A few weeks ago I was in a training and during a break I sketched out an idea I have been playing with for a long time. Can you make a single graph that can display all the big ideas of Calculus in such a way that students can work through how and why they relate to one another?
I finally set out to make it real while ticking away the hours before AP scores released (more on that punch in the face later). The result is pretty nice:
The starting function can be whatever and the x-interval is adjustable. On screen is the y-value at each end of the interval, the slope of the graph at each of the interval, a short line segment indicating the slope at each end of the interval, the area accumulated in the interval, and the value of the second derivative at any point in the interval.
The idea is to use this to have a discuss early on in the process about some properties before we put a name to them. That's why other than the y-values the other numbers are unlabeled. This combines a lot of half-hearted attempts at this I've made on the fly in the past.
I have found my students have big problems visualizing how all of this stuff works together. Concavity in particular is a weird one, what with positive/negative readouts at seemingly arbitrary locations.
Play around with it and edit as you please: Calculus in One Picture.