Newton's Method
(For finding roots of equations)
Consider the graph below of the function y = f(x).
Our goal is the find the root, r, of y = f(x). That is, the value where the graph crosses the x-axis, or where f(r) = 0. (See note below, about when this method fails.)
We start with a first guess, x1 (which we know is not the root, but which is a close guess) and construct the tangent to the graph of y = f(x) at the point (x1, f(x1)). We notice that if we find where this tangent line crosses the x-axis (at x2) , that it is a better approximation to the root than our initial guess, x1. If we continue with this process, x3 is a better guess than x2 and etc.
To generalize this process we proceed as follows.
To find x2 from x1 we first note that the equation of the tangent line f '(x1) is given by
y - f(x1) = f '(x1)(x - x1),
since the slope of the line tangent to the graph of y = f(x) at (x1, f(x1)) is given by f '(x1).
Now, since this tangent line crosses the x-axis at the point (x2, 0), the equation above becomes
0 - f(x1) = f '(x1)(x2 - x1).
If we solve this for x2 it becomes,
x2 = x1 - f(x1)/f '(x1).
If we generalize this process we come up with the Newton's method formula:
xn+1 = xn - f(xn)/f '(xn).
NOTE: Newton's method will not work unless the graph of y = f(x) has the same concavity on either side of the root, r.