Evaluate approximately the following integral

and prove that your approximation is accurate to 8 decimal places using the following strategy. To begin, use Maple to find a Taylor series approximation to the function to some degree -- for example, start with a degree 5 approximation. Evaluate the error term for this approximation by recalling from calculus that the Taylor series error for the degree Taylor polynomial of a function g on the interval [a - r, a + r] is given by

where M is the maximum value of where t ranges over the interval [a - r, a + r]. Use Maple's graphical abilities to help you obtain an upper bound for the error term. How small does the error term have to be in order for the integral of the Taylor approximation to to agree with

to 8 decimal places? Explain carefully. You may have to choose a Taylor approximation of larger degree in order for the error term to be suffciently small for the integral to be accurate to 8 decimal places.

In doing this problem you should be familiar with the Maple commands taylor and convert( , polynom). As well, you should recall that in order to evaluate the order derivative of a function g it is possible to use the command D@@n(g). In some cases, such as calculating a large iterated derivative, you may find it convenient to suppress the output of the calculation because it scrolls on for several screens. Remember that this can be done by ending the command with a colon instead of a semicolon.

Juris Steprans
Thu Mar 27 17:10:57 EST 1997