# Math Insight

### Applet: The Taylor polynomial

Illustration of an $n$th order Taylor polynomial (thin blue curve) to a function $f(x)$ (thick green curve), where $n$ can be changed from 0 to 10 using the red slider. The $n$th order Taylor polynomial centered at the point $a$ is \begin{align*} p_n(x) = \sum_{j=0}^n \frac{1}{j!} f^{(j)}(a) (x-a)^j, \end{align*} where $f^{(j)}(a)$ denotes the $j$th derivative of $f$ evaluated at $x=a$. For large $n$, the display of $p_n(x)$ usually runs off the right edge of the applet and is cut off. You can change $a$ by dragging the purple point or typing a number in the box. You can also enter a new function for $f$. If you check the “show all” box, all lower order Taylor polynomials, $p_i(x)$ for $i < n$, are plotted by thin pink curve. You can use the buttons at the top to zoom in and out as well as pan the view.

Applet file: taylor_polynomial.ggb