Engineering Calculus Notes 250

Engineering Calculus Notes 250 - partial derivative of f...

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238 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION subtracting the value of f at a point that shares one coordinate with each of these two points—say f ( x,y + y )—we can write f ( x + x,y + y ) f ( x,y ) = ( f ( x + x,y + y ) f ( x,y + y )) + ( f ( x,y + y ) f ( x,y )) and then proceed to analyze each of the two quantities in parentheses. Note that the Frst quantity is the di±erence between the values of f along a horizontal line segment, which can be parametrized by −→ p ( t ) = ( x + t x,y + y ) , 0 t 1; the composite function g ( t ) = f ( −→ p ( t )) = f ( x + t x,y + y ) is an ordinary function of one variable, whose derivative is related to a
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Unformatted text preview: partial derivative of f (Exercise 6 ): g ( t ) = f x ( x + t x,y + y ) x. Thus, we can apply the Mean Value Theorem to conclude that there is a value t = t 1 between 0 and 1 for which g (1) g (0) = g ( t 1 ) . Letting t 1 x = 1 , we can write f ( x + x,y + y ) f ( x,y + y ) = g (1) g (0) = g ( t 1 ) = f x ( x + 1 ,y + y ) x where | 1 | | x | ....
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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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