It’s about time to start talking about antiderivatives in AB Calculus. We’ve defined a few rules so far: power rule, chain with trig, chain with e^x, and chain with ln x. For several years I’ve wanted to be early about our discussion of integrals. It has been helpful for conceptual understanding later if they know about the relationship between derivatives and integrals early. In fact, knowing about integration makes Curve Sketching a lot easier.
I was just going to do some simple introduction, but then I decided we could use a moment to get up and walk around.
Start with a 53¢ pack of index cards. Write a function on the top of the card and its derivative on the bottom. Repeat about 40 times.
Cut the cards in half. For fun, include several functions that have the same derivative.
Shuffle the cards a bunch until they’re good and scrambled.
The plan is to hand them the stack of cards and tell them that each card has a match. I’m not going to say how the cards are related to each other, only that have a match in the deck. After we spend some time sorting them, we’ll talk about the results.
How are the cards related?
Could some cards match with multiple cards?
Eventually they’ll tease out the idea that one card was the derivative and one was the original. Then bam! we hit them with the idea of antiderivative, the result of working backwards from a derivative. It is highly likely that when finding some matches they will do this, knowing they have the derivative in hand, in search of the original.
The fact that multiple functions can have the same derivative is always an interesting discussion, and just like that we’ve justified the presence of +C.