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 $0 \leq k \leq n$. 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.