Subsections

• Purpose
• Getting Started
• Background
• Accuracy and Tolerance
• Exercises

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_B09.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_B09.mws, and go through it carefully. Then you can start working on the exercises.

Background

The idea of the Taylor polynomial approximation of order $n$ at $x=a$, written $P_n(x,a)$, to a smooth function $f(x)$ is to require that $f(x)$ and $P_n(x,a)$ have the same value at $x=a$. Furthermore, their derivatives at $x=a$ must match up to order $n$. For example the Taylor polynomial of order three for $\sin(x)$ at $x=0$ would have to satisfy the conditions

$$\begin{array}{ccccc} P_3(0,0) & = & \sin(0) & = & 0\\ P_3'(0... ...0) & = & 0 \\ P_3'''(0,0) & = & -\cos(0) & = & -1 \end{array}$$

You should check for yourself that the cubic polynomial satisfying these four conditions is

$$P_3(x,0) = x - \frac{1}{6} x^3.$$

The general form of the Taylor polynomial approximation of order $n$ to $f(x)$ is given by the following

Theorem 1   Suppose that $f(x)$ is a smooth function in some open interval containing $x=a$. Then the $n$th degree Taylor polynomial of the function $f(x)$ at the point $x=a$ is given by

$$P_n(x,a) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^k$$

$$= f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!} (x-a)^n$$

We will be seeing this formula a lot, so it would be good for you to memorize it now! The notation $f^{(k)}(a)$ is used in the definition to stand for the value of the $k$-th derivative of $f$ at $x=a$. That is, $f^{(1)}(a) = f'(a)$, $f^{(3)}(a) = f'''(a)$, and so on. By convention, $f^{(0)}(a) = f(a)$. Note that $a$ is fixed and so the derivatives $f^{(k)}(a)$ are just numbers. That is, a Taylor polynomial has the form

$$\sum_{k=0}^{n} a_k (x-a)^k$$

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 $[c,d]$, we define the tolerance $Tol$ of $P_n(x,a)$ to be the maximum of the absolute error

$$\mid f(x)- P_n(x,a) \mid$$

over the interval $[c,d]$. 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 $0.1$ over the given interval.
   1. $f(x) = \exp(x)$, base point $a=0$, interval $[-2,2]$.
   2. $f(x) = \sin(x)$, base point $a=0$, interval $[0,2\pi ]$.
   3. $$f(x)=\frac{1}{x^2-1}$$, base point $a=0$, interval $[-0.95,0.95]$.

2. For the function, $f(x) = x/(1-2x)$, use the TayPlot command to plot the function and a Taylor polynomial approximation of order 5 with base point $a=0$ on the same graph over the interval $-1 \leq x \leq 1$. If you increase the order of the Taylor polynomial, can you get a good approximation at $x=-1$? Can you make a good guess at the interval of convergence of the Taylor series for $f$?

3. 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.
   
   $$f(x)=e^{(x^5-x)}$$
   
   
   
   
   $$g(x)=7x+e^x+x^6$$
   
   
   
   
   $$h(x)=e^{x^5}-e^x+1$$
   
   
   
   
   $$Taylor1=\frac{x^2}{2}+8x+1$$
   
   
   
   
   $$Taylor2=\frac{x^2}{2}-x+1$$
   
   
   
   
   $$Taylor3=\frac{-x^2}{2}-x+1$$
   
   

   a)

   Enter the functions and the polynomials.

   b)

   Find the y-values of all six at $x=0$.

   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.