[32: Overkill Calculus, Gossip, and Cells at Work!]
(A/N: Oh my goodness, I did it again, I can't stop myself. That last chapter has left me feeling a sudden urge to do stuff like this. If anyone wants to offer their own explanations for Star and Stripe's quirk, feel free to do so. I want to bring many minds together to solve the energy problem. I hope you enjoy, and also learn something.)
/\\//\\//\
(A few days later)
"So, that's what will happen for the final exams. There will be a written portion and a practical portion where you will fight the teachers. Make sure to study and practice. That will be all for today." Aizawa said before leaving the class.
"Oh, no, we're done for! We have to fight the teachers‽ And I thought the written portion would be bad!" Kaminari panicked.
"Just relax. If you do well on the written portion, then you can afford to do badly on the practical portion. Come on, today is the day we're supposed to study, so follow me," Eleanor said as she got up. Not everyone followed. Some were going to study with Momo, and some felt fine on their own. The ones who went with Eleanor were Kaminari, Kyoka, Mina, Midoriya, and Ochacho. It was a small group, but Eleanor was only going to focus on math and science. Others needed help with the other subjects.
"I wonder what the place you live is like." Mina said.
"It's a lot closer than you think," Eleanor said.
As they walked out of the building, they continued onto the school grounds.
"Hey, Eleanor, aren't we going the wrong way? We're heading deeper into campus," Midoriya said.
"Relax, I know where I'm going," Eleanor said. When they came to a building, Eleanor tapped an ID on a scanner before it opened up. "Welcome to my house," Eleanor said as she entered.
"Huh? What?" Kaminari was confused.
"I live on-campus in these dorms. I'm the only one that lives here, though, but it's nice and quiet," Eleanor said as she prepared a table for the study session.
"Wait, you live on-campus? And UA has dorms?" Midoriya was shocked.
"You guys didn't know?" Eleanor asked.
"Well, in all fairness no one asked," Ochaco said.
"Alright, come on, I have stuff to teach you, so we should make use of our time," Eleanor said as she placed notebooks, pencils, and more on the table. "Take a seat, and we can get started."
Everyone took a seat. "Okay, tell me what you need help with," Eleanor said.
"Math," Mina said.
"Yeah, I think that's one we can all agree on," Kyoka said. "Can you help with this?" Kyoka showed a page from one of her homework assignments.
"A quadratic function. And it's asking you for the essential information about it. Yeah, that's straightforward." Eleanor started writing. "So, first, we take the derivative—" She was cut off instantly.
"Woah, woah, hold on a moment! We take the what? I don't remember learning all of that now," Kaminari said.
"Have you even learned anything?" Kyoka asked.
"Of course I have!' Kaminari said.
"If you say so," Kyoka shrugged as she paid attention to what Eleanor was saying. "But he is right, Eleanor. We did not cover this in our math class."
"Hmm…alright. Let me teach you calculus. Or enough calculus for this level of math," Eleanor got up and returned with a whiteboard on wheels. She grabbed a marker and started writing "Calculus made easy: Academia gatekeeps information on purpose by using fancy terms" before turning to her little group. "Alright, let's start. First, I'm going to take away the intimidation you feel." She turned to the board and wrote "d/dx" and "∫" before turning around again. She could see the look of intimidation on their faces. "Don't be intimidated, I'll turn these 'monsters' into cute and fluffy puppies. Let's start with 'd/dx.'"
"That looks like a weird fraction," Mina said
"It is not a fraction, except for the times it is. Sometimes it's treated as a fraction, but for now, it's just a symbol," Eleanor explained.
"I don't get it," Kaminari said.
"And that's okay. Let me break this down. It's not as bad as it looks." Eleanor drew a line to the "d" in "d/dx." "The 'd' represents 'a very tiny change,'" Eleanor drew a line to "dx. "And the 'dx' stands for 'a small change in x.'" Eleanor paused and turned around. "Do you follow so far?" She asked. All of them nodded so far. "Good. So, altogether, it just means how much this changes when x changes a little."
Mina blinked a little, focused as she thought there would be more. "Wait…that's it?" Mina asked when she saw that Eleanor was finished.
"Yes, that's it. That's all it means."
"Is there a way to visualize this?" Kaminari asked.
"Yes, let me do that." Eleanor drew a simple graph with a curve. "If you move a tiny bit along the graph…" She marked two points on the graph. "...this tells you how steep it is right there."
"Wait, it tells you how steep? Isn't that like slope?" Midoriya asked.
"Yes, it's like slope, just a lot more precise. And that's what a derivative is." Eleanor said.
"Oh, I get it now. That's not that bad." Kyoka said.
"Now, let me break down the integral symbol, or the 'long S,'" Eleanor started writing again while she explained. "It's just addition, but with infinitely many tiny pieces."
"...What?" Ochaco was taken aback by such a simple explanation.
"Let's put it all together. So, the 'dx' means 'a tiny bit' or 'a very small piece of x.'" She drew multiple rectangles under the curve she drew earlier. "It also represents an infinite amount of rectangles under a curve. So, integration is just adding up all those 'tiny bits of x.' And that's it. That's all what the notations mean."
"So…integration is like…super-addition?" Kaminari asked.
"Yeah, I guess you could see it like that," Eleanor said.
"Wait a minute! So, derivatives break things down into tiny changes and integrals add them back up?" Midoriya asked.
"Yes, Midoriya! They are inverse processes. You're learning! Excellent!" Eleanor exclaimed.
"That means they cancel each other out!" Mina said.
"That's mostly correct, Mina. Some information is lost when inverting the processes, but that's not something we need to get into for this level of math," Eleanor explained.
"They're not as scary anymore. I mean, I still kind of don't understand, but it's not intimidating anymore." Kaminari said.
"That's surprising." Kyoka said.
"What is that supposed to mean?" Kaminari said.
"Alright, let's focus now. So, in summary so far, you're not reacting to the concept, you're reacting to the notation and the way it's represented. Now, let me teach you some of the derivative rules that will help with quadratic functions." Eleanor started writing again. "We have the power rule, the constant multiple rule, and the constant rule. These three rules are all you will need for this level you're at. And they're simple. It'll be pattern recognition and basic arithmetic as we start practicing. So, let me explain the power rule. The power rule is used to find the derivative of a variable raised to a power. In other words, bring the power down, then decrease the power by one. With that said, let's practice. Find the derivative of x5." Eleanor explained and watched them work.
Her group started working, and in just a few moments, "I got 5x4." Ochaco said.
"That's correct." Eleanor said.
"Woah, I got the same thing!" It was so easy that even Kaminari could do it. Everyone else also got the same answer.
"Okay, now, I'll explain the constant multiple rule. If a constant, or number, is multiplied by a variable part, you can leave the number alone, differentiate the variable part, and then multiply the constant back in at the end. Now, let's apply this. Find the derivative of 4x3." Eleanor said.
They got to work. Using the previous knowledge of the power rule, and some very simple multiplication…"Is it 12x2?" Mina asked.
"Yes, correct. Did everyone get the same answer?"
Everyone nodded. "Fantastic! Now, for the last rule, the constant rule. This rule says that the derivative of any constant or number is 0. Basically, if it doesn't have a variable like x with it, it's zero. So, if I asked you for the derivative of one-million, it would be zero, since it's just a number. Now, for a little weird one you might encounter. The derivative of x is 1, remember that."
"So, is it really this easy?" Mina asked.
"Yeah, it really is. For whole numbers that is. Now, these rules also apply to fractions, so make sure you study your fractions. Also remember that the square root of a variable can be rewritten as the variable to the power of ½, and then you can use these derivative rules from there. Now, let's tie it back to quadratic functions, and really put your skills to the test." Eleanor wrote down the general form of a quadratic function. "We know that the general form of a quadratic function is f(x) = ax2 + bx + c. Now, using what we've learned and what I've taught you, take the derivative of it. Here's a hint: take the derivative of each term one by one."
They got to work again. This one took a bit longer, but the rules made it so easy that once again…"Is it f'(x) = 2ax + b?" This time, and to the shock of Kyoka, it was Kaminari.
"Yes, that's right. Good job, Kaminari. Did everyone else get that?" Eleanor asked. They all nodded and confirmed they got the same thing. "Now that you've got the derivative, I want you to solve for x. Now, it's just some basic algebra again."
They got to work again. As they did, Eleanor saw the look of realization on their faces as they realized what they had solved for. "Wait..that's…" Midoriya flipped through his notebook before seeing the formula. "...that's the vertex formula!" He exclaimed.
Eleanor grinned. "Did everyone get it?" They all nodded. "Congratualtions, you just derived a formula. Give yourself a pat on the back."
"Is this impressive?" Ochaco asked.
"Well, the concept of deriving formulas is mostly seen in higher-level studies. Until you get to that level, it's all mostly memorizing formulas." Eleanor explained. "Now, let's apply this to finding the information for quadratic functions. Kyoka, what's a problem from your homework?"
Kyoka looked at her homework. "This one gives me the function f(x) = x2 - 4x + 3 and it's asking me to find the vertex, determine the maximum or minimum, the opening direction, the roots, and the range."
"An all-in-one basically. So, yeah, calculus can be used to find all of that except the roots. For the roots, just use the quadratic formula; that's a lot faster. So, let's do this step-by-step. Once you can do this with one quadratic function, you can do it with all of them. Let's start with the vertex. This one is easy. Take the derivative, set it to zero, and solve for x, which will give you the x-coordinate. Once you have the x-coordinate, put it back into the original formula and solve it, which will give you the y-coordinate. And that's your vertex. Now, let's try it with the function from Kyoka's homework."
They got to work. "Woah, this is way easier and faster. I only have to do addition and multiplication." Ochacho said.
"For real. Why weren't we doing this before?" Kyoka said.
"I'm actually doing this!" Kaminari said.
"Is it (2, -1)?" Midoriya asked.
"Yes, correct, that's the vertex." Eleanor confirmed. She was smiling. They were learning! The power of plain language and not those pretentious, elitist-language academia bastards! Everyone got the same thing.
"Now, let's determine the maximum or minimum. This is even easier. To find the maximum or minimum, we take the second derivative of the function. In other words, the derivative of the derivative. In this case, take the derivative of f'(x) = 2x - 4. If the number is greater than 0, the parabola opens up. If it's less than 0, it opens downward. If it opens up, the vertex is a minimum. If it opens downwards, the vertex is a maximum. And by knowing what the minimum or maximum is, we can also find the range of the graph. This will solve three of the problems from Kyoka's homework in one go. Now, do it."
Just a few seconds later. "What? It's so easy! The minimum value is -1, and the parabola opens upward. And this means that the range is from -1 to infinity. Right?" Mina said.
"Yes, all correct. Just two steps give you all of that. Now, for the last one, which is finding the roots, just use the quadratic formula, or in this case, you can just factor the function for the x-values. So, yeah, that's basically it. With this method, you don't need to remember algebra formulas or tricks; just differentiate and solve. And what's better is that this method scales a lot better. So, if you get thrown a curveball like a cubic or any other higher polynomial, just differentiate and solve, but it's a basic algebra course, so don't expect it. And this is for quadratic functions, don't ignore linear and exponential functions. Though polynomials tend to be the big thing, because they want to prepare you for higher-level polynomials in the next algebra course, but calculus is a lot better than algebra for those kinds of polynomials, so I also set you up for success in the future. Also, make sure you know how to graph a function, which is also easy; just use calculus and the quadratic formula, and you'll get the points you need to connect. Let's practice a few more."
/\\//\\//\
"Hey, so why aren't we taught this way?" Mina asked.
"Don't ask me. I don't make the curriculum." Eleanor said.
"I feel that this is way beyond the intended path for us." Midoriya said.
The group was taking a break. Mina had brought a bunch of stacks, though Eleanor had also made some proper food.
"This salmon is very good, Eleanor. You cook so well." Ochaco said.
"Thanks. It's honey glaze. It's a mix of honey, soy sauce, garlic, lemon juice, water, butter, olive oil, and sriracha sauce. The salmon itself is seasoned with salt, paprika, and black pepper. It's really simple after that. Just cook it while basting it with the sauce, then put it in the oven on broil until it turns caramelized. It takes 20 minutes at most." Eleanor said as she ate.
"You seem to know quite a bit." Kyoka said.
"Well, Mina knows this. It's because I'm from Alaska. Besides moose and reindeer, salmon is everywhere, literally."
"Alaska? That's kind of cool. Is there—" Kyoka was interrupted by Mina.
"Ah, wait, Kyoka, Eleanor explained how things work over there. It's just like any other city or town. They don't have polar bears running around all over the place." Mina said.
"Oh, I see." Kyoka said.
"Well, polar bears aren't common anymore. The only wild populations left are in the high Canadian Arctic and parts of Greenland." Eleanor said. It was a real shame. She did some research and found that many iconic animals have not fared well.
"Oh, yeah, that's true. I've even heard that most rhinos only live in captivity in Africa." Kyoka said.
"I think it's the same for lions and giraffes as well." Mina said.
"Tigers as well," Kaminari said.
"Well, it's not as bad as it seems. There are still wild-populations, it's just that they are still endangered. Quirk-assisted conservation efforts have managed to keep animals in the wild, but without quirks, they probably would've gone extinct. From what I've read, there are many heroes in wildlife protection and conservation efforts. There's also anti-poaching units in Africa and other Asian countries led by heroes to protect the wild animals," Midoriya said.
"Maybe we should switch topics? This is kind of sad." Mina said.
"But it's kind of inspiring at the same time, isn't it? But, yeah, maybe we should switch to something else." Midoriya said.
"Yes! So, Eleanor, about America! What's it like? Have you been to all the places? Oh, what's that place with all the cowboys?" Mina asked.
"Oh, it's Texas! All Might has a move named after it!" Midoriya said.
"Yeah. Are there cowboys all over? You know, it'll be like..." Mina stood up and hovered her hand over her hips. "...'this town ain't big enough for the both of us, partner.' Draw!" Mina pulled out finger guns and pretended to shoot a revolver. "And they'll get on their trusty steed and run off into the sunset with the pretty girl!"
Eleanor facepalmed at such a description and scene. "That's so…" Eleanor hesitated. She knew they were just curious, but it was kind of funny thinking of it like that. Why not go along with it? "Well, I have a story. Want to hear it?"
"It's a cowboy story?" Mina asked.
"Yes, outlaws and stuff," Eleanor said
"That sounds cool!" Kaminari said.
"Okay, it's like this. There was a town called Agua Fria. A stranger went there one day, but he hardly spoke to the folks around him, he didn't have much to say. No one dared to ask his business or make a slip, because he had a big iron on his hip. When he rode into town in the morning, he came from the southside. 'He's an outlaw loose and running' came the whisper from each lip, and he's here to do business with the big iron on his hip…"
Yes. Eleanor was simply telling the same story from "Big Iron."
"...The folks were watching from the windows, everybody held their breath. They knew the ranger was about to meet his death. There were forty feet between them when they stopped to make their play…" Eleanor made a dramatic pause, and everyone was focused. "...And the swiftness of the ranger is still talked about today. Texas Red had not cleared leather, 'fore a bullet fairly ripped. And the ranger's aim was deadly with the big iron on his hip. It was over in a moment and the folks gathered around. There before them lay the body of the outlaw on the ground. He might've gone on living, but he made one fatal slip when he tried to match the ranger with the big iron on his hip." Eleanor finished.
"Whoa, that's so cool!" Kaminari said.
"It's just like what I've seen before! How dramatic!" Mina said.
"Yes, yes, I know. It comes from the 20th century, so it's super old, but still gold." Eleanor said.
"You must really like this stuff to go so far back," Ochaco said.
"I had a little phase, I guess."
"I wonder what the story would be like if there was a movie or something about it." Mina said.
"Our break is almost over, what subject is next?"
"Oh, biology, it's confusing. There are a bunch of words that sound the same, but mean completely different things." Kaminari said.
"The immune system is hard for me," Mina said.
"It's okay, we can cover that." Eleanor said.
"Wait, before we do that, and since it's just a few of us. Who do you think is the most attractive?" Mina whispered as she started to gossip. "Eleanor, who do you think is the most attractive boy here?"
Eleanro raised an eyebrow. "Boy? Let's see. Hmm…well…I haven't given it any consideration."
"Then consider it now!"
"My focus has been elsewhere. And it's too restrictive."
"Huh? What does that mean?"
"The parameters of your question leave much to be desired."
"The parameter is 'cute boy!'"
"Too restrictive for me."
"You're no fun."
"Come on, you have to study and I have to teach." Eleanor got up and went back to the whiteboard, and flipped it over to the blank side.
Everyone shrugged at why Eleanor seemed to be evasive, but they wouldn't get into personal details or think too much about it. It's not their business.
"Alright, so, what did you mean when you said everything looks the same, Kaminari?"
Kaminari went through his notebook. "Here, look. Mitosis and meiosis, and translation and transcription. How do I tell the difference?"
"Alright, let's start with what they are. Mitosis is the process by which a cell replicates itself with the same DNA. One cell becomes two identical cells. This happens for growth and repair. Mitosis only happens for non-reproductive cells, unless it's a unicellular organism, which uses mitosis for asexual reproduction. You'll also need to know the phases of this process: prophase, metaphase, anaphase, and telophase. Now, Meiosis is a little different. Meiosis produces sex cells, or gametes: sperm in males and eggs in females. So, one cell becomes four, each with half the original information and genetically distinct. It achieves this by mixing DNA in a process called crossing over. It does this for genetic diversity. Think of it like this: mitosis is copying homework exactly, and meiosis is copying homework and changing it up a bit, so the teacher doesn't notice."
"Oh, I get it now!" Kaminari said.
"Yeah, of course you did with a comparison like that." Kyoka said.
"Let's move on to translation and transcription. These two are steps in the same protein synthesis process. It starts with transcription. So, the original information is stored in the DNA. Transcription turns DNA into RNA, specifically messenger RNA, or mRNA. It's the same genetic information in a different format. Now, the protein that does this is RNA polymerase; It builds a complementary mRNA strand from the DNA template. That's transcription. Now, the next part is translation. Translation is the process of turning mRNA into protein. This process is carried out by the ribosome, which either floats freely in the cytoplasm or is bound to the endoplasmic reticulum. Think of this like this: transcription is copying a part of a script, and translation is turning that part into a performance." Eleanor explained.
"I see now. I got it. It's not that hard," Kaminari said.
"Now, I'm only teaching you what you'll most likely be asked on a high-school-level biology test. This stuff gets insanely complex," Eleanor said.
"Oh, so it's not easy," Kaminari's mood deflated a little.
"For this level, this is all you most likely need to know. Now, let's get to the immune system. Pay attention, Mina, you said this gave you trouble. Let's start with the steps of an immune system response. Let's frame it this way: It starts with detection, then signaling, then response, and finally remembering. First is identifying the threat; this part is the innate immune system. The cells that are part of this are neutrophils, macrophages, dendritic cells, natural killer cells, mast cells, eosinophils, and basophils. Though for a basic biology class, neutrophils and macrophages will likely be what is asked for, so we'll focus on those. It's simple. Neutrophils and macrophages eat the pathogen in a process called phagocytosis. Both of them also release cytokines that activate a larger immune response. They're basically the first responders. Here's where they differ: macrophages take parts of the pathogen and present it to other immune cells for a larger response. This is step one. In short, neutrophils and macrophages detect a target, eat the pathogen, release cytokines, and in the case of macrophages, present parts of the pathogens to other cells to initiate a larger response. Do you follow so far?"
"So…the neutrophils and macrophages are like security guards? They just go around looking for any bad stuff? Like a mall cop, or something?" Kaminari asked.
"Yes, basically. Good, you're learning. Let's move to the next part: signaling. This part is simple, and I already explained it. The cytokines I mentioned earlier call for more cells; basically, calling for backup. The macrophages present parts of the pathogens to T-cells, which kicks off the process of the adaptive immune system. Now, the combination of the cytokines and the macrophages presenting the pathogens to the t-cells, activates the t-cells and b-cells. That's the signaling. Next, is the response."
Eleanor started writing and explaining the next part.
"B-cells produce antibodies, which they use to tag threats. T-cells eliminate pathogens or infected cells directly. This is the immune system's response part. This is where your body produces more fighters to fight pathogens. That's what you need to know for the response. There's a lot more to it, but this is what should appear on a biology test. Now, the final part: remembering. After the threat is dealt with, the vast majority of activated immune cells die, except for a few that become memory B cells and T cells. These memory B cells and T cells allow faster, stronger responses the next time your body encounters the same pathogen. These cells are why vaccines work. And in the 18th century, a doctor named Edward Jenner developed the first smallpox vaccine, which eventually led to its eradication. Memory B-cells and T-cells are the reason the vaccine works."
"Woah…this is really interesting. Why do I understand it more than when the teachers explain it?" Mina asked
"Because a classroom is a rigid, one-size-fits-all instruction. It assumes everyone learns at the same rate, which is clearly not true, and sometimes the whole class will be held back by one student doing bad. The reason you all seem to be learning well is because, since we're a small group, I can more easily engage with each of you to cover your different paces and learning styles, and I'm also not forced to follow a curriculum. Like, no offense, but even Kaminari is learning well," Eleanor said.
"It's 'cause you make it sound so easy and simple," Kaminari said.
"It's because I'm explaining things in a way you understand. In a regular classroom, teachers often aren't able to do that," Eleanor explained.
"So…is school purposely making us dumb?" Mina asked.
"Alright now, that's not the case either. You're not dumb, you just need the information represented in a different way, or reframed in plain language. It's just the way that a classroom is set up," Eleanor said.
"Oh, I see." Mina said.
"Well, I think that'll be enough for today," Eleanor said. "Make sure to use the knowledge I gave you, especially calculus, that'll help you for a long time in your math classes." Eleanor started to pack everything up. "Have a good evening."
Kaminari, Kyoka, Mina, Midoriya, and Ochaco each said their goodbyes before leaving, carrying techniques that were overkill for their level and might confuse their teachers.