The purpose of this lab is to use Maple to give you some experience with Taylor polynomial approximations to functions, including some applications.

cp ~bfarr/Taylor_start.mws ~

You can copy the worksheet now, but you should read through the lab
before you load it into Maple. Once you have read to the exercises,
start up Maple, load
the worksheet `Taylor_start.mws`, and go through it
carefully. Then you can start working on the exercises.

You should check for yourself that the cubic polynomial satisfying these four conditions is

The general form of the Taylor polynomial approximation of order to is given by the following

satisfies

over the interval . The

- For the following functions and base points, determine what
order is required so that the Taylor polynomial approximates the
function to within a tolerance of over the given
interval. Include explanations how you got your answers in your report.
- , base point , interval .
- , base point , interval .
- , base point , interval .
- , base point , interval .

- Prove Theorem 2. Remember that an example is
not a proof! (Note - this should be done by hand. part of using Maple
is knowing when not to use it.)
- For the third function in exercise 1, , consider the
Taylor polynomial with base point . Can you choose the order so that the
Taylor polynomial is a good approximation (within , say) to
at ? How about at ? Discuss the difference in
the behavior of the Taylor polynomials at these two points. Can you
divide the real line up into regions, one where the approximation is
good and one where it is bad?
- You know that the derivative of is . Can you
find a relationship between the derivative of the order
Taylor polynomial with base point for and and a Taylor
polynomial of some order with base point for ?
- You know that the indefinite integral of is
. Can you find a relationship between the indefinite
integral of the order
Taylor polynomial with base point for and and a Taylor
polynomial of some order with base point for ? (Hint
- your answer should be yes.)
- One of the applications of Taylor polynomials is in
approximating integrals that cannot be done analytically. One such
integral appears frequently enough in applications that it has been
given a name. It is the error
function,
,
which is defined by

The idea of using Taylor polynomials to approximate the integral is to replace the with its Taylor polynomial, which can be easily integrated. Can you find the minimum order required to approximate to within an accuracy of ? Use zero for the base point of the Taylor polynomials. (Hint - Maple has a command`erf`for the error function.)

2000-09-22