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Subsections
The purpose of this lab is to use Maple to introduce you to
Taylor polynomial approximations to functions, including some
applications.
To assist you, there is a worksheet associated with this lab that
contains examples and even solutions to some of the exercises. You can
copy that worksheet to your home directory with the following command,
which must be run in a terminal window, not in Maple.
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.
The idea of the Taylor polynomial approximation of order at
, written , to a smooth function is to require
that and have the same value at .
Furthermore, their derivatives at must match up to order
. For example the Taylor polynomial of order three for at
would have to satisfy the conditions
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
Theorem 1
Suppose that
is a smooth function in some open interval
containing
. Then the
th degree Taylor polynomial of the
function
at the point
is given by
We will be seeing this formula a lot, so it
would be good for you to memorize it now! The notation
is used in the definition to stand for the value of the
th derivative of at . That is,
,
, and so on. By convention,
. Note that is fixed and so the derivatives are
just numbers. That is, a Taylor polynomial has the form
which you should recognize as a power series that has been truncated.
To measure how well a Taylor Polynomial approximates the function over
a specified interval , we define the tolerance of
to be the maximum of the absolute error
over the interval . The Getting started worksheet has
examples of how to compute and plot the absolute error.
 For the following functions and base points, determine what
minimum order is required so that the Taylor polynomial approximates the
function to within the specified tolerance over the given
interval.

, base point , interval ,
tolerance of .

, base point , interval ,
tolerance of .

, base point , interval ,
tolerance of .

, base point , interval ,
tolerance of .
 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 again 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?
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William W. Farr
20030404