dx, (b) ZZ x 2 + y 2 1 ln( x 2 + y 2 ) dxdy, sketch the domain of integration and evaluate the integral. 7. Find the volume of the region bounded by the cylinder x 2 + y 2 = 2 y , the paraboloid x 2 + y 2 = z and the plane z = 0. 8. Compute ZZZ R xz dV , where R is the solid tetrahedron with vertices (0 , , 0) , (1 , , 0) , (1 , 1 , 0) , (0 , 1 , 1) . 2...
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- Spring '08