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 $x_0, x_1, x_2, \ldots$ 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 $\vert x_i - x_{i+1}\vert < 0.0005$.

1. Use Newton's method to approximate all of the solutions of $f_{-1}(x)=0$ where

$$f_{-1}(x) = x^4 + 4x^3 - 8x + 1$$

Use $x_0 = -0.3$ for one of your starting values-- the others are up to you.

2. Use Newton's method to approximate all of the solutions of $f_5(x)=0$ where

$$f_5(x) = x^4 + 4x^3 - 8x - 5$$

Use $x_0 = -0.3$ for one of your starting values-- the others are up to you.

3. Use Newton's method to approximate all of the solutions of $f_{16}(x)=0$ where

$$f_{16}(x) = x^4 + 4x^3 - 8x - 16$$

Use $x_0 = -0.3$ for one of your starting values-- the others are up to you.

4. The behavior of the approximations you found starting at $x_0 = -0.3$ should ``be different'' for $f_{-1}$, $f_5$, and $f_{16}$-- 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).