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 ?
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 nth
is also called the nth Maclaurin polynomial for f.
As an example, find the
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.
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 .
Determine the degree of the