Calculus I—Day 3

An evening class just before a day off, and I had almost full attendance. We had our first quiz, which went off without too much grumbling, and I managed to present a couple of easy proofs. Aside from running out of time, my overall impression is that the class went quite well.

What I Taught

I began the class by working through some homework problems (as requested by the students) and another example involving the Squeeze Theorem. This time, I had the board space cleaned up properly, and was able to push the arguments through in a more linear, easy to follow manner. I don’t know if it actually cleared anything up for anyone, but a few of the students who were most vocally confused on day 2 seemed to follow along better this time around.

The rest of the lecture (minus 15 minutes at the end for a quiz) was devoted to the concept of continuity. I started with the abstract definition: a function \(f\) is continuous at a point \(a\) if (1) \(a\in\mathcal{D}(f)\), (2) \(\lim_{x\to a} f(x)\), and (3) \(\lim_{x\to a} f(x) = f(a)\). In the text, continuity is defined only by the third property, with the first two being given as an immediate consequence of the definition. I prefer this statement, as it highlights the kind of things that can go “wrong.”

I gave examples of the definition largely in terms of graphs of functions that fail to be continuous at certain points. I did this to show functions with specific types of discontinuities, i.e. removable, infinite, and jump discontinuities. In the case of jump discontinuities, I was also able to discuss the idea of one sided limits, and to introduce the greatest integer function (i.e. \(\lfloor x\rfloor\)), which is the archetypal step function, and which will come up again.

When I first brought up continuity, several students recalled the intuitive idea that a function is continuous if its graph can be drawn without lifting the pen. This notion seems inconsistent with the local definition given above, hence the idea of continuity on an interval, which was the next definition presented.

At this point, I had planned an example, but I was worried about time, so I skipped it and instead moved onto a theorem about continuity. I stated that continuous functions are well-behaved in that sums, products, and quotients of continuous functions are also continuous. I proved it for products and noted that the remaining conclusions follow from the limit laws in a similar manner.

I finished the class by stating (without proof) that most of the functions that we will be dealing with are continuous on their domains. These are things like polynomials, trigonometric functions, and logarithmic functions. I concluded the class by giving an example of how this statement could be combined with the theorem in the previous paragraph to compute a large number of limits, then passed out a quiz.

I had intended to get to two more theorems about the compositions of continuous functions and finish with the Intermediate Valued Theorem, but I ran out of time, so I extended the deadline on the homework, and promised to start with those after the holiday.

What Worked

While I ended up not coving everything that I had wanted to cover, I am glad that I took the time to go over an extra Squeeze Theorem example at the beginning of class. The students were still awake, and most of them were engaged, so it seems that it was a good use of time, even if it meant not getting to everything on schedule. This really isn’t that big a deal, anyway, as I have built a few review days into the calendar, which can just as easily be used to catch-up at the end of a section if need be.

Moreover, there was good back-and-forth in class. Nearly all of the my students have encountered the idea of a continuous function in a previous class (algebra or pre-calculus, most likely), but never have seen the more formal treatment. They asked good questions about how their intuitions matched the definitions, and seemed to understand how the formal definition (in terms of limits) matched up with their intuition (in terms of graphs of functions). Best of all, there was some good peer-to-peer interaction, which I think helped a great deal.

What Didn’t Work

I skipped an example about continuity on an interval because I was worried about time. I wish I hadn’t done that. On the plus side, I can always get to it in the next class.

More broadly speaking, we are only three days into class, and already the issue of time management seems to be a recurring theme. I really need to get over that. Yes, the summer session class goes by very quickly, and yes, that is stressful, but rushing doesn’t help. It will be better for the students if I take the time now to build up a foundation for the rest of the class.

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