Engineering Calculus Notes 250

Engineering Calculus Notes 250 - partial derivative of...

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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 = δ 1 , we can write f ( x + △ x,y + △ y ) − f ( x,y + △ y ) = g (1) − g (0) = g ′ ( t 1 ) = ∂f ∂x ( x + δ 1 ,y + △ y ) △ x where | δ 1 | ≤ |△ x | ....
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