## About the Mean Value Theorem

Let \(f\) be a continuous function on a closed interval \([a, b]\). The Mean Value Theorem tells us that there exists some point \(c\) between \(a\) and \(b\) so that the tangent line to \(f\) at \(c\) is parallel to the line through \((a, f(a))\) and \((b, f(b))\). That is,
\[ f'(c) = \frac{f(b) - f(a)}{b-a} \]

## Using the Applet

After choosing a function type, slide the circles marked "a" and "b" to change the interval. The applet will find one of the values of \(c\) that satisfy the theorem (there may be more than one!). The red line indicates the tangent line to the function at \(c\), while the solid blue line shows the secant through \((a, f(a))\) and \((b, f(b))\).

## About this Applet

This applet was created using JavaScript and the Raphael library. If you are unable to see the applet, make sure you have JavaScript enabled in your browser. This applet may not be supported by older browsers.