#worked calculus example

Being in space, you would have assumed that you wouldn’t have to worry about things like math and calculus. It wasn’t like you assumed it wouldn’t be involved in space travel, but you assumed that you wouldn’t have to deal with it with this exact space trip, considering you were abducted by aliens and all. You’d assumed that you were going to be eaten, like everyone else who’d been captured. Yet you and a very small number of others hadn’t yet and were instead being put through a number of tests. A few were physical in some way, but most had tested your intelligence. You couldn’t help but feel like a rat in a maze, being tested on how fast we could change our behavior after being electrocuted. Were these aliens, the Black Arms as you now knew they were called, studying you? If they were, then what for?

This specific test was a test of your ability to do calculations. Vaguely you thought back on that comic that was rather popular a while ago, about someone being forced to do algebraic equations to get into heaven, and couldn’t help but feel that this was something like that. The first few questions weren’t too hard for you, you’d never been bad at math. But when you got into the calculus section, you started to get nervous. You’d only learned so much calculus before you’d been, well, abducted by aliens. And one part in particular always go to you. Integrals in general just hadn’t clicked with you yet. You knew how to do it, but you couldn’t always do it right. It didn’t help that the massive ruler of the Black Arms, Black Doom, was watching everyone testing. You were sure it couldn’t be, but it was almost like he was watching you more than the others.

The next question was about just one of these. Of course it was. It asked what the indefinite integral was of the cube root of seven minus tangent t, all multiplied by secant squared of t. Nothing seemed to be going your way. Why did they even to know if you could do integrals anyway? Surely, they were just going to eat you in the end anyway! Why did they need you to work alone too? Wouldn’t this test be just as effective if you could all work together? You tried to solve the problem anyway, if you only could get a little help…

The floating Black Doom started to move through the space, and to your horror, he came towards you! Had you displeased him in some way? You were quite certain this was when you would be eaten, when he said in his low voice, “You struggle. Why do you persist?”

You glanced around, hoping there would be someone else nearby to respond, but no one did, so you answered nervously, “I don’t know how to do this bit. I hadn’t learned it in class yet. I think I can get it though, if someone…” You paused for a moment, considering whether or not this was a test too. Should you say it? Was that against the rules? No one else spoke, and some others clearly struggled too.

He seemed to read your mind as he let out a low laugh. “It is no shame to require assistance,” he said, “The Black Arms are a hive mind, there is a constant cooperation and collaboration between us. I’ve not seen one of your kind request assistance, not in a non-life threating situation. It takes intelligence to be capable of learning and wisdom to be willing to ask questions. Certainly, you are a promising candidate. Come with me, I will show you how to solve problems such as these.”

…what? Black Doom, ruler of the Black Arms, probably a demi-god, destroyer of worlds, was offering to teach you how to do calculus? Called you intelligent and wise? This couldn’t be real. Shakely, you stood and followed the towering alien out of the room. You still weren’t sure you weren’t going to be eaten, but you followed obediently. And how did he know what you were going to ask for help? Could he read your mind?

“Yes,” he said, though you hadn’t spoken a word. You walked with him down a long corridor now. “We’ve partially attuned you and your fellows to the hive mind. You cannot connect to it, but we can examine your thought processes through it. We use it to determine how well you complete the tests to reveal which of you are most suitable.”

“How are you able to do that?” you asked, knowing there’s no point in leaving it unsaid now. You tried not to think about what “suitable” referred to.

“It’s in your food. A psychokinetically active additive makes your small minds more open to us. We need you all alive for now, so you must eat to live.”

“So, it’s like the radioactive compounds in PET scans, but to see our thoughts instead of our brains?”

He glanced at you, giving a judgmental gaze. “They do not have those in the rest of the civilized universe. Most do not put radioactive compounds in their bodies willingly.”

“Oh.” You walked in silence down the long corridor, until you followed him into an antechamber with a giant tablelike structure in the center.

“I will show you how to complete problems like these,” he declared, moving to one side of the table and indicating that you should join him by scaling a massive seat that seemed more like a monolith than a chair. You managed, with a reasonable amount of effort to climb on top of it. “Good,” he said as you gazed up at him, “You struggled with this.” You nodded as he indicated the problem in question.

“It simply a matter of substitution,” he said, using a massive claw to etch the problem on the table.

“You must simply replace one part of the integral with a separate variable, making it easier to integrate.”You weren’t quite sure you knew what he meant, but you nodded along as he etched a separate variable, u, into the table.

He continued. “It would be more difficult to integrate the cube root of 7 tangent t than of a single variable, such as u. To integrate with respect to u, you must also transform dt to du.” He then wrote a derivative equation:

“Why would we take the derivative of u?”, you asked, confused at the seemingly irrelevant step.

“Have patience,” Black Doom snapped, then continued, “If you’d not interrupted, I would have been able to inform you that the derivative of u is taken to isolate du and dt, and their equivalent portions of the equations.” As he took the derivative of 7 minus 7 tangent t, it made more sense; the majority of the derivative was the factor outside of the radical.

“Multiplying both sides by dt isolates du and dt, making it possible to replace secant squared t and dt with an equivalent expression in that makes the integral possible to integrate in terms of u rather than t.”

When he lay it out before you, it seemed far simpler than you thought it would be. “So, you can take a simpler integral rather than a more complicated one by replacing parts of it with equivalent and simpler variables?”

“Correct,” If Black Doom had a mouth, you could have sworn smiled at you. “Your species’ capacity to learn so rapidly is something we valued in you. If I were to name a positive characteristic in your inferior species, it would be that.”

You gave a weak smile in response, somewhat concerned by his phrasing. You watched him write the edited integral.

This integral was far more reasonable to deal with, a simple matter really. You etched the integrated solution on the table in your smaller hand writing with your writing utensil, something between a pencil and marker.

Black Doom nodded approvingly. “There is but one more step,” he said, “to replace u with what it was originally substituted for.”

“That’s the solution then?” you confirmed.

“Indeed,” he replied. “You are certainly a most suitable candidate. Very adaptable and obedient. I shall keep you in mind for later. Now return to your fellows. Do not bother with completing more of this test, your progress has been deemed satisfactory.”

You nodded and rushed towards the exit, then paused for a moment. “Thank you,” you said, trying not to make your voice shake.

Black Doom looked at you curiously for a moment, after which he said, “As you should. Now begone!”

As you turned and returned to the first chamber you were in, you gave the test you’d completed in a pile of other completed tests and returned to a group of your fellow abductees.

“Oh my chaos!” one gasped as you returned, “You’re alive! We thought for sure you got eaten!”

You shook your head. “No,” you said, “Actually, Black Doom explained calculus to me.”

“You’re clowning? You’re not clowning? I sense clowns.”

“No, I’m serious.” You turned away. “I think there’s something more going on here, though,” you whispered, “I’ve told you before I think they’re studying us, right?”

The others nodded, and some of the Black Arms nearby almost imperceptibly tensed, shifting.

You glanced around, “I think these tests are for more than just study,” you said softly, “I think there’s a bigger goal here. We’re candidates for something. These are just the preliminary tests to determine which of us is best for the real experiment.”

The rest of the group recoiled. “Surely not?” one chuckled nervously. “I mean, the aliens wouldn’t have told you their real plan, right?”

You opened your mouth to double down, but one of the Black Arms jerked their head and reached for a phaser at the edge of your vision. “You’re probably right,” you said, keeping your eye on the alien, “just a theory.”

The group visibly relaxed and so did the surrounding Black Arms. You didn’t though. What had happened still didn’t feel real to you. But as much as you knew something else was going on, you pushed it to the back of your mind, in a place you hoped the Black Arms couldn’t find.

***