Work Text:
Online schooling was, in general, a great thing for Kanade. It allowed her to do all her work at her own pace and at her own time, which, combined with the easier course load and lack of proper tests or finals, meant she could spend more of her time composing.
Now, this did mean that she could potentially… fall behind, per se, as there was no physical presence actually pressuring her to stay on track. Some may say she was well versed in the art of “speedrunning a week and a half’s worth of work in a day” by slaving at it from the moment she woke up to the start of their daily Nightcord meeting.
And so it really wasn’t a surprise when she opened her math work, understood approximately none of it, saw the sun beginning to peek through the crack in her curtains, and ultimately decided to put it off for another day in favor of going to sleep.
Another day became another two days. Then four days, a week, two weeks, until it’d been three entire weeks and Kanade had done everything but her math. Every time she opened up that damned math worksheet, she’d be reminded of its contents and decided that literally anything else was more worthy of her time. Which… may have led to her being a week ahead in Classics, two weeks ahead in History, three days ahead in English, and an entire three weeks behind in Math.
Which was great. Kanade definitely wanted to spend an entire day (or three, considering how her disdain for the subject made her much slower at it) doing nothing but math. Yay.
Taking a deep breath, she steadied herself before opening the file once more. Surely it would make much more sense now that she’d spent three weeks ruminating on it.
1. Use implicit differentiation to find an equation of the tangent line to the ellipse x2/2 + y2/8 = 1 at (1, 2).
What. Does. That. Mean. How can differentiation be implicit? Doesn’t implicit just mean implied? How do you imply the derivative of something???
Whatever. Maybe the next few questions will be better, and she can just return to the first one later.
2. Use implicit differentiation to find an equation of the tangent line to the hyperbola x2/6 - y2/8 = 1 at (3, -2).
Again with the implied derivatives. Okay. But also, what the hell is a hyperbola?
Fine. She’ll just skip down to the bottom of the worksheet and work her way back up. Everything will seem a bit easier once she gets into the flow of things.
15. Prove (Theorem 3.3) that d/dx [xn] = nxn-1 for the case in which n is a rational number. (Hint: Write y=xp/q in the form yq=xp and differentiate implicitly. Assume that p and q are integers, where q > 0.)
Wow. Those are indeed words on a page. Kanade’s never been more confused. Did she miss a video or something that taught all this?
Quickly, she scans her teacher’s instructions for this worksheet and finds… nothing. Scans it again. Wait. There, right in the middle of the longest paragraph known to mankind, is one hyperlinked word. Yes! She’s saved! She clicks it, and is redirected to a twenty minute long YouTube video.
So, we’ve got the equation x squared plus y squared is equal to 1. I guess we could call it a relationship. And if we were to graph all of the points x and y that satisfied this relationship, we get a unit circle —
This is the slowest talking man Kanade has ever heard in her life.
— how we can figure out the slope of the tangent line at any point of this unit circle. And what immediately might be jumping out —
Weren’t tangent lines covered two units ago? How is there always more with this kind of thing?
— for any x value we actually have two possible y’s that satisfy this relationship right over here. So you might be tempted to maybe split this up —
Twenty minutes later, and somehow Kanade is more confused than when she first started. Running away to be a hitchhiking bard in the middle of the countryside is sounding more and more appealing by the minute. Unfortunately, that would leave her without her trusty composing setup, and said composing setup is very necessary to make a song that could save Mafuyu one day, and — wait. Mafuyu. Her lovely, amazing, extremely talented girlfriend who happens to be quite good at math.
Maybe, just maybe, she could ask Mafuyu for some help. Ah, but she’s always so busy with her own work, and Kanade would hate to be just like one of Mafuyu’s classmates who begs for her notes or assistance because they weren’t paying attention in class. The alternative, however, is rewatching that video until it makes sense, and Kanade may as well go insane before that has any chance of happening.
Sighing, she returns to the worksheet, presses print, and takes the still-warm paper and a pencil to Mafuyu’s room. Like always, the door is closed, so Kanade knocks and waits for a soft “come in” before entering.
Mafuyu is situated at her desk, laptop open to a half-filled document. Her eyes soften just a touch as her head tilts like a confused puppy.
Cute.
No, wait, she’s here for a reason.
“Are you busy?” Kanade asks, tucking the worksheet against herself.
“Not particularly. I was revising some of the lyrics I shared yesterday based on your feedback. Why?”
“I, I uh —” a blush began to form on Kanade’s cheeks as she ducked her head in shame, letting her hair create a curtain around her. “I may need a little help with my math work…”
Mafuyu nods, scooting over to make space for Kanade next to her. For whatever reason, the sight makes Kanade’s stomach twist uncomfortably, and she feels the overwhelming need to apologize and leave.
“I’m sorry, you’ve probably got your own stuff you want to do and I really shouldn’t be bothering you with things like this —” her rambling is cut short by a hand cupping her cheek, lifting her head up so that she has no option but to make eye contact with Mafuyu. A confused, slightly frowning Mafuyu.
“I already agreed to help you. If you felt the need to ask for assistance, you probably needed it.” Mafuyu looks like she wants to say more but falls silent, moving her hand from Kanade’s cheek to her wrist and tugging slightly.
Silently, the pair make their way back to the table, where Mafuyu seems quite insistent on hooking her chin over Kanade’s shoulder to peer down at the paper laid down in front of them.
“What part do you not understand?” Mafuyu questions, taking her right arm and wrapping it around Kanade’s waist.
“The uh, well, kind of… everything?” Kanade isn’t sure if the growing redness on her cheeks is from the embarrassment of not understanding the math or from the way Mafuyu’s drilling small circles into her side with her thumb. Maybe both. Definitely both.
“I see. Well, we can start here then.” Mafuyu uses her free hand to point at the first problem.
It takes quite a bit of explaining on Mafuyu’s end for Kanade to begin understanding, but she powers through it nonetheless. Eventually, she finishes all fifteen problems on the worksheet and subsequently slumps down on the table.
“Have you finished?” Mafuyu asks, peering over at Kanade.
She’d returned to working on her laptop once Kanade started to get the hang of things but had refused to let them fully separate, intertwining their hands together. How she was typing with only her left hand, Kanade had no clue, but the contact was still… nice. Grounding, if anything. Mafuyu would squeeze her hand a little whenever Kanade was working her way through a difficult problem (which was all of them), and the small amount of pressure was enough to keep her from being overly frustrated.
“With this worksheet, yeah,” Kanade responds, lifting her head up from the table to allow Mafuyu to take the paper out from under her.
“Okay. Everything here seems to be correct. How much more do you have?”
Oh. Uhm. How is she supposed to explain to Mafuyu that she may or may have not ignored this for three weeks? And since she didn’t do this paper, she didn’t do any of the other papers for this class in fear of understanding it even less, meaning there was a huge pile of similarly sized worksheets waiting to be done.
Kanade gulps. “Uhm. Three weeks worth?”
“Three weeks?” Mafuyu never really sounds happy, but she especially does not sound happy right now. “Go print them out and bring them here. We can work on them for another hour before having dinner.”
Kanade sulks her way back to her room and begins the process of printing everything. For whatever reason, her printer decides to stop printing on both sides of the page, only making the growing pile of paper seem even larger.
When she returns to Mafuyu’s side, she’s greeted with a quick kiss and a stern reprimand to simply ask if she needs help instead of procrastinating for so long. Kanade agrees, if a bit half-heartedly, and solemnly returns to her worksheet.
1. Find the derivative of y=arccos(-x3)
This was going to be a long, long day.