## 2.13 Newton's Method

### Summary

Tangent lines are used to find the root of an equation f(x) = 0, where f is a differentiable function. The procedure, called Newton's Method, defines a sequence of points on the x-axis that converge to the root. Although the method can fail, it works in a surprisingly large number of cases.

By the end of your studying, you should know:

• The geometric interpretation of Newton's Method.
• The formulas that define Newton's Method.
• How to apply Newton's Method.

On-screen applet instructions: Type a starting value followed by the Enter key. Try different starting values and observe the effect. Can you explain what happens when the starting value is 1?

### Examples

Consider the equation

and say you intend to solve it using Newton's method. If the initial value chosen is not 0, what will happen when you apply Newton's method?

Using Newton's Method, find the equation of the line of maximum slope tangent to the curve y = sin(x) that passes through the origin. Give the answer to 6 decimal places.

Use Newton's Method to approximate to 6 decimal places the solution to the equation

Newton's Method

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 2.13 Newton's Method (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

#### Interesting Application

Although we are studying real-valued functions of a real variable, Newton's Method can be applied to complex numbers as well. The results yield some interesting pictures that show so-called Newton Basins, the set of iterative points leading to the solution.

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel