## Learning Objectives

A student will be able to:

- Approximate a function by the method of linearization.
- Know Newton’s Method for approximating roots of a function.

## Linearization: The Tangent Line Approximation

If is a differentiable function at , then the tangent line, , to the curve at is a good approximation to the curve for values of near (Figure 8a). If you “zoom in” on the two graphs, and the tangent line, at the point of tangency, , or if you look at a table of values near the point of tangency, you will notice that the values are very close (Figure 8b).

Since the tangent line passes through point and the slope is , we can write the equation of the tangent line, in point-slope form, as

Solving for ,

**Figure 8a**

**Figure 8b**

So for values of close to , the values of of this tangent line will closely approximate . This gives the approximation

**The Tangent Line Approximation (Linearization)**

If is a differentiable function at , then the approximation function

is a linearization of near .

**Example 1:**

Find the linearization of at point .

*Solution:*

Taking the derivative of ,

we have and

This tells us that near the point , the function approximates the line . As we move away from , we lose accuracy (Figure 9).

**Figure 9**

**Example 2:**

Find the linearization of at .

*Solution:*

Since , and we have

**Figure 10**

## Newton’s Method

When faced with a mathematical problem that cannot be solved with simple algebraic means, such as finding the roots of the polynomial calculus sometimes provides a way of finding the approximate solutions.

Let's say we are interested in computing without using a calculator or a table. To do so, think about this problem in a different way. Assume that we are interested in solving the quadratic equation

which leads to the roots .

The idea here is to find the linearization of the above function, which is a straight-line equation, and then solve the linear equation for .

Since

or

We choose the linear approximation of to be near . Since and thus and Using the linear approximation formula,

Notice that this equation is much easier to solve than Setting and solving for , we obtain,

If you use a calculator, you will get As you can see, this is a fairly good approximation. To be sure, calculate the *percent difference* between the actual value and the approximate value,

where is the accepted value and is the calculated value.

which is less than .

We can actually make our approximation even better by repeating what we have just done not for , but for , a number that is even closer to the actual value of . Using the linear approximation again,

Solving for by setting , we obtain

which is even a better approximation than . We could continue this process generating a better approximation to . This is the basic idea of *Newton’s Method*.

Here is a summary of Newton’s method.

**Newton’s Method**

- Guess the first approximation to a solution of the equation . A graph would be very helpful in finding the first approximation (see figure below).
- Use the first approximation to find the second, the second to find the third and so on by using the recursion relation

**Example 3:**

Use Newton’s method to find the roots of the polynomial

*Solution:*

Using the recursion relation,

To help us find the first approximation, we make a graph of . As Figure 11 suggests, set . Then using the recursion relation, we can generate , .

**Figure 11**

Using the recursion relation again to find , we get

We conclude that the solution to the equation is about .

## Multimedia Links

For a video presentation of Newton's method **(10.0)**, see Math Video Tutorials by James Sousa, Newton's method (9:48).

## Review Questions

- Find the linearization of at .
- Find the linearization of at .
- Use the linearization method to show that when (much less than ), then . Hint: Let
- Use the result of problem #3, , to find the approximation for the following:
- Without using a calculator, approximate .

- Use Newton’s Method to find the roots of .
- Use Newton’s Method to find the roots of .

## Texas Instruments Resources

*In the CK-12 Texas Instruments Calculus FlexBook® resource, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/book/CK-12-Texas-Instruments-Calculus-Student-Edition/section/3.0/.*

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