Calculus I—Day 17

We started a new chapter! Yay! I’m not sure that I really like including the two sections I taught last night on the exam that I have to give on Friday, but those were the orders from on-high, and I am a lowly adjunct, so I’m going with it for the time being (and there is something to be said for consistency from one class to another—I just wish it were consistent in a slightly different manner).

Anywho, Mean Value Theorem! Woo!

What I Taught

Last night’s lecture was the last of the really theory heavy lectures that I have planned for the term. In 95 minutes, I got from the definition of the derivative (plus the rules that we have developed for computing derivatives) to the Mean Value Theorem.

The path started by building some graphical intuition for defining minimums and maximums, with a distinction between local and global phenomena. I made the definition formal, then stated the Extreme Value Theorem (without proof). The Extreme Value Theorem segues nicely into Fermat’s Theorem, which I was able to state and prove. I noted that Fermat’s Theorem implies that points where the derivative of a function is zero are special, then provided a basic technique for finding global minimums and maximums of continuous functions on closed intervals (essentially applying Fermat’s Theorem plus some knowledge of continuity). We worked a couple of examples, then moved on.

I next presented Rolle’s Theorem. I did not provide a complete proof, but roughly sketched out an outline. I then stated the Mean Value Theorem, which says that if a function is continuous on a closed interval, then the derivative at some point is equal to the slope of the secant line through the endpoints. I tried to build some intuition. I did this by providing two examples: the first was the graph of some wavy function, on which I noted that there were points where the tangent line was parallel to the secant line. The second example involved average velocity: if I pass two speed traps 10 miles apart going the speed limit, but my average velocity over those 10 miles exceeds the speed limit, then there must be some time between the speed traps at which I was exceeding the speed limit.

With some intuition built, I was able to prove the Mean Value Theorem. I barely got done before class ened, and missed out on two quick (but interesting) corollaries. We didn’t talk about homework at all.

What Worked

I felt really good about the lecture. It was fun to teach, I managed to include some reasonable physicality into my lecture (i.e. walking back and forth across the room to demonstrate that my velocity has to be zero at one (or more) places—it makes me look kind of silly, but gets a response and increases engagement), and I managed to get to the main point without losing too many people.

I was also able to make the presentation of the theorems somewhat multimedia (in a good way!). There were lots of nice pictures on the board, a few graphs in Maple to give intuition, and a couple of calculator exercises for the students. Again, I believe that this enhanced engagement.

What Didn’t Work

Two problems leap to mind: first this was (at the end of the day) another talking head lecture (“Me talk, you listen!”), and second there were some computation issues (though I am going to try to get out of taking the blame for those).

The first problem is one that I think I am going to dismiss out of hand. Yes, it would be pedagogically better to give the students more of the cognitive load and make them contribute more to the process of learning. On the other hand, the material that I needed to cover in the time I was allowed to cover it was dense. Even acting as a talking head, I didn’t finish everything that I had wanted to finish. While it might be bad pedagogy in general, my students are (supposedly) adults, and should be capable of taking some responsibility for their own learning. If they took good notes and spend some time reviewing them (and/or the textbook), they will be fine.

So, is droning on about the Mean Value Theorem for 95 minutes great pedagogy? Not really. But in the current context, I think that it is acceptable.

The second problem is one that is becoming increasingly frustrating and I am still not sure how to handle it. We are to a point in the term when many of the problems that I present require a great deal of background knowledge: basic algebra skills; the ability to work with trigonometric, inverse trigonometric, exponential, and logarithmic functions; comfort with limits; and to ability to compute derivatives (as functions or at specific points).

I am willing to provide some hand holding, but at this point in the term, when I say “The derivative of a function \(f\) at a point \(c\) is given by \(
\lim_{h\to c} \frac{f(c+h)-f(c)}{h},
\)” I really don’t want to take the next five minutes reviewing the definition of the derivative, especially when the students asking the questions missed the two or three days of lecture in which the definitions were developed.

I am really at a loss for what to do. Should I just plan on more handholding, and prepare that ahead of time? Should I curtly refer students to their (or their colleague’s) notes? Should I throw a fit? I feel like a few students are slowing down (and frustrating) the entire class, and I don’t like it.

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