# Taylor Polynomials and the Error

## Maple Usage for Taylor Polynomials

Maple has a command called **taylor** that will generate Taylor polynomials. However, the form of
its output is not convenient and so two commands which will better suit our purposes have been
placed in the **CalcP** package. These are **Taylor** and **TayPlot**. Remember, that in order to use
these two commands, you must first load the **CalcP** package as indicated below.
**> with(CalcP):**

If you are getting an error message with this command, you probably haven't copied a needed initialization file to your home
directory. Look here to see how to do that.
To generate the Taylor polynomial *P*_{3}(*x*; 0) for sin(*x*), the syntax would be the following

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

The Taylor polynomial *P*_{4}(*x*; /6) for sin(*x*) would be produced by this command.
**> Taylor(sin(x), x=Pi/6, 4);**

For values of x sufficiently close to the base point *x=a*, the larger *n* gets, the better the Taylor polynomial *P*_{n} approximates the values of *f*.
The **TayPlot** command will give you the graph of *f*(*x*) and the graphs of *P*_{n}(*x*; *c*) for a set of *n*
values you select. That way you can see if the graph of *P*_{n}(*x*; *c*) becomes more like the graph of *f*(*x*)
as larger values of *n* are used. To get a single plot which superimposes the graphs of *P*_{1}(*x*; 0), *P*_{3}(*x*; 0),
and *P*_{5}(*x*; 0) on the graph of sin(*x*) over the interval from - to , you would use the following syntax.

**> TayPlot(sin(x), x=0, {1,3,5}, x=-Pi..Pi);**

It is also possible to specify the *y* range for the plot as is done below.
**> TayPlot(sin(x), x=0, {1,3,5}, x=-Pi..Pi, y=-1.2..1.2);**

More than three Taylor polynomials may be plotted at a time. But, if too many are plotted, the
plot will be too full to be useful.
## Error

Let *R*_{n}(*x*; *c*) = *f*(*x*) - *P*_{n}(*x*; *c*).
It can be instructional to graph |*R*_{n}(*x*; *c*)| on the interval under consideration. Designate the
maximum of |*R*_{n}(*x*; *c*)| on the interval by *E*_{n}. The graphing can be accomplished in this way

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

From the graph it can be seen that *E*_{3} ~ 0.008.
Suppose you need to find *n* such that when *P*_{n}(*x*; 0) is used to approximate sin(*x*) you will have
*E*_{n} < 0.005 on the interval [-1, 1]. Guess at *n* and make a try.

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

In looking at this plot you can see that the curve goes out of the region at the top of the window.
This means that *E*_{3} > 0.005. So, experiment a little more; use a larger value of *n*. Try again, using
5 instead of 3. Now the curve goes out the sides of the window. This says that *E*_{5} < 0.005.