Ken settled into his chair behind the last table in the Atkinson lecture theatre and pulled out his pen and notebook. There were two more long tables ahead before the viewing screen and the lecturer’s table at the front. The class fit in comfortably. “Do you know if all our classes are in here?” he asked.
Quintessa was sitting on his right. “I heard something like that, but I suppose it’ll be on our timetables, so we’ll find out now.”
While she was speaking, Mathematician von Rejk walked in and dropped a pile of paperwork onto the lecturer’s table. “John Dustborn was supposed to be handling paperwork for the first ten minutes, but he’s introducing the third years to Feynman diagrams. I don’t like paperwork, so this will take substantially less than five minutes. These are your syllabus outlines. They’re all the same for now: the administrative nightmare of electives only starts later.” He dropped a stack of pages in front of Mark, who passed them down the row.
“These are your timetables. I’m not sure why we bother printing them, since every morning you’ll have theory sessions in here and every afternoon is applications in the lab. Nonetheless.” He dropped another stack of pages onto the table. “What else? Your lecturers will tell you what you need to hand in. You can ask your class tutor for help if you’re afraid of the lecturer — who is you class tutor by the way?”
There was a brief silence in which Ken wondered who was supposed to answer that. Melinda seemed to make the decision and said, “Ricardo Arcos, sir.”
Mathematician von Rejk grunted. “Last I was here, he wasn’t much older than yourselves, but I daresay he’s turned out alright. Wouldn’t ask him to help with a welding project, mind.” He looked distant for a moment. Ken wondered what story he was remembering.
“But enough of that!” von Rejk brought his fist down onto the table with a crash. “You are here to learn science: science and mathematics. It’s difficult to do much of one without the other. Now, I’m not much of a one for fussing with definitions.” Ken didn’t find that hard to believe, given the singed workshop gear under von Rejk’s red gown.
“However, without some definitions we won’t get anywhere and I need you to understand something about the calculus if we’re going to make progress. There will be no proofs and this will irritate some of you.” He paused and looked them over. “Which of you exactly is it going to irritate?”
Ken raised his hand immediately. Knowing something was true was no fun if he didn’t know why it was true. Quintessa and Kelly Jean were as quick as he was. Mark was just behind them. Ken smiled when Melinda tentatively joined them.
Von Rejk counted hands. “An even split, huh?
Let me point out that Becky Liang will be taking you for Analysis lectures. You’ll do all of this in painstaking detail and irritate the other half of us. That might make you feel better.” He grinned. “In any case, I need you to have a working knowledge of the calculus, both integral and differential, before we can do anything. Becky won’t get you there for weeks, so this is the crash course.” He turned to the screen behind him and drew two crossed lines for axes and a squiggly curve between them.
“This much you’ve all seen at school. I have some quantity — call it y to keep it general — which depends on some other quantity, which we’ll call x. We can make that concrete in endless ways. The speed of a spaceship depends on the time it’s spent accelerating. You give me an example.” He jabbed a finger at Mark.
“Um, uh, um, the temperature of a gas depends on the pressure?”
“Good. You.” He pointed to Verashni.
“The position of a moving object depends on the time it’s been moving for.”
“Good. You.” He pointed to the back. Ken hadn’t expected to be picked on and blanked for a moment.
“Um, well, I guess the height of an object determines its gravitational potential energy.”
“Yes, so the energy depends on the height. Good.
“We can draw a relationship like that by asking what happens at a particular x value.” He made a mark on the horizontal line. “When x has this value — this much time has passed or this is the height of the object — I put the point higher or lower depending on what the value of y is there. All with me so far?”
Everyone nodded. They’d covered this in high school, if not quite so vigourously.
“Good. Now, what happens if I change the value I’m looking at for x? You.” He jabbed his finger at Bryony.
“Um, y will change too?”
“I — I don’t know.” Bryony looked at him helplessly.
“Very good. Never be afraid to admit it when you don’t know something. Terrible mistake, that.” He brandished his marker at the class. “Now, do you agree that this is a question worth asking?”
Everyone nodded again.
“Why?” After a few moments silence he pointed to Kelly Jean. “You.”
“Well, uh, obviously you’d want to know how things are going to change, so you can predict what will happen.”
“Give me a concrete example.”
“If your position depends on time, you want to know how long it will take to change you position to wherever you’re going.”
“Good. Somebody else. You.” He picked out Jaxon.
“Sometimes you can break a mechanism if you change the current you’re feeding it too fast, so there’s got to be something in there that depends on how the current changes.”
“Alright. Lots of ideas in there, but good. You.” He jabbed the marker at Buhle.
“If you know exactly how something changes, you would know where it starts getting smaller again, couldn’t you? So you could find the maximum. That seems useful.”
Von Rejk paused. “Very nice reasoning there. Optimisation is more than useful, I assure you.
“You.” He pointed to Verashni.
“Well, speed is how much your position changes as time goes on, right?”
“Careful now.” He wagged the marker reprovingly. “How much your position changes is just how far you’ve travelled. The rate at which it changes is your speed — or rather, your velocity. Speed is a wishy washy kind of word.”
Verashni ducked her head in embarrassment.
“Don’t be afraid of making mistakes, either,” von Rejk said to them. “Think how much all of you just learned thanks to her attempt.” Ken wasn’t sure Verashni was appreciating how helpful she’d been. Whatever Mathematician von Rejk said, he was inclined to be careful, although he wasn’t sure von Rejk would leave him much choice in the matter.
He was safe for now, as the mathematician turned back to the screen. “Since we’re in agreement that this idea of change is going to be helpful, let’s give it a name. The rate at which y changes as we change x is called the derivative of y with respect to x. I’ll write it dy/dx.” He scrawled the symbols on the screen. Ken noted them down.
“In general, is the derivative going to depend on x?”
“Yes,” Quintessa said. “The way y changes is different at different x values.”
“Good. When won’t the derivative depend on x?”
“Um.” Quintessa stared at the ceiling for a few moments. “A straight line?”
“Good.” Von Rejk pulled out a pocketwatch and consulted it. He nodded before he put it away.
“I’ll spend the rest of the lecture giving you some rules for calculating these derivatives. Like I said, Mathematician Liang will prove these with you later on. For now, make sure you’ve got them written down somewhere.”
At the end of the lecture Ken had a list of disappointingly unpatterned formulae. “Can you see the logic to how this works?” he asked Quintessa.
“Some of it makes sense. If there’s a repeating pattern in the original, you see the same kind of repetition in the derivative, see?” She pointed out the formulae. “But we’ll have to wait for Mathematician Liang’s lectures for the whole story.”
Ken was flipping to the first page of his notes to look over the formulae again when Kelly Jean said loudly, “If you’re going to peer over my shoulder at my private notes all lecture you can go and find somewhere else to sit. It’s ridiculous and I won’t accept it.”
“I wasn’t,” Jaxon protested. “Maybe I looked once or twice when the screen was indecipherable, but they’re just lecture notes.”
“Once or twice? You copied down every word I wrote and you have no right to! Those are my notes.”
Jaxon seemed to have a knack for finding trouble, Ken thought. “Let her be, Jaxon. There’s plenty of space up here and it’s not worth fighting over.”
Kelly Jean glared at him.
“It’s not!” Ken protested. “There’s plenty of space.”
Jaxon gathered his things and moved to the back desk. “It’s ridiculous trying to make out what’s on the screen at that speed,” he said. “We ought to get notes or something.”
“I didn’t think it was too bad,” Ken said. “Maybe you just need to get used to his handwriting.”
Jaxon shrugged. “Well, I guess I learned who not to sit next to.”