During our warm-up activity today, we looked at a function and identified critical points, relative, and absolute extrema for this function:

It was kind of neat talking about this and the extreme value theorem from last time. Since the domain is not defined over a closed interval, the EVT doesn't guarantee the existence of an absolute maximum or minimum value. The students seemed to really get the idea this year that this function specifically has no absolute maximum over the domain because it is an open interval - last year there was a lot of confused faces on this idea. There were a couple really insightful comments about whether there would be an open interval domain over which the function **did** have an absolute maximum, even though the hypothesis wasn't satisfied. The theorem just tells you whether or not you are guaranteed to find one, not that there isn't one at all. Really good stuff, and I'm proud of the way everyone was chiming in to talk about what they understood.

The most important thing was that this led perfectly into introducing the idea of an existence theorem. This idea is different from other theorems (especially in comparison to geometry) that students have learned because the information it gives you is not as specific as "alternate interior angles are congruent" or "the remainder of polynomial P(x) upon division by (x - c) is P(c)". All it does is tells you whether you __can__ find what the theorem says is there. I didn't plan on having this discussion today, but it was perfect for then introducing the mean value theorem, and I will definitely repeat it in the future.

I then gave my students this geogebra applet to play with today.

Download link here.

The students understood pretty quickly what they had to do, and didn't seem to have a hard time. It was kind of interesting to watch them rediscover the concept of forming a tangent line using two points, as that concept has been a bit overshadowed by other things as we looked at derivative rules before the test they took last week. Some students moved P and Q so that they were tangent, and then adjusted the domain using C and D to find a domain over which the tangent line and line AB were parallel.

From this, I showed them what the slope of line AB represented (average rate of change over the interval) and came up with the right side of the MVT. We then talked about what the slope of the tangent line they identified represented - a couple immediately referenced the derivative of the function. What is the relationship between parallel lines? What would make it so that you couldn't find this value? Ideas of continuity and differentiability jumped out. There it was: the entire mean value theorem.

Last year I presented the students with the MVT, and then we drew graphs to represent what it was saying. They kind of got it, but it wasn't a sticky idea. I was doing all the developing. This approach today started with something visual that __they__ were doing, that they could understand intuitively, and then that intuition was applied to develop an abstract concept out of that understanding.

I continued doing what I had done last year - answering some multiple choice questions about the MVT (See here for today's handout) analytically, and I immediately lost a couple students. So I showed them how to throw the new function into Geogebra and adjust the domain to match the problem. They could then solve the problems graphically - they immediately located the points to be able to answer the questions.

The group is a mix of AP and non-AP exam bound students. I __will__ introduce them all to the analytic ways of identifying these points, and we did some of it today. It was really nice that the moment things got a bit too abstract, I could push students to identify how the question being asked was the same as the idea of the MVT, and they were then able to solve it.

Without the technology, these students would have been done for the rest of the period. Those that could handle the algebra, would. Those that couldn't would spend the rest of the period feeling like they were in over their heads. Introducing how to use the technology to really understand what was being said by the abstract theorem enabled many more students to get in on the game. That made me feel all warm and fuzzy inside. The rest of the class focused on definitions of increasing functions using the derivative, something that was made incredibly easy by referring back to the activity at the beginning of the period.

We'll see how well they remember the ideas moving forward, but it felt great knowing that, at least for today's lesson, everyone in the room had a way into the game.

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