Subsections

# Taylor Polynomials

## Purpose

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

## Background

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.

## Accuracy and Tolerance

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 .

## Maple Commands

To use the Taylor and TayPlot commands you need to load the CalcP7 package.
>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.

## Exercises

1. 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.
1. , base point , interval .
2. , base point , interval .
3. , base point , interval .
4. ,base point , interval . (Hint: You may want to use axes=none at the end of this plot command to see what is happening in the plot.)
2. For the function, ,first plot the function using the regular plot command then 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 .
1. If you increase the order of the Taylor polynomial, can you get a good approximation at ?
2. Can you make a good guess at the radius of convergence of the Taylor series for ?
3. Repeat all of exercise 2 using and the same ranges.

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?