## Newton's Method for Finding Roots

### The Algorithm for Newton's Method

Rather than deal with each particular function and starting point, let's find a pattern for a general function . For the starting point , the slope of the tangent line at the point is so the equation of the tangent line is . This line intersects the -axis when , so

is . This line intersects the -axis when , so and . Starting with and repeating this process we have ; starting with , we get and so on.

In general, if we start with , the line tangent to the graph of at the point intersects the -axis at the point , our new estimate for the root of .

##### Algorithm for Newton's Method:

(3) Repeat step (2) until the estimates are "close enough" to a root or until the method "fails".

When the algorithm for Newton's method is used with , the function at the beginning of this section, we have so

The new approximation, , is the average of the previous approximation, , and divided by the previous approximation, . Problem 16 asks you to show that this pattern, called Heron's method, approximates the square root of any positive number. Just replace the "" with the number whose square root you want.

**Example 1: **Use Newton's method to approximate the root(s) of . Solution: so

The graph of for (Fig. 8) indicates only one root of , and that root is near so pick Then Newton's method yields the values (the underlined digits agree with the exact root).

If we had picked , Newton's method would have required iterations to get digits of accuracy. If , then iterations are needed to get digits of accuracy. If we pick , then the values of are not close to the actual root after even iterations, . Picking a good value for can result in values of which get close to the root quickly. Picking a poor value for can result in values which take longer to get close to the root or which don't approach the root at all.

**Note: An examination of the graph of the function can help you pick a "good" .**

**Practice 3: **Put and use Newton's method to find the first two iterates, and , for the function

**Example 2: **The function in Fig. 9 has roots at and . If we pick and apply Newton's method, which root do the iterates, the , approach?

Solution: The iterates of are labeled in Fig. 10. They are approaching the root at .

**Practice 4: **For the function in Fig. 11, which root do the iterates of Newton's method approach if (a)