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Taylor Polynomial

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The Taylor polynomial

$\displaystyle p_n(x) = f(a) + f'(a) (x-a) + \cdots + \frac{f^{(n)}(a)}{n!} (x-a)^n $

interpolates the derivatives of a function $ f$ at a point $ a$ up to order $ n$, i.e., $ p_n^{(k)}(a)=f^{(k)}(a)$ for $ k= 0,\ldots ,n$. If $ f$ is $ (n+1)$-times continuously differentiable,

$\displaystyle f(x) = p_n(x) + R,\quad R = \frac{f^{(n+1)}(t)}{(n+1)!} (x-a)^{n+1} \,, $

for some $ t$ between $ a$ and $ x$.

Example:

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  automatically generated 7/20/2016