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

cp /math/calclab/MA1023/Taylor_start.mws ~/My_Documents

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 .
- , base point , interval

- For the third function in exercise 1,
, 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 ? By increasing the order of the Taylor polynomial in your plot, can you make a good guess at the radius of convergence of the Taylor series for ?A theorem from complex analysis 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 unbounded? Is the distance between this point and the base point consistent with your guess of the radius of convergence from the plot?

- For the fourth function in exercise 1,
, plot the graph of this function along with its Taylor polynomial approximation of order 4 with base point over the interval
and
. Can you increase the order so that the Taylor polynomial is a good approximation to at ? How about at ? can you make a good guess at the interval of convergence of the Taylor series for ?
- Can you investigate exercise 1 a bit further to decide whether or not the end points are included in the interval of convergence for exercise 2 and 3?

2008-02-06