Engineering Calculus Notes 359

Engineering Calculus Notes 359 - shows that these two...

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3.7. HIGHER DERIVATIVES 347 (Note that the signs attached to the four values of f ( x,y ) correspond to the signs in Figure 3.29 .) Each of the ±rst-order di²erences x f ( y 0 ) ( resp . x f ( y 0 + y )) is an approximation to ∂f ∂x at ( x 0 ,y 0 ) ( resp . ( x 0 ,y 0 + y )), multiplied by x ; their di²erence is then an approximation to 2 f ∂y∂x at ( x 0 ,y 0 ), multiplied by y x ; we shall use the Mean Value Theorem to make this claim precisely. But ±rst consider the other way of going: the di²erences along the two vertical edges y f ( x 0 ) = f ( x 0 ,y 0 + y ) f ( x 0 ,y 0 ) y f ( x 0 + x ) = f ( x 0 + x,y 0 + y ) f ( x 0 + x,y 0 ) represent the change in f ( x,y ) as x is held constant at one of the two values x = x 0 ( resp . x = x 0 + x ) and y increases by y from y = y 0 ; this roughly approximates ∂f ∂y at ( x 0 ,y 0 ) ( resp . ( x 0 + x,y 0 )), multiplied by y , and so the di²erence of these two di²erences x y f = y f ( x 0 + x ) − △ y f ( x 0 ) = [ f ( x 0 + x,y 0 + y ) f ( x 0 + x,y 0 )] [ f ( x 0 ,y 0 + y ) f ( x 0 ,y 0 )] = f ( x 0 + x,y 0 + y ) f ( x 0 + x,y 0 ) f ( x 0 ,y 0 + y ) + f ( x 0 ,y 0 ) approximates 2 f ∂x∂y at ( x 0 ,y 0 ), multiplied by x y . But a close perusal
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Unformatted text preview: shows that these two second-order dierences are the sameand this will be the punch line of our proof. Actually, for technical reasons, we dont follow the strategy suggested above precisely. Lets concentrate on the rst (second-order) dierence: counterintuitively, our goal is to show that 2 f xy ( x ,y ) = lim ( x, y ) (0 , 0) y x f y x . To this end, momentarily x x and y and dene g ( t ) = x f ( y + t y ) = f ( x + x,y + t y ) f ( x ,y + t y ); then g ( t ) = b f y ( x + x,y + t y ) f y ( x ,y + t y ) B y....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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