It was quite windy today. It normally takes me about 10 minutes to bike to class, door-to-door. Today, it took almost 20, thanks to a very stiff headwind (and heavier-than-usual traffic). Got to class just in the nick of time, all out of breath and such. Whee!
What I Taught
First, I went back and discussed the second derivative test. I stated it, justified it graphically, and tried to give the students some indication as to why one might prefer it over the first derivative test (yes, you have to take two derivatives, but you only have to evaluate those derivatives at a single point, rather than consider intervals). I think that they were basically with me.
Next, we finished the discussion of l’Hospital’s Rule. This involved lots of examples. Lots and lots of examples. I worked through some of them, and the students worked through a few more.
Finally, I concluded the mini-unit that we have been in all week by outlining what the text calls “Guidelines for Curve Sketching.” This involves a relatively long list of important points and features that need to be found in order to sketch a reasonable approximation of the graph of a function. I emphasized that the goal was to get the major things right (local minima and maxima, concavity, symmetry, asymptotes, and so on).
Before the end of class, I gave them a “Curve Sketching Exercise” (not a quiz—I told them on Tuesday that there would not be a quiz yesterday). In 20 minutes, working in groups, they managed to finish about half of the problem, so I told them to take it home and finish it overnight. I will give them some time to finish up today, then give a grade in the quiz category.
There were a lot of opportunities for students to work alone and in groups. As I circulated around the classroom, I saw a lot of right answers, and a lot of discussion leading to better answers. As usual, it was time consuming, but worth the time.
Additionally, while I am not happy that I had to go back and discuss the Second Derivative Test out of context, it did give me the chance to see if my students really understood the First Derivative Test, and I managed to get them to explain to me the advantages of each (with only a little prompting). This gives me quite a bit of confidence that they are getting the ideas.
What Didn’t Work
I was a little surprised that it took more than 20 minutes to graph a function. Even more surprising to me was that most of the groups managed to get through less than half of the checklist in that 20 minutes. I am worried that this bodes poorly for the next exam, on which they are definitely going to have to work a problem involving curve sketching.
I think this means that I should work another example today…