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

\\filer\calclab

when you hit enter, you can then choose MA1023 and then choose the worksheet

Taylor_start_B11.mw

Remember to immediately save it in your own home directory. Once you've copied and saved the worksheet, read through the background on the internet and the background of the worksheet before starting 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 .

>with(CalcP7);

The exponential function can be approximated at a base point zero with a polynomial of order four using the following command.

>Taylor(exp(x),x=0,4);You might want to experiment with changing the order. To see and its fourth order polynomial use

>TayPlot(exp(x),x=0,{4},x=-4..4);This plots the exponantial and three approximating polynomials.

>TayPlot(exp(x),x=0,{2,3,4},x=-2..2);Notice that the further away from the base point, the further the polynomial diverges from the function. the amount the polynomial diverges i.e. its error, is simply the difference of the function and the polynomial.

>plot(abs(exp(x)-Taylor(exp(x),x=0,3)),x=-2..2);This plot shows that in the domain x from -2 to 2 the error around the base point is zero and the error is its greatest at x = 2 with a difference of over one. You can experiment with the polynomial orders to change the accuracy. If your work requires an error of no more than 0.2 within a given distance of the base point then you can plot your accuracy line y = 0.2 along with the difference of the function and the Taylor approximation polynomial.

>plot([0.2,abs(exp(x)-Taylor(exp(x),x=0,3))],x=-2..2,y=0..0.25);We knew this would have some of its error well above 0.2. Change the order from three to four. As you can see there are still some values in the domain close to x = 2 whose error is above 0.2. Now try an order of 5. Is the error entirely under 0.2 between x = -2 and x = 2? Larger orders will work as well but order five is the minimum order that will keep the error under 0.2 within the given domain.

- 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 multiple Taylor polynomial approximations of various orders with base point on the same graph over the interval ; use a y-range from to . If you increase the order of the Taylor polynomial, can you ever get a good approximation at ? Can you make a good guess at the radius of convergence of the Taylor series for ? - Consider the function
. Plot the graph of this function along with its Taylor polynomial approximation of order 4 with base point over the interval
. Limit the range of your plot from 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? If you increase the order of the Taylor polynomial, can you ever get a good approximation at ?

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?

2011-11-12