Key Concepts

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