Tutorial_7 - Page 1 Chapter 5 FUNCTIONS OF SEVERAL...

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Unformatted text preview: Page 1 Chapter 5 FUNCTIONS OF SEVERAL VARIABLES Example 1. Compute the first-order and the second-order partial derivatives of the following functions: (a) f ( x, y ) = 2 y x 2 . f x =- 4 y x 3 , f y = 2 x 2 ; f xx = 12 y x 4 , f yy = 0 , f xy = f yx =- 4 x 3 . (b) f ( x, y ) = ( x 2- xy + y 2 ) 2 . f x = 2( x 2- xy + y 2 )(2 x- y ) , f y = 2( x 2- xy + y 2 )(2 y- x ). f xx = 2(2 x- y ) (2 x- y ) + 2( x 2- xy + y 2 )(2) = 12 x 2- 12 xy + 6 y 2 ; f yy = 2(2 y- x )(2 y- x ) + 2( x 2- xy + y 2 )(2) = 16 y 2- 12 xy + 6 x 2 ; f xy = f yx = 2(2 x- y )(2 y- x ) + 2( x 2- xy + y 2 )(- 1) = 12 xy- 6 x 2- 6 y 2 ; (c) f ( x, y ) = e xy 2 . f x = y 2 e xy 2 , f y = 2 xy e xy 2 . f xx = 0 e xy 2 + y 2 y 2 e xy 2 = y 4 e xy 2 ; f yy = 2 x e xy 2 + 2 xy 2 xye xy 2 = (2 x + 4 x 2 y 2 ) e xy 2 ; f xy = f yx = 2 y exy 2 + y 2 2 xye xy 2 = e xy 2 (2 y + 2 xy 3 ). Example 2. Find the critical points of the function x 3- 3 xy + y 3- 5. Classify them using the Second Derivative Test....
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This note was uploaded on 08/24/2011 for the course ENGINEERIN 1122 taught by Professor Stadnik during the Spring '11 term at University of Ottawa.

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Tutorial_7 - Page 1 Chapter 5 FUNCTIONS OF SEVERAL...

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