This applet shows some of the features of Newton's method.[br]When Newton's method works, it converges quickly,[br]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.[br]Each illustrates some features:[br][list][br][*][math]f(x)=2x \cos(x)-.84[/math] [br]This is a nice function with lots of roots.[br][*][math]f(x)=x^4-2x^3-x^2-2x+2[/math] [br]This is a nice polynomial with two roots.[br][*][math]f(x)=x^4-3x^3+x^2+2x-2[/math] [br]Although this looks similar to the previous problem, starting at some points causes an endless loop that never converges.[br][*][math]f(x)-x^2+1[/math] [br]This problem clearly has no roots. It shows what Newton tries to do in such a case.[br][*][math]f(x)=cbrt(x)[/math] [br]This is a classical problem where Newton's method does not work.[br][*][math]f(x)=\cos(x)-x/5[/math] [br]This function has an obvious root. It also has a lot or relative extrema so there are i lots of bad starting points.[br][*]Finally, you can enter your own function.[br][/list][br][br]With each functions there are some interesting questions to ask:[list][br][*] How big is the region around a root where the function will converge quickly, say within 15 steps?[br][*] 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?[br][*] Are there regions where we don't find a root, even with lots of iterations?[br][/list]