Subsections

• Purpose
• Getting Started
• Background
• Accuracy and Tolerance
• Maple Commands
• 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 similar to some of the exercises. You can access this worksheet by typing in the search field (magnifying glass next to start menu):

\\storage.wpi.edu\academics\math\calclab\MA1023\Taylor_start_A19.mw

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

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

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

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

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

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 $[c,d]$.

Maple Commands

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

> with(CalcP7):

We know that a linear approximation to a function at a point is the tangent line at that point. You can see that the first order Taylor polynomial is the tangent line. A second degree polynomial would be a better approximation as it not only has the same slope as the function, but the same concavity at the base point. The example below shows the exponential function $f(x) = \exp(x)$ approximated at a base point zero with a polynomial of order 1 and order 2 and plotted on the same graph as $f$ using the following commands:

> f:=x->exp(x)
> tp1 := Taylor(f(x),x=0,1)
> plot([f(x),tp1],x=-2..2)
> tp2 := Taylor(f(x),x=0,2)
> plot([f(x),tp2],x=-2..2)

The approximation is better as you increase the order of the Taylor polynomial. To see this effect, you might want to experiment with changing the order and observe the change in absolute error. Suppose you want to use the Taylor polynomial for $f(x) = \exp(x)$ with base point $x=0$ to approximate $f(1)$ with error no greater than 0.1. The commands below show how you can check the absolute error as you increase the order of the Taylor polynomial.

> tp := Taylor(f(x),x=0,1)
> evalf(abs(f(1)-subs(x=1,tp)))
> tp := Taylor(f(x),x=0,2)
> evalf(abs(f(1)-subs(x=1,tp)))
> tp := Taylor(f(x),x=0,5)
> evalf(abs(f(1)-subs(x=1,tp)))

The command below allows you to see the exponential function and three approximating polynomials on the same graph.

> TayPlot(f(x),x=0,{1,2,3},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. To see how well the Taylor polynomial approximates a function over an interval, we can plot the error.

> plot([abs(f(x)-Taylor(f(x),x=0,3)),0.1],x=-2..2,y=0..0.15)

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 = 0.1 along with the difference of the function and the Taylor approximation polynomial.

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 function $f(x)=\ln(x+1)$,

   a)

   Find the first order Taylor polynomial for $f(x)$ about the base point $x=0$ and plot it on the same graph as $f(x)$ over the interval $[-2,2]$. Then, determine the absolute error when approximating $f(1)$ using this first order Taylor polynomial.

   b)

   Find the second order Taylor polynomial for $f(x)$ about the base point $x=0$ and plot it on the same graph as $f(x)$ over the interval $[-2,2]$. Then, determine the absolute error when approximating $f(1)$ using this second order Taylor polynomial.

   c)

   Find the minimum order Taylor polynomial for $f(x)$, with base point $x=0$, needed to approximate $f(1)$ with absolute error no greater than 0.1.

   d)

   Find the minimum order Taylor polynomial for $f(x)$, with base point $x=0$, needed to approximate $f(-0.9)$ with absolute error no greater than 0.1.

   e)

   Find the minimum order Taylor polynomial for $f(x)$, with base point $x=0$, needed to approximate $f(-0.99)$ with absolute error no greater than 0.1.

2. For the function, $$f(x) = \frac{1}{5+2x}$$, use the TayPlot command to plot the function and multiple Taylor polynomial approximations of various orders with base point $x=0$ on the same graph over the interval $-5 \leq x \leq 5$; also use a y-range $-5 \leq y \leq 5$.

   a)

   If you increase the order of the Taylor polynomial, can you get a good approximation at $x=3$?

   b)

   Can you make a guess for the interval or radius of convergence. That is, an interval of $x$ values for which the Taylor polynomial converges to the function and outside that interval, does not converge to the function. Explain how this interval can be verified based on what you know about convergence of geometric series.