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2007-10-05 The Error in the Approximation delta f = df

# 2007-10-05 The Error in the Approximation delta f = df -...

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The Error in the Approximation Ll f :: df Let f(x) be differentiable at x = Xuand suppose that i:u is an incrementof x. We have two ways to describe the changein f as x changesfrom Xo to xo + ~x: The true change: The differential estimate: ~f = f(xo + ~x) - f(xo) df = f'(xo)~x. How well does df approximate ~f? We measurethe approximation error by subtracting df from ~f: Approximation error = ~f - df = t:.f - f' (xoMx = /(xo + ~:) - f(xo), - f'(xo)~x t!.f ( f(XO + ~x) - f(xo) f ' ( ») = - Xo ~X ~x . , Cal! [~is pm f = €. ~x. As ~x -7- 0, the difference quotient f(xo + ~x) - f(xo) ~x approaches f' (xo) (rememberthe definition of
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Unformatted text preview: f' (xo)), sothequantity in parentheses becomes a very small number (which is why we called it E). In fact, € ~ 0 as ~x ~ O. When ~X is small, the approximation error € t:.x is smaller still. 6.f = f'(xo)~x + E ~x . --,-' ~ ' ' true estimated error change change If y = f(x) is differentiable at x = xo, and x changes from Xo to Xo + ~x. the change ~y in f is given by an equation of the form ~y = f'(xo)6.x + E ~x in which € ~ 0 as ~x ~ O....
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