Engineering Calculus Notes 360

Engineering Calculus Notes 360 - y ) x so for some t (0 ,...

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348 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Now, y x f = g (1) g (0) and the Mean Value Theorem applied to g ( t ) tells us that for some ˜ t (0 , 1), this diFerence = g ( ˜ t ) = b ∂f ∂y ( x 0 + x,y 0 + ˜ t y ) ∂f ∂y ( x 0 ,y 0 + ˜ t y ) B y or, writing ˜ y = y 0 + ˜ t y , and noting that ˜ y lies between y 0 and y 0 + y , we can say that y x f = b ∂f ∂y ( x 0 + x, ˜ y ) ∂f ∂y ( x 0 , ˜ y ) B y where ˜ y is some value between y 0 and y 0 + y . But now apply the Mean Value Theorem to h ( t ) = ∂f ∂y ( x 0 + t x, ˜ y ) with derivative h ( t ) = 2 f ∂x∂y ( x 0 + t x, ˜
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Unformatted text preview: y ) x so for some t (0 , 1) b f y ( x + x, y ) f y ( x , y ) B = h (1) h (0) = h ( t ) = 2 f xy ( x + t x, y ) x and we can say that y x f = b f y ( x + x, y ) f y ( x , y ) B y = 2 f xy ( x, y ) x 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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