The purpose of this lab is to use Maple to introduce you to 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

which you should recognize as a power series that has been truncated.

over the interval . The

- For the following functions and base points, determine what
minimum order is required so that the Taylor polynomial approximates the
function to within a tolerance of over the given
interval.
- , base point , interval .
- , base point , interval .
- , base point , interval .

- For the function,
, use the
`TayPlot`command to plot the function and a Taylor polynomial approximation of order 5 with base point on the same graph over the interval . If you increase the order of the Taylor polynomial, can you get a good approximation at ? Can you make a good guess at the radius of convergence of the Taylor series for ?A theorem from complex variables says that the radius of convergence of the Taylor series of a function like is the distance between the base point ( in this case) and the nearest singularity of the function. By singularity, what is meant is a value of where the function is undefined. Where is undefined? Is the distance between this point and the base point consistent with your guess of the radius of convergence from the plot?

- 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.)

2004-04-09