The third week was devoted to differentiation. I motivated the derivative briefly with it being a concept that captures the ideas of
- rates of changes
- linearization (tangent lines)
- finding local max and min
Then we dug into the definition and the derivatives of the elementary functions and the rules of differentiation. All this is of course standard stuff, very technical and mechanical, a kind of drudgery that has to be done. I proved (at a level that ought to be convincing to the audience) some of the rules:
- Derivatives of small natural number powers of x. The students then did “square-root of x” and “1 over x” as part of the weeks guiding problem(s). Hand-waving took care of general power functions.
- I differentiated exponential functions b^x, doing the crucial limit in Matlab on the OH-projector. It is nice to see the limit approach 1 as you chose b = 3, 2.7, 2.71, 2,718 et cetera. Real nice – this went down well I think.
- I also did the natural logarithm.
- Once again I learned the bitter lesson that arguments for the Leibniz rule, an in particular the chain rule, are hard to get across. Suddenly there is an explosion of formalism, and this even though I shaved off a lot of hair. Well, hard to get across, but not impossible. I think I see how to do it in a clear way next time around.
So it was a week of quietly working along. More importantly though, I realize that I’m actually teaching in a different way now, after my trip to the US. It was in the making before – I mean, I did go there for a reason – but quite a few things that I’ve been thinking off for the last few years or so, came into sharper focus. I will devote the next post to that.
BTW, working with a naive theory (common sense) of limits works well for a course like this. It saves a lot of time. The cases where it breaks down has to be dealt with as they occur.