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Linear Approximation

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Suppose that $ f$ has continuous second partial derivatives. Then

$\displaystyle f(x+\Delta x) = f(x) + f'(x)\Delta x + R $

with

$\displaystyle R = \frac{1}{2} f''(t)(\Delta x)^2 = O((\Delta x)^2) $

and $ t$ lies between $ x$ and $ \Delta x$ .
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  automatically generated 4/ 7/2008