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08spr-f - Math 51 Final Exam June 6 2008 Name Section...

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Math 51 Final Exam — June 6, 2008 Name : Section Leader: Fai Joseph David Anca Bezirgen (Circle one) Chandee Cheng Fernandez-Duque Vacarescu Veliyev Section Time: 10:00 11:00 1:15 2:15 (Circle one) Complete the following problems. You may use any result from class you like, but if you cite a theorem be sure to verify the hypotheses are satisfied. In order to receive full credit, please show all of your work and justify your answers. You do not need to simplify your answers unless specifically instructed to do so. You have 3 hours. This is a closed-book, closed-notes exam. No calculators or other electronic aids will be permitted. If you finish early, you must hand your exam paper to a member of teaching staff. If you need extra room, use the back sides of each page. If you must use extra paper, make sure to write your name on it and attach it to this exam. Do not unstaple or detach pages from this exam. Please sign the following: “On my honor, I have neither given nor received any aid on this examination. I have furthermore abided by all other aspects of the honor code with respect to this examination.” Signature: The following boxes are strictly for grading purposes. Please do not mark. 1 16 9 12 2 12 10 15 3 16 11 12 4 12 12 15 5 12 13 15 6 12 14 15 7 13 15 15 8 8 Total 200
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Math 51, Spring 2008 Final Exam — June 6, 2008 Page 2 of 16 1. (16 points) Let f ( x, y ) = x 2 y - 4 xy + 1 2 y 2 + 1 . (a) Calculate formulas for the gradient of f and the Hessian matrix of f at the point ( x, y ). (b) Find all critical points of f . (c) For each critical point, determine if it corresponds to a local maximum, local minimum or saddle point of f . Show your reasoning. (d) Write the quadratic approximation (that is, the degree-2 Taylor polynomial) for f at the point ( x, y ) = (0 , 0).
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Math 51, Spring 2008 Final Exam — June 6, 2008 Page 3 of 16 2. (12 points) Suppose S is the surface in R 3 given by the equation xz 3 + yz 2 + x 2 y = 18. (a) Find an equation of the plane tangent to S at (1 , 2 , 2). (b) Using linear approximations, estimate the z -coordinate of the point on the surface S that has x = 1 . 1 and y = 1 . 96.
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Math 51, Spring 2008 Final Exam — June 6, 2008 Page 4 of 16 3. (16 points) Let f ( x, y ) be a function on R 2 . Suppose that f (1 ,
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