"It's really good they aren't making me go I think I'd be really annoying. Anyway, yeah, you hafta know a lot of kinds of engineering to be very good at being a demon because you need to make sure the things you're building work, what d'you want to start with -"

"Math, I guess? I think you need to know math for that, and all people will teach me is multiplication."

"Do you know how to graph stuff? Calculus is really easy to explain from that."

"Yeah. Had to learn on my own, though." She sounds proud of that.

"Good for you!" Here's a holoscreen with a graph. "So imagine that we put a tracker on some guy and its GPS - do you even have GPS - okay, anyway, our tracker can't tell us how far he's gone but it can tell us how fast he's going, here's the graph of how fast he's going. We want to figure out how far he's gone. D'you see how -"

"You have to multiply the speed by the time?" she guesses. "But it changes all the time..."

"Yes, it does. So let's make some simplifying assumptions - pretend he's only got so much acceleration, his speed increases linearly until he's going this fast and then it flattens out -"

"So after here you can just multiply this together with how long he's at this speed, but before—" She squints. "If you could divide the triangle in little pieces you could do the same, sorta?"

"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!!!!