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);

Tue Feb 6 14:48:55 EST 1996