Taylor's Theorem
An approximation method for differentiable functions.
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Study Taylor’s Theorem
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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