Example 1.  Use Newton's method to find the roots of the cubic polynomial  .  
1 (a) Fast Convergence.  Investigate quadratic convergence at the simple root  ,  using the starting value  

Solution 1 (a).

Graph the function.

The Newton-Raphson iteration formula  g[x]  is

Investigate quadratic convergence at the simple root  ,  using the starting value  

First, do the iteration one step at a time.  
Type each of the following commands in a separate cell and execute them one at a time.

Notice that convergence is fast and the sequence is converging to the simple root    

At the simple root    we can explore the relationship    for  k  sufficiently large.

This will be done by investigating the ratio    for  k  sufficiently large.

Evaluate the quantity   at the root  .

Which is close to the computed value    

(c) John H. Mathews 2004