"Yep, you could. You can do that even if it's a super bumpy line -" gesture, now it's a super bumpy line - "can't you?"

"I guess? But it's harder, and it wouldn't be exactly the same, would it?"

"Wouldn't - start with the straight one, and then we'll make it more complicated and check why it's not the same -"

"Okay, so how do you do it with the straight one?"

"For the part that's flat, you said you'd just multiply how long by how fast. So imagine that the part that's slanted were stairstepped, instead - you could do the same thing -"

"Yeah, it would just be bits at different speeds right?"

"Exactly!" Flap-flap - "what do you get if you pretend a slanted line is a staircase with four steps?"

"Four bits, you just multiply them and add them like this, but you get more than your should 'cause there's all those bits where we're pretending he's going faster than he really is—"

He nods vigorously - "you're smart you're going to get it really fast - so if there are eight steps is the error smaller? -"

"Yeah? 'Cause we're pretending he's going faster but not as much faster as before, and there are more parts where we're not pretending at all."

"So there's a mathematical trick we can use to check the answer we'd get if we had infinite rectangles. It pretty much works by looking what we'd have at eight, and sixteen, and thirty-two, and a million, and noticing that they get closer and closer to the real answer."

"—oh that's really cool, so it's like you get an infinity of rectangles and they're all really like lines? Wait—" She squints at the graph. "If you're adding the lines aren't you just getting the area under there?"

"Yep. Which is easy to find for the line case, you probably already know how to find the area of a triangle. But it works for the area under any other kind of curve, too - uh, as long as the line's not jagged, it can have steep turns but not actually anything that'd be the equivalent of instantaneous changes in speed -"

"I guess it wouldn't make sense for something to suddenly change speed without going through the speeds in between."

"Yeah, and things that do act that way you can't handle with this method, but it'll work on every single kind of curve that isn't like that. D'you want to learn the case for straight lines and easy curves, or go right to the general solution -"

Bounce. "Show me the general solution."

Bounce bounce here is the fundamental theorem of calculus!!!!

Oh wow this makes all kinds of sense it's so cool "You're really smart!" And it is so great not being talked down to—although her mother never did, but the teachers and everyone else did and now she's sad.

He totally doesn't notice that because he is too busy excitedly explaining - "and there's a different kind of problem that turns out to be related to this one, and that is the problem of telling how much he's accelerating at a particular instant off a graph of his speed -"

Okay she can follow and become not-sad again by paying attention to the explanation.

And she will think it's pretty cool that the finding-the-tangent-to-a-line process is the exact reverse of the area-under-a-curve one, right, because it's pretty cool.

Wow it's awesome and more complicated than anything she's ever done ever but so cool.

"You'll probably want to play with it a bunch to get a feel for it, I can make a textbook if you want to do actual formal problems or you can just poke the computer into giving you new curves and then play with them."

"Can I do both?"

"Yeah, definitely!" Here's a Federation calculus textbook, here's a not-chip-locked computer she can play with. He flops contentedly on the floor.