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_C09.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_C09.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 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 ? Use what you know about convergence of geometric series to confirm your guess for the interval of convergence. - You read in the begining of the background that a Taylor Polynomial must have the same y-value, at the base point, as the original function. Also, the derivatives at that point must be the same. Three functions and three second-order polynomials are given; determine which function goes with which Taylor Polynomial by following the steps below.

- a)
- Enter the functions and the polynomials.
- b)
- Find the y-values of all six at .
- c)
- Find the first derivative at 0 of all six.
- d)
- Find the second derivative at 0 of all six.
- e)
- Which Taylor Polynomials go with which functions.

2009-02-04