Today I finished reading a section of my calculus book and worked on some of the associated exercises. I’ve decided not to skip section exercises entirely, but to be selective with them. It still feels hard to know which exercises will be most instructive, without doing them. Yet trying to evaluate that might be a useful exercise in itself. One thing that is important in tutoring is the ability to look at a problem and quickly outline the steps to its solution in your mind so that you can guide the student.
One of the problems I worked on today was finding the equation for a parabola given three points on that parabola. I made a start on it, but I think a different approach may be needed. In particular, I think the simultaneous equations involved will probably be easier to solve if I couch them in the $y=ax^2+bx+c$ form rather than the $(y-k)=a(x-h)^2$ one (or the $(x-h)^2=4p(y-k)$ one). Tomorrow I’m going to think a bit about how these relate to one another. For instance, where does the $p$, so useful in finding the focus and directrix, end up in $y=ax^2+bx+c$?