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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 di²erences are the same—and this will be the punch line of our proof. Actually, for technical reasons, we don’t follow the strategy suggested above precisely. Let’s concentrate on the ±rst (second-order) di²erence: counterintuitively, our goal is to show that ∂ 2 f ∂x∂y ( x ,y ) = lim ( △ x, △ y ) → (0 , 0) △ y △ x f △ y △ x . To this end, momentarily ±x △ x and △ y and de±ne 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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