m2sol-20C-su2004

m2sol-20C-su2004 - Name: Student Number: Math 20C. Midterm...

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Unformatted text preview: 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 , 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 , 3 , 4 i = 1 5 h , 3 , 4 i . Then, D u f (0 , 2 , 1) = 1 4 h 1 , 2 , 4 i 1 5 h , 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 ....
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This note was uploaded on 04/30/2008 for the course MATH 20C taught by Professor Helton during the Summer '08 term at UCSD.

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m2sol-20C-su2004 - Name: Student Number: Math 20C. Midterm...

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