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m2sol-20C-su2004

# m2sol-20C-su2004 - Name Math 20C Midterm Exam 2 Student...

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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 ). f ( x, y, z ) = 1 2 x + 2 yz h 1 , 2 z, 2 y i . (b) Find the directional derivative of f at (0 , 2 , 1) in the direction given by h 0 , 3 , 4 i . f (0 , 2 , 1) = 1 2 0 + 4 h 1 , 2 , 4 i = 1 4 h 1 , 2 , 4 i . u = 1 9 + 16 h 0 , 3 , 4 i = 1 5 h 0 , 3 , 4 i . Then, D u f (0 , 2 , 1) = 1 4 h 1 , 2 , 4 i · 1 5 h 0 , 3 , 4 i = 1 20 (6 + 16) = 11 10 . Therefore, D u f (0 , 2 , 1) = 11 / 10. (c) Find the maximum rate of change of f at the point (0 , 2 , 1). |∇ f (0 , 2 , 1) | = 1 4 |h 1 , 2 , 4 i| = 1 4 1 + 4 + 16 = 21 4 . Therefore, the maximum rate of change of f at (0 , 2 , 1) is 21 / 4. # 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 f ( x, y ) = e - ax cos( y ) - e - y cos( x ) is solution of Laplace’s equation f xx + f yy = 0.
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m2sol-20C-su2004 - Name Math 20C Midterm Exam 2 Student...

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