The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.

When to Use Newton's Method

We wish to approximate a zero of f(x), where f'(x) is continuous and not close to zero, as in Figure 5.9.1.

Step 1

Sketch the graph of f(x), and choose a point x1 near the zero of f(x). xl is the first approximation.

Step 2

Compute f'(x).

Step 3

Compute the second approximation


Step 4

For a closer approximation repeat Step 3. The (n + 1)st approximation is given by


As a rough check on the accuracy, compute f(xn) and note how close it is to zero.

Steps 3 and 4 can be done conveniently on a hand calculator.

Warning: Since Newton's method involves division by f'(x1) avoid starting at a point where the slope is near zero. Figure 5.9.3 shows that when the slope is close to zero, the tangent line is nearly horizontal and the approximation may be poor.


Figure 5.9.3

Example 1: Approximating a Zero
Example 2: Approximating a Fifth Root
Example 3: Approximating an Intersection of Two Graphs

Last Update: 2006-11-25