Free Marks I: Calculus

[Trigger Warning: Contains potential headaches!]

"How about this? Since I'll be working on the midterm paper." Rinae-sensai lifted her finger excitedly as if she had some bright idea. 

"If you manage to solve this math problem. You'll get a free 5% mark in the midterm!" She explained, brimmed with enthusiasm and the students started to murmur in amazement. 

"Hehehe, I know you'll be excited about it." She turned to write down the problem on the board. It read "dy/dt-y = 1, y(0) = 0, find y(t) =???"

"Please find 'y' for me!" Rinae-senpai smiled mischievously as the class gasped and groaned as the mix of desperation and frustration hung in the air.

"This is definitely foul." 

"We can't solve this with a normal integration method. We only learn how to do one variable at a time!"

"I have never seen this type of problem before." The students burst into complaining at the complexity of the problem, it was beyond their high-school curriculum for Calculus. 

"This is one of the Ordinary Differential Equations or ODEs. Anyone would like to try for the class?" Rinae taunted, waving the markers on her hands. Most students had already given up. 

"I want to try it!"

"Can I try it?"

Simultaneously, Ayane and Shouichi spoke and raised their hands. It surprised the teacher a little. 

"Hooo. We have two brave souls! Come on out, you two can show me on each side of the board." Her dark-colour eyes glittered with ecstasy.

"She must be a genius freak," Shouichi remarked inwardly. 

They both took out one marker each and started vigorously writing down their solutions, squeaking sound reverberating across the room. The classmates were flabbergasted by their fast scribble. 

"If anyone can solve this, it has to be Morimura-san."

"Is Seito-san doing it right? I have never seen anything like that before."

"Wow, those two are really going at it."

Quiet chatters from the students could be heard in the background.

After 10 minutes, the high-pitched sounds of markers stopped. 

"My! You both got it right! Despite using different approaches." Rinae praised them with pure astonishment, clasping her hands.

The final answer was "y(t) = Ce^(t)-1 and C = 1". 

"Starting with you, Morimura-san, right? What do you think about this problem? Fufufu" The teacher giggled gleefully. 

"This problem is really advanced but not beyond the high school's Calculus," Ayane responded to her teacher, her face frigid cold as ever, "To begin, I employed the integrating factor of u(t)=e^(t) on both sides of the equation and used the integral product rules to rewrite it into an integrateable form. After the integration, I rewrote it into the solution as shown here." Ayane explained, her speech was rapid yet crystal clear. 

The classmates gasped in awe and were absolutely captivated by Morimura Ayane's intellect.

"100 points for Morimura-san! Good job! Now now, Seito-kun's turn." The teacher switched to point her hand toward Shouichi. 

Clearing his throat he stepped up, "So I applied Laplace transform to the equation then rewrote it to be in capital "Y(s)" equal something... like this," He pointed at the equation on the board, paused to make sure everyone was still following. 

"I had to perform the partial fraction decomposition to break down the equation into two terms. Once that's done, I applied the Inverse Laplace transform to get the final equation... hence the answer," Shouichi carried on with confidence in his tone.

"Wow! 100 points for Seito-kun!" Rinae stared at him with seductive eyes. "But you are using University level's Calculus. This is no longer high-school mathematics, Seito-kun." She stepped closer, her large chest almost touched him.

"How did you learn about this?" She probed with pure curiosity under her glasses.

"Eh? I guess I've read it somewhere…" He modestly brushed her off, averting his face away from her massive balloons.

In truth, he had been secretly studying the engineering textbooks his father had left behind. The complexity of those books felt like a challenging puzzle, stimulating his mind and serving as a way to exercise his intricate thought process.

Rinae turned to the class with a grand, theatrical gesture. "Well, I guess you all remember the answer, right?" She announced with a satisfied smile.

"As promised, this will be an extra question on the midterm," She emphasised. The joyful class erupted in cheers and applause, clapping enthusiastically for both Ayane and Shouichi.

***

The class finished for the day without any issues. Ito Emi, Aoyama Shinji, Morumura Ayane, and Seito Shouichi walked as a group along the school grounds. Shouichi and co. Already decided to continue their study session at Yoko Sweet Shop. As for Shouichi's girlfriend, Misaki, she was picked up by her tutor plus carer plus driver, Wendy-san and headed straight to a private session at her house. 

"Eh? Isn't that Wanabe Ririku-senpai?" Ayane asked loudly, noticing a pink-haired girl washing her hair at the handwashing station.

"Good afternoon, Wanabe-senpai!" Shouichi called her out and waved at her. The girl appeared to be awfully tired and brooding, which contradicted the cheerful and vigorous version they met at the temple. 

"Good afternoon, underclassmen..." She spoke, dragging her words. 

"What's with the hair?" Ayane asked with a puzzled expression. 

"Ah? I think I might have slept on the glue or something… some idiots were not cleaning properly after their work in the clubroom." Ririku explained while rubbing her unique pink-coloured hair. 

"Eh? You sleep in the clubroom?" Ayane was surprised that Ririku-senpai was different from the image in her mind. 

"Hahaha. I can't stand 8 hours of lectures… I need to sleep in between… or die tiring, I mean trying." Ririku explained, brightening her mood up a little. 

"It's really rare to see you at school, senpai. I swore I'd never met you here before. Not counting your announcement." Shouichi chimed in. 

"Yeah… I'm actually an introvert, I rarely go anywhere besides my clubroom and my class." Ririku clarified with ease. 

"Oh, that made so much sense. Hahaha," Shouichi giggled. 

"Alright, boys and girls. Have fun. I'm going home to sleep now." Ririku bidded her farewell and walked lifelessly toward the school entrance. 

"I suppose people have different sides to them, and personalities can change depending on the situation," Ayane concluded with a gentle smile. The group nodded in unanimous agreement before continuing on their way.

***

"Good Afternoon, you got quite a crowd here, hmm? Shou-kun." Yoko greeted them as the group entered through the shop's door. 

"Good afternoon, Yoko-san!" The group greeted her with friendly expressions. 

"We'll be in your care, today. Yoko-san." Shinji let out a smug and gave her his strange black card with both of his hands. 

"Eh? What is that senpai?" Emi questioned him with curiosity in her expression. 

"Oh, I just got it from my old man," Shinji spoke, rubbing his silver hair. 

"Wow, Shinji-kun. Did your father just give you an AMEX black card?" Yoko inspected the credit card closely. Shouichi curiously drifted close to look at the card in his mother's hand and raised his brows puzzledly. 

"What is AMEX?" Shouichi pried, rubbing his jaw.

"Let me enlighten you... Ahem* It's a credit card called "American Express Black Card". Super exclusive and invitation-only for privileged ones... From what I've heard you need to have a net worth of over 10 million dollars to be invited," Ayane responded with her broad spectrum of knowledge, nerdily in her gesture,

"Damn, Shinji. How wealthy is your family?" Shouichi's inner voice asked.

"Hehehe. I didn't realise that." Shinji modestly laughed it off, despite astounding gasps from his friends, "May I buy you guys some snacks and drinks with it?" Shinji suggested with a playful wink.

"Yay!" Emi exclaimed with joy, jumping excitedly, her assets were bouncing up and down. 

"I wouldn't mind the freebies," Ayane spoke and chuckled to herself.