Engineering Calculus Notes 256

Engineering Calculus Notes 256 - 244 CHAPTER 3. REAL-VALUED...

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Unformatted text preview: 244 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION If we substitute into this the expression for the affine approximation → → → → → T− 0 f (− 0 + △− ) = f (− 0 ) + d− 0 f (△− ) x x x x → x x we obtain the following version of Equation (3.4): → → → → → → f (− 0 + △− ) − f (− 0 ) = d− 0 f (△− ) + △− ε, x x x x x x → − → where ε → 0 as △− → 0 . x → → Using the representation of d− 0 f (△− ) as a dot product, we can rewrite x x this in the form → −→ → → → → → f (− 0 + △− ) − f (− 0 ) = ∇ f (− 0 ) · △− + △− ε, x x x x x x → − → where ε → 0 as △− → 0 . x → In a similar way, we can write the analogous statement for − (t), using p →→ − = − (t ): ˙ p0 v → → − (t + △t) − − (t ) = − △t + |△t| − , → → p0 p0 v δ → − → − where δ → 0 as △t → 0. → Now, we consider the variation of the composition f (− (t)) as t varies from p t = t0 to t = t0 + △t: → − → −→ → → → → f (− (t0 + △t)) − f (− (t0 )) = ∇ f (− 0 ) · − △t + |△t| δ + △− ε p p x v x → − → −→ → −→ → → = ∇ f (− 0 ) · (− △t) + |△t| ∇ f (− 0 ) · δ + △− ε. x v x x Subtracting the first term on the right from both sides, we can write → −→ → → → f (− (t0 + △t)) − f (− (t0 )) − ∇ f (− 0 ) · (− △t) p p x v → − → − → −→ △x ε. = (△t) ∇ f (− 0 ) · δ + |△t| x △t Taking the absolute value of both sides and dividing by |△t|, we get → −→ 1 → → → f (− (t0 + △t)) −f (− (t0 )) − ∇ f (− 0 ) · (△t− ) p p x v |△t| → → − → −→ △− x ε = ∇ f (− 0 ) · δ ± x △t → → − → −→ △− x |ε| . δ+ x ≤ ∇ f (− 0 ) △t ...
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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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