07spr-f

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

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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).
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.
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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