Newton's method: finding roots with tangents

Home in on a root by following tangent lines, and watch the number of correct digits double. About 8 minutes, University.

What you will do

1. A root is where f(x) = 0. Most equations cannot be solved exactly. Newton's method finds an approximate root by following tangent lines down to the axis.
2. One tangent at a time. From a guess x_n the tangent crosses the axis at x_n+1 = x_n - f(x_n)f'(x_n).
3. Fast convergence. The number of correct digits roughly doubles each step. That is why it is so popular.
4. When it fails. If the slope is 0 the tangent never reaches the axis. The engine reports that instead of dividing by zero.