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

$$\mid f(x)- P_n(x,a) \mid$$

over the interval [c,d]. 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 0.008, 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 $\sin(x)$ had a tolerance of 0.005 on the interval [-1,1]. 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);