>>6
yeah, there are some functions where if you apply newton's method, it can overshoot the root. Sometimes it'll still find it, sometimes it'll overshoot it more and more, and end up bouncing back and forth between larger and larger negative and positive numbers. But if you set it up right with the right kind of function, you can prove that it'll converge, and you can get a sufficient amount of iterations required to obtain a desired small error. Think about the functions you are using geometrically, and find a good place to start it so that overshooting never happens. Then you can try to get some bounds on how fast it'll converge.