#### Polynomial
Approximations of Elementary Functions

** **

We begin this section by
deriving several polynomial approximations of the function near the point .
Such approximations are said to be **centered
at** .

As a first step, it seems
reasonable to require that both and .
Use these requirements to find a first-degree polynomial approximation (of the
form ) of at .

Now impose an additional
requirement that and find a second-degree polynomial
approximation (of the form ) of at .

Can you find a third-degree
approximation ?

#### Taylor and Maclaurin Polynomials

In the event that the
approximating polynomial is expanded about some number *c* that is different than zero, it is convenient to write the
polynomial in the form

Using this form makes it
easier to determine what the coefficients should be. We arrive at the following **definition**:

If *f* has *n* derivatives at *c*, then the polynomial

is called the *n*th
Taylor
polynomial for *f* at *c*. If ,
then

is also called the *n*th
Maclaurin polynomial for *f*.

As an **example**, find the Taylor
polynomials through for centered at .

Find the Maclaurin
polynomials and for .

Use a fourth Maclaurin
polynomial to approximate .

**Remainder of a Taylor Polynomial**

To measure the accuracy of
the approximations we have been discussing, we introduce the concept of a **remainder** as follows:

The maximum value of is called the **error** associated with the approximation.

**Taylor****’s Theorem.** If a function *f* is has derivatives in an interval *I* containing *c*, then for each *x* in *I* there exists *z* between *x* and *c* such that

,

where .

A useful consequence of this
is that .

Can you spot the
oh-so-obvious connection to a result from Calculus 1?

As an **example**, compute the third Maclaurin polynomial for .
Now use Taylor’s
Theorem to approximate by and determine the accuracy of the
approximation.

Determine the degree of the Taylor polynomial centered at that should be used to approximate so that the error is less than 0.001.