Math380_Summer07_Exam2 - minimum value on D(b(2pts Explain...

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Name: Math 380 – Exam 2 July 16, 2007 50 points possible 1. Let F ( x, y, z ) = yz x 2 + y 2 + z 2 , - 2 xz x 2 + y 2 + z 2 , xy x 2 + y 2 + z 2 . (a) (4pts) Compute the divergence of F . (b) (2pts) Interpret the divergence of F physically (or geometrically). 2. (a) (2pts) Let f : U R n R be C 2 . State the Second-Order Taylor Formula for f about a point a in R n . (b) (1pt) What does it mean to say that the error term in the Second-Order Taylor Formula is of second-order? (c) (2pts) Let A be an n × n symmetric matrix. Define the quadratic form associated with A . (d) (3pts) Explain how quadratic forms were used to derive the Second-Derivative Test for scalar-valued functions. (Warning: I am not asking you to state the test.) 3. Let f ( x, y, z ) = x 3 + y 3 + z 2 - xy . (a) (4pts) Show that 1 3 , 1 3 , 0 is a critical point of f . (b) (6pts) Determine the nature of the critical point 1 3 , 1 3 , 0 . 4. Consider T : U R 2 R where T ( x, y ) = x 2 - y + 200 and D = { ( x, y ) | x 2 + y 2 1 } . (a) (3pts) Explain, without calculating anything, why T must attain a maximum and a
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Unformatted text preview: minimum value on D . (b) (2pts) Explain why T does not attain a maximum or a minimum value in the interior of D . (c) (6pts) Use Lagrange multipliers to find the maximum and minimum values of T ( x,y ) on the boundary D ( x 2 + y 2 = 1). 5. Let Z Z D ( x + y ) 2 e ( x-y ) dxdy where D is the region bounded by y = 0, y =-x , and y = x-4. (a) (3pts) Sketch the region of integration, D , in xy-space. (b) (1pt) Find an appropriate change of variables u = g ( x,y ) and v = h ( x,y ) to simplify the integral. (c) (3pts) Sketch the region of integration, D * , in uv-space. 5. Continued. (d) (8pts) Use the Change of Variables Theorem to set-up, but do not evaluate , the integral in uv-space equal to Z Z D ( x + y ) 2 e ( x-y ) dxdy ....
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