This set of exercises will be mostly experimental. So, fire up your favorite computational environment. This text will include examples using both Python and Mathematica.

1

Continuing with the example of \(f(x)=x^2-2\) explored in Example 1.1.3, compute ten Newton iterations for several values of \(x_0\text{.}\) Be sure to choose both positive and negative values and values that are both large and small in magnitude.

2

In the previous exercise, what happens when \(x_0=0\text{?}\) Draw a graph to illustrate the situation.

3

Let \(f\) be a quadractic function that has two, distinct, real roots but that is otherwise arbitrary. Using a geometrical understanding of the real Newton's method, show why an initial seed \(x_0\) always leads to a sequence that converges to the closer of the two roots of \(f\text{.}\)

4

Let's modify Newton's original example just a little bit to consider