I'm glad to see the current discussion(s) taking place. I don't contribute much but I read and enjoy it all. I think this list does serve many purposes.
Most contributors seem to be interested in the "nuts and bolts" of calculus so I'd like to pass on an experience I had yesterday which might fit that category. Brian Felkel and I are just finishing up a new on-line business text with live links to lots of stuff. We use Excel and Maple (any cas will do) in an integrated every-day fashion. I called our assistant Dean in the College of Business yesterday and presented the following (His name is Tim Burwell and he has a B.S. in math, Ph. D. in economics and much experience in business so I like to get his advice).
I explained to Tim that we have just finished teaching the part where integral calculus is explained and used, and, in Monday's class I had posed the question of how wide one would need to make airline seats to accommodate 99% of the people getting on the plane. After a short discussion, they had worked the problem using Excel and Maple - time elapsed: about 15 minutes. Brian and I wanted to know how much integration by hand we should cover (its getting near the end of our term). His response was as follows:
"Since you have covered the concepts needed, a short discussion from a historical perspective might be OK. No more than how one gets the anti derivative of x^n, however".
I was a little surprised and pursued it with him. It appears he has in mind about 15 minutes discussion. His point was: if they have the concepts we use all the time and ever need to do any integration of formulas by hand in a graduate course, it is more time and cost effective to teach them at that point.
Having taught calculus and business classes for many years, I guess I should not have been surprised, but I was. Twenty years ago (10?) this would have been heresy and might still be at some schools.