Extra Example 1.  Investigate Newton's method for finding the roots of  .
Given the initial seed    determine if the sequence    converges to one of the roots  .  

Explore Extra Solution 1.

Enter the function    and find its roots.

We color each point   in the rectangle   in the following fashion.  Color the point yellow, blue, or red, if the sequence of iterations produced by the seed    converges to , respectively.  The iteration formula is

The formulas for  f[z] and  g[z]  and approximations to the roots are used in the function  newton[x,y]  to make the density plot.

Make a plot showing the initial seeds  ,  and the nature of the sequence   that is produced by it.

(c) 2006 John H. Mathews, Russell W. Howell