We want to find the roots of the equation f(x) = 0 using a ritual called Newton's Method.
To do this we start ...

>Why?
Huh?
>Why would we want to find roots?
For some examples, check here or here or ...
>Okay, so what's next?

First we guess at a root, say x0. That gives us a point on the curve y = f(x), namely:   ( x0,f(x0) ). Then we move along the tangent to the curve at that point ... until we reach the x-axis at, say x1. The relationship between x0 and x1 is:   x1 = x0 - f(x0) / f '(x0) You can see that by staring intently at the figure:   As you can see, x1 is closer to the root than our guess of x0. Neat, eh? >Is that always the case? Uh ... no, not always. You gotta make a reasonable guess. Anyway, we now assume that x1 is our next guess and repeat ... moving along the tangent to the next guess, x2 = x1 - f(x1) / f '(x1) Then repeat the same ritual to get x3 = x2 - f(x2) / f '(x2). Then repeat to get ... >Yeah, I get it. And if you're lucky you get closer and closer to the actual root, eh? Yes, like so: click for an example where we're finding a root of f(x) = sin(x) = 0 starting with a guess: x0 = 2. The various Newton iterates, them's x1, x2, x3 etc., converge to π = 3.14159...

>I assume there's a spreadsheet.
Yes. Just click on the picture to download the spreadsheet::

After downloading, try a lousy initial guess. That's great fun!