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.

## Getting Started

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 /math/calclab/MA1023/Taylor_start.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.mws, and go through it carefully. Then you can start working on the exercises.

## 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 . The Getting started worksheet has examples of how to compute and plot the absolute error.

## 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

2. For the third function in exercise 1, , 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 ? By increasing the order of the Taylor polynomial in your plot, can you make a good guess at the radius of convergence of the Taylor series for ?

A theorem from complex analysis 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 unbounded? Is the distance between this point and the base point consistent with your guess of the radius of convergence from the plot?

3. For the fourth function in exercise 1, , plot the graph of this function along with its Taylor polynomial approximation of order 4 with base point over the interval and . Can you increase the order so that the Taylor polynomial is a good approximation to at ? How about at ? can you make a good guess at the interval of convergence of the Taylor series for ?

4. Can you investigate exercise 1 a bit further to decide whether or not the end points are included in the interval of convergence for exercise 2 and 3?