Example 3.  Plot the absolute error    over the interval  , and estimate the maximum absolute error over the interval.
Compute the error bound    and observe that    over  .  

Solution 3.

Aside.  If you really looked at that error bound closely, then you discover a curious fact.

How can the error bound be smaller than the error at x=0 ?  Because we really need to use the interval   when we found  .  Being extra careful we find  , then we would have the error bound

Now we have "fixed it."

(c) John H. Mathews 2004