Newton's Method for Approximating Zeros - Expii

Newton's Method is a geometric method to approximate the zeroes of any function, by using derivatives. The process is relatively simple: Suppose we want to estimate a zero of \(f(x)\). First, choose any guess for the zero, and call it \(x_0\). Then, calculate \(x_1,x_2,x_3\), and so on using the iterative formula [xn=x{n-1}-\frac{f(x{n-1})}{f'(x{n-1})},] chosen so that \((x_n,0)\) is the intersection of the \(x\)-axis with the tangent line to the graph \(y = f(x)\) at the old \(x\)-value \(x = x_{n-1}\). In many but not all cases, each \(x_i\) gives a better estimate for the zero than the previous value \(x_{i-1}\), and so by repeating this process enough times we can arrive at a reasonably close approximation.