1. Predictions. Let students use what they know to try to predict accumulation based on simple functions. displacement for velocity; water in pool based on rate, et cetera. See if students can come up with some kind of system for accurately approaching an answer.
Introducing the FTC is really a problem of introducing definite integrals more than one of a discrete topic. So we start at the beginning–I’ve got a couple of activities that I like for this, both from AP Central. One based on finding areas under curves with over and underestimates, the other based on velocity and displacement. The overall goal is to realize that smaller slices give us better estimates, and move into some different types of estimates (TABLE PROBLEMS!!) I’d like to add a focus on estimation–that if we know a starting point, and how much something has changed, we’re in good shape to find out pieces of the function. Run some AP Graph based integral problems with defined areas. Work on formalization of integral notation, math practice and predicting values.
After teasing out the definition of a definite integral, we can begin talking about how they are connected to derivatives, and what happens when they bump up against each other. We would have already talked about undoing basic derivatives (finding a function that gives a derivative of f'(x)) for basic polynomials, trig, et cetera, but now can begin thinking about them in a more formal sense…
and! Begin talking about functions defined by integrals. I like the idea of a starter problem getting students to explain the meaning of int(f'(x),0,x), and try to put into words what it means to take the derivative of that. I like the guided practice in the ‘Functions Defined by Integrals” module. Here we can work a couple more free response problems involving functions defined by integrals–particularly focusing on getting students to interpret their meaning, and figure out what the derivative of such a function would be.
At some point in here, we need to formalize the FTC in both parts, so might be a good time for some brief lecture on the topic.
1. AP Central FTC Focus