Visualizing Taylor Polynomials
Overview
If \(f(x)\) has a power series representation around \(x=a\) with radius of convergence \(R>0\), then we get the Taylor series representation \[f(x)\approx \sum_{n=0}^{\infty} \dfrac{f^{(n)}(0)}{n!}(x-a)^n\] for \(x\) within distance \(R\) of \(a\)
If we only compute a finite number of terms, then we get the Taylor polynomial approximation \[f(x) \approx f(0) + f'(0)(x-a) + \dfrac{f''(0)}{2!}(x-a)^2 + \dots + \dfrac{f^{(n)}(0)}{n!}(x-a)^n\]
Instructions
In the interactive below, you
- select a function \(f\) from the drop down list
- select a center for your approximation \(a\)
- select the degree of the Taylor polynomial approximation.
The function will be graphed along with its Taylor approximation.