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Calculus I—Day 14
After three weeks of instruction, we finally got to our first applications (i.e. “story problems”): exponential growth and decay models. As the problems themselves can be solved in terms of algebra techniques applied to exponential functions, it seems odd to teach this in a calculus class. On the other hand, the framework for understanding what the model says comes from differential calculus, so perhaps now is an appropriate place in the curriculum to talk about such models.
Calculus I—Day 13
At this point in the term, we have developed essentially all of the major abstract and theoretical tools for differentiation, and have a few little loose ends to tie up before we start looking at applications of the derivative (e.g. applied mathematics problems, curve sketching, and so on).
Calculus I—Day 12
As with most people that have studied mathematics beyond high school, I first encountered implicit differentiation in my initial semester of calculus. It seemed like magic then. In the intervening years, I have seen the idea and the underlying theory pop up again a few times, and it still seems like some kind of magic. I have worked through the appropriate proofs, and am continually astonished that it works. I probably shouldn’t admit this in public, but even today I feel like I have no intuition regarding the proof. The whole argument is a bit of a black box.
On the bright side, I didn’t have to prove the theorem to my calc I students (and, frankly, the proof probably shouldn’t be done in public—it should only be performed behind closed doors by small groups of hooded monks as a kind of hazing ritual for young math majors). For them, it is sufficient to see the power of the result.
Calculus I—Day 11
At the beginning of the term I made the decision / was instructed not to teach the \(\varepsilon\)-\(\delta\) definition of a limit. This class really is directed at engineering and natural sciences majors, so that is, I think, a reasonable (if somewhat disappointing) strategy. Calculus is—in the eyes of the university administration—a cookbook class that teaches a bunch of recipes to non-math majors.
Because I didn’t give a precise definition of the limit, we reached the end of what can easily be proved, and have moved into the realm of learning recipes. Mind you, I’m not complaining—simply pointing out that the tone and content of the course has made a subtle shift that will continue for the next several weeks. We are now in plug-and-chug mode!
Calculus I—Day 10
The big event of the day was returning exams. I’m not sure why, but I always feel kind of uncomfortable when I hand back exams to students that performed poorly. I’m note sure what to do with that…
Anyway, the exam is over and done. On to the next thing!
Calculus I—Day 9
I gave an exam during the last class, so rather than the usual format of “What I Taught”/”What Worked”/”What Didn’t Work”, I am going to talk about the exam itself.
The exam consisted of 8 multiple choice questions, 8 true/false questions1The true/false questions require not just an answer of true or false, but also an explanation of the answer, both allowing for partial credit and a better demonstration of a student’s mastery of the theory in question, and 6 free response questions. The free response questions were all meant to be fairly straight-forward computations, the true/false questions sought to highlight theory, and the multiple choice questions had a mix of both.
Calculus I—Day 8
I got locked out of the classroom a few days ago (fortunately, there was one other person in the building at 5:30 on a Wednesday afternoon in the middle of July, and he happened to have a key which opened the door, so all was not lost), and because the university is run by a massive bureaucracy, it is going to be another week before I get a key. Thus, in order to ensure that I can get into the room, I have been wandering over to the classroom an hour or two early (while there are still people with keys hanging around) to make sure I can get in. Surprisingly, on the day before an exam, several students had the same idea. Heh.
Calculus I—Day 7
An exam is coming up soon, and the students are beginning to panic a bit about quizzes. I keep trying to tell them that the quizzes are a pretty low key affair (they don’t make up a huge portion of the final grade, and I tend to grade leniently and give lots of feedback), but I have enough type-A overachievers that it seems to be a constant concern. Don’t get me wrong—I prefer to have really motivated students, I just with that they were motivated by something other than grades.