You see above you the graph of y = x3 - x. Newton's method works just fine here. Mess with the applet until you see how it works. Make sure to try it on the "STEP" setting, so you can really see what is going on. One thing to try: Can you find a value of x0 that is SO stupid that it takes Newton's method more than ten iterations to find a root? Where are the dumbest choices of x0?

There are two bad cases to investigate next. First, the case that we discussed in class, where Newton's method starts to buzz around a root that isn't there. The second case, even stranger, is when Newton's method gets in a loop.

  1. In the case of y = x3 - x, for what initial guesses will Newton's method converge on the root 0? Are there any initial guesses for which Newton's method will utterly fail? If so, what are they?
  2. In the first case, are there values of x0 that are positive for which Newton's method still finds the root? For the values you tried, did it eventually find it?
  3. In class, you received four reasons that Newton's method might fail. Did the second case fall into any of those categories? If so, which one?
  4. Why would Newton's method probably not be super-helpful in finding all the roots of f(x)=0.1(x-1)(x-1.001)(x-1.002)?