Example 1.  Consider the function  .  
1 (b).  Investigate the error term for the Maclaurin polynomial of degree n = 10 over the interval  [-0.5, 0.5].  
Solution 1 (b).

Background. Lagrange form of the Remainder.  The Lagrange form of the error is    where  c  is known to exist and lies somewhere between  0  and  x.

First we need to bound the size of the term    for values of  c  in the interval  .  This can easily be done graphically, but to do it analytically with derivatives is quite messy.  We choose to look at the following graph to see what is happening.

How big does    get ?  Looking at the graph we can estimate it to be  .  

That's good enough.

How big does the error    get ?  Notice that .  
We will use the bound the first portion  and  then bound the portion    over the interval  [-0.5, 0.5]  by evaluating it at    

Now multiply the two numbers together to find the error bound for Lagrange's remainder formula.

      

This is a little larger than the actual maximum error we found. Remember is an "error bound."

(c) John H. Mathews 2004