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 ~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.

## 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 and, furthermore, that 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. The following easier theorem should help you to see where the formula comes from.

Theorem 2   Suppose is a smooth function in some open interval containing and that is a positive integer. Then defined by

satisfies

## 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. Include explanations how you got your answers in your report.
1. , base point , interval .
2. , base point , interval .
3. , base point , interval .
4. , base point , interval .

2. For the third function in exercise 1, , consider the Taylor polynomial with base point . Can you choose the order so that the Taylor polynomial is a good approximation (within , say) to at ? How about at ? Discuss the difference in the behavior of the Taylor polynomials at these two points. Can you divide the real line up into parts, one where the approximation is good and the others where it is bad?

3. You know that the derivative of is . Can you find a relationship between the derivative of the order Taylor polynomial with base point for and and a Taylor polynomial of some order with base point for ?

4. You know that the indefinite integral of is . Can you find a relationship between the indefinite integral of the order Taylor polynomial with base point for and and a Taylor polynomial of some order with base point for ? (Hint - your answer should be yes.)

5. One of the applications of Taylor polynomials is in approximating integrals. In the first lab, we encountered the error function , which is defined by

The idea of using Taylor polynomials to approximate the integral is to replace the with its Taylor polynomial, which can be easily integrated. Can you find the minimum order required to approximate to within an accuracy of ? Use zero for the base point of the Taylor polynomials.