The purpose of this lab is to use Maple to introduce you to higher order Taylor polynomial approximations to functions.

In the previous lab, we introduced quadratic Taylor polynomial approximations. In this lab, we investigate higher-order Taylor polynomials.

The idea of the *Taylor polynomial approximation of order n at
x=a*, written , to a smooth function is to require
that and have the same value at

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

The general form of the Taylor polynomial approximation of order **n**
to is given by the following

We will be seeing this formula a lot, so it
would be good for you to start memorizing it now! The notation
is used in the definition to stand for the value of the
**k**-th derivative of **f** at **x=a**. That is, ,
, and so on. By convention, . Note that **a** is fixed and so the derivatives are
just numbers. The following easier theorem should help you to see
where the formula comes from.

Maple has a command called `taylor` to generate these Taylor
polynomial expansions, but the form it produces is not the most
convenient, so two commands have been written as part of the `
CalcP` package, which should be loaded with the following command.

> with(CalcP):

The two procedures are called `Taylor` and `TayPlot`. The
syntax for `Taylor` is

`Taylor(`*f,x=a, n*`);`,

where **n** is the order, **f** is an expression or a procedure, and **a**
is the base
point. The following examples should make the use of this procedure
clear. There is also help available with the command `?Taylor`.

> Taylor(sin(x),x=0,3);

> Taylor(sin(x),x=0,15);

> Taylor(sin(x),x=Pi/6,4);

> Taylor(exp(x),x=0,5);

The result of this command is a polynomial expression that can be plotted, differentiated, etc.

It seems intuitive that the larger **n** is, the better the Taylor
polynomial will approximate . To help you investigate this, a
procedure `TayPlot` has been written which plots and a set of
Taylor polynomials simultaneously. The syntax for this command is

`TayPlot(`*f,x=a,{n1,n2,n3, ...},x=b..d,ops*`);`,

where **f** and **x=a** are as above, *x=b..d* is the usual **x** plot
range specifier, and *ops* are (optional) options that `TayPlot`
passes to the `plot` command. The set *{n1,n2,n3, ...}*
consists of integers corresponding to the Taylor polynomial
degrees desired. For example,

> TayPlot(sin(x),x=0,2,3,5,x=-Pi..Pi);

> TayPlot(sin(x),x=0,2,3,5,x=-Pi..Pi,y=-1.2..1.2);

are both valid calls of `TayPlot`. Both plot and the
2nd, 3rd, and 5th order Taylor polynomial approximations. In the
second `TayPlot` command, the **y** range has been set to fit the
behavior of the function. You can plot
more than three Taylor polynomials if you want, of course. You can
also use a letter other than **x** for your independent variable. Help
for `TayPlot` is available with the `?TayPlot` command.

Use `Taylor` to compute the following Taylor polynomials. Use `
TayPlot` to help you visualize your results.

- ,
**a=0**, order 1, order 2, order 3. - ,
**a=0**, , order 4, order 5, order 6. - ,
**a=0**, order 8, order 13, order 21.

To measure how well a Taylor Polynomial approximates the function over
a specified interval , we define the tolerance **Tol** of
to be the maximum of

over the interval . You can actually see how the error of a Taylor polynomial varies over an interval with a Maple command like

> plot(abs(sin(x)-Taylor(sin(x),x=0,3)),x=-1..1);

For this example, the tolerance, **Tol**, is about , which you can
find out by looking at the graph.

Now suppose you were asked to determine the order required so that the Taylor polynomial approximation to had a tolerance of on the interval . One simple method for doing this graphically is shown below.

> plot(abs(sin(x)-Taylor(sin(x),x=0,3)),x=-1..1,y=0..0.005);

If you look at the plot, you see that the curve goes out of the plot
on the top of the window. This means that the tolerance is not
satisfied. If the order is increased to **5**, as in the following
example, then the curve goes out of the plot on the sides, meaning
that the tolerance is satisfied.

> plot(abs(sin(x)-Taylor(sin(x),x=0,5)),x=-1..1,y=0..0.005);

- For the following functions and base points, determine what
order is required so that the Taylor polynomial approximates the
function to within a tolerance of over the given
interval. Include explnations how you got your answers in your report.
- ,
**a=0**, interval . - ,
**a=0**, interval . - ,
**a=0**, interval . - ,
**a=1**, interval .

- ,
- Prove Theorem . Remember that an example is not a proof!
- For the third function in exercise 1, , consider the
Taylor polynomial about
**a=0**. Can you choose the order so that the Taylor polynomial is a good approximation (within , say) to at**x=2**? 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 two parts, one where the approximation is good and one where it is bad?

Tue Jun 27 15:37:50 EDT 1995