MAT131 - Optional project

*due before Monday 12/7 to get any credit*

This optional project investigates Newton's method, by doing several examples. In each case, you can use a calculator, or a computer, or even do them by hand. You need not show intermediate calculations, but write down the sequence of approximations you get for each trial. In every case, you can stop when the approximation you get is good to 3 decimal places (that is, when .

**1.**
Use Newton's method to approximate all of the solutions of where

Use for one of your starting values-- the others are up to you.

**2.**
Use Newton's method to approximate all of the solutions of where

Use for one of your starting values-- the others are up to you.

**3.**
Use Newton's method to approximate all of the solutions of where

Use for one of your starting values-- the others are up to you.

**4.** The behavior of the approximations you found starting at should ``be different'' for , , and -- indeed,
two of them probably behave differently from all your other choices. Explain
what is unusual about them. Can you give a possible explanation? (look at
the graphs of the functions for a hint).

Scott Sutherland 2002-08-29