After just that I say, let’s start something new: I’m tired of only being able to compute areas of circles polygons (really only triangles and rectangles, about the limit of my memory). Let’s, given a function f, and bounds a and b, try to find the area ‘under’ f. Then talk about computing this via Riemann sums, but point out how difficult that is, and wish there was a better way. Then, I say, hey, just for grins, let’s let F(x)=int_a^x f(x) dx, and think about it’s derivative. Walking through the proof of FTC, we find F’=f, and I have them remind me that this is called antidifferentiation.

@Christine: I thought it went fabulously. You remind me that I should make a followup post. I’ll try to get to that this week.

Of course, encouraging your students to fall in love with engineering students and talk about math is not a real practical suggestion, so how did your introduction go over?

-Christine

So start with constant velocity, then linear velocity (and sometimes nice quick applications involving stopping distances for cars at various initial velocities).

At some point, the generalization to accumulating the rate-of-change of a quantity over time gets introduced, which then suggests that this Riemann sum process is an inverse of the finding a rate of change process that they’ve already mastered. Voila, a plausible handwaving of the Fundamental Theorem.

Generally, I like the ask questions first teach skills later approach.

How are you going to make the leap to the areas under the curve? Seems like they’ll be good at doing the formula stuff, do you just do last year’s lesson after the new strategy this year?

@Daniel: Ah, learning it first in Physics… you don’t really get exposed to the undergirdings of the concept. Blah. But thank you so much for your perspective. Last year I only spent a little time on Riemann Sums and how they relate to integration (because they’re so tedious), but I will take a goodly amt of time thanks to your suggestion.

@jd: After what I saw yesterday, I think I’m never going back.

@Andy: Interesting… you show them how to do position/vel/acc problems BEFORE showing them integration. I never thought of that, because I’m always like “displacement/position is the area under the curve!” I like the graphical relationship when showing that, but I’ll have to think this through.

Thanks everyone for your extensive and thoughtful comments. They are really helpful!