Newton's Method for estimating square roots. Sometime ago I wrote a program that used Newtons Method and derivatives to approximate unknown square roots (say $\sqrt 5$) from known square roots like $\sqrt  4$.I have since lost the calculator and the book I got the equation from. 
Edit Researched a bit let me see if I have this right.
First I start with my known $$\sqrt 4=2$$ then I subtract. Thus. $2-\frac {4-5}{(2(\sqrt (4)}$
Then I take that answer call it $r_{t1}$ and plug it back in so that I have $r_{t1} -\frac {{r_{t1}}^2-5}{2(r_{t1})}$ Rinse lather repeat..
Right?

To find a square root of $a$ using Newton's Method, we can write:
$$f(x) = x^2 - a$$
This is because the roots would be:
$$f(x) = x^2 - a = 0 \implies x^2 = a \implies x = \pm ~ \sqrt{a}$$
Apply Newton's iteration:
$$x_{n+1} = x_n - \dfrac{f(x)}{f'(x)} = x_n - \dfrac{x^2-a}{2x}$$
Select an $x_0$ and iterate away.
You can find a worked example here.
