Spreadsheet Calculus: Newton's Method

Sometimes you need to find the roots of a function, also known as the zeroes. Sometimes finding the zeroes is pretty easy. Other times, that isn't the case. Take for example the 6th degree polynomial shown below. How do you find the zeroes to that equation? You could make a rough estimate with the graph, or you could use Newton's method.

Step 1 How it works

Newton's method uses tangent lines to approximate the zeroes of a function. It's difficult to explain only in word how it works, take a look at the picture below. The graph is e^-x. We start at [5,148] and follow the line tangent to the curve. When the tangent line hits the x-axis (y=0) we do it again, at the x-value we found by following the tangent line. Eventually we would expect to reach a 0. In this case, we never would since e^-x never touches the x-axis. It illustrates the point though.
Convince yourself that f(a)=f'(a)(a-b). f'(a) is the derivative of the function at point a. It's the slope of the function at a. If you multiply this by a certain number (a-b), b is undefined, it will equal the function at that point. If we know the value of the function at point a, and the derivative of the function at point a, then we can rewrite the equation in terms of b (the undefined point on the x-axis.) Rewritten as such:
b=a-f(a)/f'(a). Using this equation, we can use any point on the x-axis, find the next point, then use that one, and repeat until we hit one of the zeroes.