If you've studied calculus at all, you learned early on a sad but true fact of life: There are  many more calculus problems that can't be solved than there are problems that can be.  Getting a neat, closed-form solution to a tough problem is always satisfying, and makes a  mathematician or physicist feel like a real hero. But in that real world, no closed-form  solution ever seems to exist for the problem that's facing us at the moment. So when an  integral or derivative can't be found … when that elusive closed-form solution doesn't exist …  what do we do?  

Simple: let the computer do it. Computers may not be able to perform calculus directly, but  they're great at adding and multiplying. If we can reduce calculus to small steps involving  such simple operations, there's virtually no calculus problem the computer can't solve,  numerically. That's what this article is all about. Before we're finished, you should not only  know how to solve such problems, but also the foundations behind the math … not just the  “how,” but the “how come?” You'll see the general approaches and ideas that underlie all of  the numerical methods. In the process, rn also show you one of my favorite “speed secrets,”  a neat method for generating integration and interpolation formulae to any desired order.

Presented at Embedded Systems Conference 1993 in Santa Clara

Click below to download the paper:
ESC_1993_Vol1_Page69_Crenshaw_Calculus by Numbers.pdf