Key Concepts

• Newton’s method approximates roots of [latex]f(x)=0[/latex] by starting with an initial approximation [latex]x_0[/latex], then uses tangent lines to the graph of [latex]f[/latex] to create a sequence of approximations [latex]x_1,x_2,x_3, \cdots[/latex].
• Typically, Newton’s method is an efficient method for finding a particular root. In certain cases, Newton’s method fails to work because the list of numbers [latex]x_0,x_1,x_2, \cdots[/latex] does not approach a finite value or it approaches a value other than the root sought.
• Any process in which a list of numbers [latex]x_0,x_1,x_2, \cdots[/latex] is generated by defining an initial number [latex]x_0[/latex] and defining the subsequent numbers by the equation [latex]x_n=F(x_{n-1})[/latex] for some function [latex]F[/latex] is an iterative process. Newton’s method is an example of an iterative process, where the function [latex]F(x)=x-\left[\frac{f(x)}{f^{\prime}(x)}\right][/latex] for a given function [latex]f[/latex].