Expii

# 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.