GUIDING QUESTION:
How can I compute a solution to an equation?
What are fixed points and what do they have to do with the root finding problem?
Enclosure vs. fixed point iteration methods
Enclosure methods (like the bisection method): guaranteed but slow convergence.
Fixed point methods: faster, but no guarantees.
We begin by reviewing the Mean Value Theorem.
Mean Value Theorem (MVT)
Let $f$ be continuous on $[a,b]$ and suppose $f$ is continuously differentiable on $(a,b)$.
There is a number $c$ in $(a,b)$ such that
\begin{equation}
f'(c)=\frac{f(b) -f(a)}{b-a},
\end{equation}
or equivalently,
$$f(b)-f(a) = f'(c)(b-a).$$
MVT
“ Instantaneous speed at some point in interval is equal
to average speed over entire interval. ”