07spr-f - Math 51 Final Exam June 8 2007 Name Section...

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Math 51 Final Exam — June 8, 2007 Name : Section Leader : Theodora Peter Eric Henry Baosen (Circle one) Bourni Kim Schoenfeld Segerman Wu 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 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 14 pts 9 8 pts 2 12 pts 10 10 pts 3 8 pts 11 12 pts 4 12 pts 12 10 pts 5 16 pts 13 16 pts 6 12 pts 14 14 pts 7 12 pts 15 12 pts 8 20 pts 16 12 pts Total 200 pts
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Math 51, Spring 2007 Final Exam — June 8, 2007 Page 2 of 17 1. (14 points) Let f ( x, y ) = 1 2 x 2 + 3 2 y 2 - xy 3 . (a) Find all the critical points of f . For each, specify if it is a local maximum, a local minimum, or a saddle point, and briefly show how you know. (b) Write the quadratic approximation (that is, the degree-2 Taylor polynomial) for f at the point ( x, y ) = (1 , 1).
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Math 51, Spring 2007 Final Exam — June 8, 2007 Page 3 of 17 2. (12 points) Consider the function f ( x, y ) = 50 - x 2 - y 2 . (a) Find an equation that defines the level set of f through the point ( x, y ) = (3 , 4). Sketch and label the curve and point on the axes below. (Be sure to include the scales on your axes.) (b) Calculate f , the gradient of f , at the point ( x, y ) = (3 , 4) and indicate it on your diagram above. (c) Calculate the directional derivative of f at the point (3 , 4) in the direction of the vector (2 , - 1).
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Math 51, Spring 2007 Final Exam — June 8, 2007 Page 4 of 17 3. (8 points) Suppose S is the surface in R 3 given by the equation xy + yz + xz = 1 . (a) Find the equation of the tangent plane to S at the point ( x, y, z ) = ( - 1 , 2 , 3). (b) Use linear approximation to estimate the value of z for the point on S where x = - 1 . 01 and y = 2 . 02.
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Math 51, Spring 2007 Final Exam — June 8, 2007 Page 5 of 17 4. (12 points) (a) Assume h ( x, y ) = g ( x 2 + y 2 ), where g is a function of one variable. Find h x (1 , 2) + h y (1 , 2), given that g (5) = 3.
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