An approximation method for differentiable functions.

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Concept |
Content |

Polynomials |
A basic class of functions that consists a linear combinations of monomials. |

Power Series |
A sequence of partial sums of polynomial functions. |

Higher Derivatives |
Taking derivatives of derivatives of differentiable functions. |

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The aim of this part is to approximate a sufficiently smooth (that
means sufficiently often differentiable) function by a polynomial. More
precisely, we will perform the approximation by \[f(x_0+h)\approx\sum_{k=0}^n\frac{f^{(k)}(x_0)}{k!}h^k=f(x_0)+f'(x_0)h+\frac{f^{\prime
\prime}(x_0)}2h^2+\frac{f^{\prime\prime\prime}(x_0)}6h^3+\ldots+\frac{f^{(n)}(x_0)}{n!}h^n.\]
We will also estimate the approximation error.

**Theorem 1** (Taylor’s formula). *Let \(I\) be an interval and assume that \(f:I\to\mathbb{R}\) is \(n+1\)-times differentiable. Let \(x_0\in I\) and \(h\in\mathbb{R}\) such that \(x_0+h\in I\). Then there exists some \(\theta\in(0,1)\) such that \[f(x_0+h)=\underbrace{\sum_{k=0}^n\frac{f^{(k)}(x_0)}{k!}h^k}_{\text{``Taylor
polynomial''}}+\underbrace{\frac{f^{(n+1)}(x_0+\theta
h)}{(n+1)!}h^{n+1}}_{\text{``remainder term''}}.\] The
number \(x_0\) is called **expansion
point*.

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