This applet shows some of the features of Newton's method.
When Newton's method works, it converges quickly,
The applet lets you focus on either the full sequence of points or following through step by step.

The Applet comes with number of functions preloaded.
Each illustrates some features:

This is a nice function with lots of roots.

This is a nice polynomial with two roots.

Although this looks similar to the previous problem, starting at some points causes an endless loop that never converges.

This problem clearly has no roots. It shows what Newton tries to do in such a case.

This is a classical problem where Newton's method does not work.

This function has an obvious root. It also has a lot or relative extrema so there are i lots of bad starting points.

Finally, you can enter your own function.

With each functions there are some interesting questions to ask:

How big is the region around a root where the function will converge quickly, say within 15 steps?

Are there places where the root found is not stable, that is where a small change in the starting point gives a big change in the root found?

Are there regions where we don't find a root, even with lots of iterations?