Taylor Polynomials

The $n$-th Taylor polynomial of $f(x)$ at $x=0$ is the degreee $n$ polynomial, denoted by $P_n(x)$ that best approximates $f(x)$ near $x=0$ in the sense that $f(x)$ and $P_n(x)$ have the same $k$-th derivative at $x=0$ for all $0\le k\le n$. These polynomials are given by: $$P_n(x)=\sum _{k=0}^n\frac{f^{(k)}(0)}{k!}{x}^k$$ The following allows you to explore the Taylor polynomials of a function $f$ interactively. Input a function and move the sliders to change the value of $n$ to see how the Taylor polynomials change as $n$ changes.

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