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.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 $n$ 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. $f(x) = \cos(2x)$, base point $a=0$, interval $[0,2\pi ]$.
   3. $f(x) = \ln (x+1)$, base point $a=0$, interval $[-0.9,0.9]$
   4. $f(x) = 1/x^2$, base point $a=2$, interval $[1,3]$.

2. For the function, $f(x) = x/(x+3)$, use the TayPlot command to plot the function and a Taylor polynomial approximation of order 5 with base point $x=0$ on the same graph over the interval $0 \leq x \leq 4$. If you increase the order of the Taylor polynomial, can you get a good approximation at $x=4$? 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 $f$?

   A theorem from complex analysis says that the radius of convergence of the Taylor series of a function like $f$ is the distance between the base point ($a=0$ in this case) and the nearest singularity of the function. By singularity, what is meant is a value of $x$ where the function is undefined. Where is $f$ 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 third function in exercise 1, $\ln (x+1)$, plot the graph of this function along with its Taylor polynomial approximation of order 4 with base point $a=0$ over the interval $-1 \leq x \leq 2$ and $-2 \leq y \leq 1$. Can you increase the order so that the Taylor polynomial is a good approximation to $f(x)$ at $x=2$? How about at x=0.5? Discuss the difference in the behavior of the Taylor polynomials at these two points. Can you divide the real line up into two parts, one where the approximation is good and one where it is bad?