### Applet: Quadratic approximation to a function

The function $f(x)$ (thick green curve) is approximated by a quadratic function $Q(x)$ (thin blue curve) and linear tangent line $L(x)$ (thin red line) at the point $x=a$. The quadratic function is equal to the linear function plus $c(x-a)^2$, where $c$ is specified by typing a value in its box. One can enter an expression involving derivatives like $f''(a)$ in the box for $c$. The idea is to find a value for $c$ that gives the best quadratic approximation as you zoom in around the point at $x=a$ (purple point). You can change $a$ by dragging the magenta point or typing a value in the box. You can also type in a new function for $f(x)$. Values of $f$, its derivatives, and the linear and quadratic approximations are shown. You can use the buttons at the top to zoom in and out as well as pan the view.

Applet file: quadratic_approximation.ggb

#### Applet links

This applet is found in the pages

#### General information about Geogebra Web applets

This applet was created using Geogebra. In most Geogebra applets, you can move objects by dragging them with the mouse. In some, you can enter values with the keyboard. To reset the applet to its original view, click the icon in the upper right hand corner.

You can download the applet onto your own computer so you can use it outside this web page or even modify it to improve it. You simply need to download the above applet file and download the Geogebra program onto your own computer.