m2-20C-su2004

# m2-20C-su2004 - f x y = e-ax cos y-e-y cos x is solution of...

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Name: Student Number: Math 20C. Midterm Exam 2 July 23, 2004 Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No credit will be given for illegible solutions. 1. (8 points) Consider the function f ( x, y, z ) = x + 2 yz . (a) Find the gradient of f ( x, y, z ). (b) Find the directional derivative of f at (0 , 2 , 1) in the direction given by h 0 , 3 , 4 i . (c) Find the maximum rate of change of f at the point (0 , 2 , 1). # Score 1 2 3 4 Σ 1

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2. (8 points) Find any value of the constant a such that the function

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Unformatted text preview: f ( x, y ) = e-ax cos( y )-e-y cos( x ) is solution of Laplace’s equation f xx + f yy = 0. 2 3. (8 points) Let f ( x, y ) = 12 xy-2 x 3-3 y 2 . (a) Find all the critical (stationary) points of f . (b) For each critical point of f , determine whether f has a local maximum, local minimum, or saddle point at that point. 3 4. (8 points) Use Lagrange multipliers to fnd the maximum and minimum values oF the Function f ( x, y ) = x 2 + y 2 subject to the constraint 1 4 x 2 + 1 9 y 2 = 1. 4...
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m2-20C-su2004 - f x y = e-ax cos y-e-y cos x is solution of...

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