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 explain this in terms of what you know about the convergence of power series? - Consider the function
. Plot the graph of
this function along with its Taylor polynomial approximation of order
with base point over the interval . Limit the
range of your plot to . By increasing the order of the Taylor
polynomial in your plot, can you make a good guess at the interval of
convergence of the Taylor series?

2002-11-15