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