# +C

We covered indefinite integrals yesterday in Calculus. I’m happy with how I teach them to add C to all of their indefinite integrals:

“What’s the antiderivative of $x^2 + 3$?”

$\frac{1}{3} x^3 + 3x$!” they joyfully exclaim. We spent all yesterday on this, so they think they’ve got it.

“Hmmm. I disagree. I think it’s $\frac{1}{3} x^3 + 3x+5$.”

They stare at me, then at the board. “Oh – because when you take the derivative, the 5 goes away. Why 5? Couldn’t it be 10? or 6?” they ask.

And so we talk about how you have to add a constant, but you don’t know which one, so you have +C  and let that represent a whole family of antiderivatives. So really, the answer is $\frac{1}{3} x^3 + 3x+C$. And that’s a good conversation, and they almost always remember to add +C – possibly because I talk about how I lost so many points on a test for forgetting it, and because I harp on it. But while they remember, I’m not sure that they understand.

This year, I’m remembering to use Desmos a lot more. And so I had them pull out Desmos and actually graph the thing. Yes, add a slider for C. Now hit play. What happens?

I liked that they got to actually see the different functions that this +C represents, and how we’re taking the same function and moving it up and down the y-axis. This may be useful when it comes time to do slope fields; I’m not sure how yet, but I think I can pull it in.

Next up, we talked about differential equations and the difference between general and particular solutions. I had them graph a point and had them manipulate C until the function went through it; then we found the value of C algebraically.

Revolutionary? No. But a few more lightbulbs than usual went off. Anytime I can move something out of the realm of my verbal descriptions and onto an actual picture in front of them, I want to.