Calculus BC is presenting an interesting challenge. Pacing is *really* hard to nail down. With a such a small, equally capable group, stuff that was normally a week in AB is taking us a day, tops. It reminds me of when I had a class of two (yes, two, though it eventually became ten) students one time. It's really easy to go "you got it?" and get 100% agreement.

Compounding the matter, we are wandering in all kinds of weird directions because, hurricane. My initial motives behind BC were to write a narrative around coordinate systems: rectangular, parametric, vector, and polar. When we missed 10 days, I rewrote the narrative entirely to get something productive out of Hurricane School. With a super fast grading period (an entire post of its own), I've been winging it for a month.

Other than logs/exponentials, we've knocked out all the derivative rules. We covered nearly the entirety of curve sketching (concavity, extrema identification, etc) in like 2 days, and we even introduced integrals via Riemann and trapezoidal sums. Next on the list is algebraic integration, but first a dabble into the abstract idea of integration, computing areas geometrically. But there are a lot of concepts at play here. For this, I started with six graphs:

I covered several situations, all based on some reflections I made a couple years ago. If you really want to demonstrate you know Calculus, you have to do it symbolically or all the algebra is just irrelevant.

We spent a couple days with these graphs. Prior to playing with them, I showed them something I rigged up in Desmos:

I move a slider and I have them keep an eye on the black number (reading 9.28 in this shot). I stop the slider at various benchmark points along the way and we note the value of the black number. At no point do I tell them what the black number means. Eventually they reason out that the black number represents the area at any given time from x = -10 to x = 17. Further, they're able to determine how areas increase or decrease the total depending on position above/below the x-axis.

### Concept 1: Integral as Area

Students take a'(t) and f'(t) and determine the area from left to right. We talk about how the function could be subdivided to accomplish this (rectangles, trapezoids, triangles, circles, etc). I redefine their answers as the integrals of a'(t) and f'(t) from far left to far right.

### Concept 2: Area is Relative

Students use b'(t) and I define an arbitrary starting point, in this case I chose t = 0. This axis is measured in steps of 3, so I have them determine an integral from 0 to 30 and from 0 to -27. One student wondered out loud that 0 to -27 seemed off, as if the limits of the integral were backwards. I shrugged and played dumb.

I went back to the Desmos graph and we talked about "reversing" contributions. If I slid "backwards" in time, any area that was previously added is taken away from the total, and any area taken away is now added back. Thus, if our integrals moves left along the x-axis, the notion of what increases or decreases the total is reversed. We then mark the value of b'(t) from 0 to -27 as negative overall. We have gone "backwards" in time 27 units.

### Concept 3: End Points are Flexible

With g'(t), I define a new function k(x). I define k(x) as the integral of g'(t) from an arbitrary starting location (in this case I chose -3), and an arbitrary ending location "x." I ask, given that definition, how would we determine k(3)? k(-8)? They think about it for a bit and see that "x" gives them flexibility to define the endpoint of the integral, and k(3) and k(-8) represent choices on how much area to find. In my scenario where the reference point for the integral is t = 3, k(-8) will follow our "backwards" in time model.

### Concept 4: Absolute Extrema

Lastly, with c'(t) and h'(t), I bring them back to curve sketching. Based on these pictures, where would c(t) and h(t) have extrema? Currently we know how to identify minimums, maximums, and points of inflection. I introduce the concept of a relative min/max with these pictures. And then branch out into determine absolute minimums and maximums. For c'(t), I offered an initial condition at the far left, and we calculated area at four places: the far left, local max, local min, and the far right. Those numbers determined, we could see how end points come into consideration when making a conclusion about what's "absolute" and what's not.

Effectively, three days and six pictures give kids a symbolic look at almost all the major concepts in the AB course.