Math206Test4

# Math206Test4 - Mid-term Exam # 4 December 3, 2007 MthSc 206...

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Mid-term Exam # 4 December 3, 2007 MthSc 206 – 013: Calculus of Several Variables Instruction: PRINT your name clearly in capitals and underline your last name . NAME: EXAM SOLUTION 1. (8pts) For the function f ( x, y ) = x 2 + y 2 , answer the following questions. (a) (2pts) Find the gradient of f ( x, y ) at (1 , 1). Solution: Since f x = 2 x and f y = 2 y , it is immediate that f x (1 , 1) = 2 and f y (1 , 1) = 2 . Thus f ˛ ˛ (1 , 1) = h 2 , 2 i . (b) (2pts) Find the direction in which the function increases most rapidly at (1 , 1). Solution: The gradient at (1 , 1) is f ˛ ˛ (1 , 1) = h 2 , 2 i . Thus, its direction is u = h 2 , 2 i |h 2 , 2 i| = h 2 , 2 i 2 2 + 2 2 = D 1 2 , 1 2 E . (c) (2pts) Find the direction in which the function decreases most rapidly at (1 , 1). Solution: Its direction is given by - u = D - 1 2 , - 1 2 E . (d) (2pts) Find the directions of zero change in the function at (1 , 1). Solution:

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## This note was uploaded on 04/03/2008 for the course MTHSC 206 taught by Professor Chung during the Spring '07 term at Clemson.

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Math206Test4 - Mid-term Exam # 4 December 3, 2007 MthSc 206...

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