Function Type:

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, $f'(c) = \frac{f(b) - f(a)}{b-a}$
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 $$(b, f(b))$$.