Resolving Doubts

The bell had barely cut through the classroom when the teacher for Class A wisely set down their chalk. Without any redundant instructions or plans to drag out the lesson, they turned and walked off the podium with a light, almost "make a run for it" pace.

Only a few "grind-lords" in the back row refused to admit defeat, clutching their exercise books as they hurried in pursuit. Soon, the faint sound of rapid-fire questions echoed in the hallway—clearly determined to carry out the creed of "if you aren't grinding yourself to death, you aren't grinding hard enough."

"Uesugi Hara, what's the logic behind this problem?" Morishita Ran's voice came from the front. As she turned around, she was still clutching an open notebook, her fingertip pressed against the prompt of a calculus problem.

Uesugi Hara glanced at the problem and explained the key logic in a few brief sentences. Before finishing, he even circled a few similar problems in his own workbook and handed it over. "Finish these and let me check them."

Morishita Ran's eyes widened as she took the book, her expression suggesting, How could you be this kind of person?—a mix of disbelief and shock.

Uesugi Hara didn't pay it any mind. He turned to Kamuro Masumi beside him and curled his lips. "Kamuro, after 'swimming' in the ocean of knowledge for so long, do you feel like you're turning into a 'freshwater fish'?" The mockery in his tone was impossible to hide.

"What kind of new insult is this?" Kamuro Masumi slumped over her desk, her voice tinged with a breakdown that was halfway to laughter.

She had been mulling over the lesson—the teacher's explanation had felt like a clear roadmap just a second ago, easy to follow, but in the blink of an eye (no, she was certain she hadn't even blinked!), it was like she had suddenly stumbled into a fog, losing all sense of direction. Where the hell is the path?!

Seeing her state, Uesugi Hara stopped teasing and pulled his own notebook from his drawer to hand to her. The edges were a bit frayed, but the contents were written with meticulous care—the lesson's main points were highlighted in red, small diagrams were drawn next to key formulas, and even the teacher's offhand remarks were annotated in blue.

"Stop staring at thin air." He pointed to the section on "Function Monotonicity and Curve Transformation." "The teacher just said, 'The sign of the derivative determines whether the function increases or decreases, just like the slope determines the direction of a climb.' Where are you stuck?"

Kamuro Masumi stared at the analogy circled in red, frowning. "But he suddenly jumped to 'the second derivative affects the concavity of the curve.' I can't figure out how to connect 'concavity' to the 'slope' from before."

Uesugi Hara picked up a pen and drew a small mountain on a piece of scratch paper. "Think of it as climbing a mountain. The first derivative is how steep the path under your feet is—positive means you're climbing up, negative means you're walking down. Now, what about the second derivative?" He drew a little stick figure on each side of the mountain. "On the left side, the higher you climb, the shallower the slope becomes, like you're reaching the summit—that's a 'convex function.' On the right side, the higher you climb, the steeper it gets, like you're just starting out—that's a 'concave function.' Simply put, it's about whether the 'slope itself is getting steeper or shallower,' which has nothing to do with whether you're going uphill or downhill."

Kamuro Masumi stared at the two little figures for a few seconds, then suddenly went "Ah," and tapped her finger on the notebook. "So in that example problem earlier, when the second derivative was positive, the curve was 'bulging' upward? Just like that right-hand slope?"

"At least it wasn't a waste of drawing." Uesugi Hara tossed his pen back into his case. "Now look at the example the teacher gave again. Does it look like a map now?"

Kamuro Masumi hugged Uesugi Hara's notebook and flipped through a few pages. The thoughts that had been a tangled mess just moments ago suddenly smoothed out, and even her expression relaxed. "It seems... really clear now. Your notes are much easier to understand than the teacher's board work."

Uesugi Hara raised an eyebrow, just about to reply with "That goes without saying," when he saw Morishita Ran leaning over with her completed exercises, her pen tip pointing at the final problem. "Uesugi Hara, I can't seem to draw the auxiliary line for this one correctly. How do you find the 'symmetry point' you mentioned?"

When he turned to look, Kamuro Masumi was already calculating the example problem on her own. Her pen tip scratched rhythmically against the paper, pausing occasionally before moving again—she no longer looked like someone on the verge of a breakdown.

"Here," Uesugi Hara took Morishita Ran's notebook and drew a small right-angle symbol next to the figure. "The axis of symmetry is the y-axis. If you flip this vertex over, you can calculate the landing coordinates directly without forcing it." He wrote as he spoke, tapping his fingertip on the paper. "It's like folding a piece of paper; the place where the two points overlap is the symmetry point."

Morishita Ran went "Oh," her eyes lighting up as she lowered her head to quickly correct her steps. Kamuro Masumi looked up then, holding her pen and pointing at Uesugi Hara's notebook. "That problem about the extreme points—the 'little mountain peak' and 'little valley' you drew, does that mean the maximum and minimum values?"

"Why else would I draw them as decorations?" Uesugi Hara glanced at her, but still explained, "The peak is the maximum, the valley is the minimum. Remember: 'climb first then descend is a peak, descend first then climb is a valley.' It's easier to remember than the definition."

Kamuro Masumi couldn't help but laugh. "You sure know how to take shortcuts." But she had already started drawing little mountain peaks in her own notebook.

The sunlight from outside the window slanted in, landing right on the three open notebooks. The questions from the "grind-lords" in the hallway hadn't ceased, but the classroom held a rare, relaxed atmosphere—the sound of pen tips gliding across paper mixed with Morishita Ran's soft mutters of "so that's how it is" and the sound of Kamuro Masumi's finally steady breathing.

Seeing that both Morishita Ran and Kamuro Masumi were buried in their calculations and no longer looking up to ask questions, Uesugi Hara closed his workbook and stood up to walk to the other side of the classroom. Yamamura Miki was staring blankly at her open textbook, her fingertips unconsciously picking at the corner of her notebook, not even noticing when he arrived at her desk.

"Yamamura-san." Uesugi Hara's voice was softer.

Yamamura Miki jerked her head up, her cheeks instantly flushing a light red. She hurriedly moved her fingers away from the notebook, her fingertips stained with a bit of pencil lead. "U-Uesugi-kun?"

"I saw you frowning at your textbook for a long time," Uesugi Hara said, handing her his notebook, the red-ink highlights standing out clearly against the white paper. "If there's anything you didn't understand, you can come to me anytime. Take these notes for now; they have the logic for the example problems the teacher just went over."

Yamamura Miki's fingers paused on the cover of the notebook before she took it gently. Her fingertips accidentally brushed his, and she recoiled as if electrocuted, her voice thin as a mosquito's. "Th-thank you, Uesugi-kun..." She lowered her head and flipped through a few pages, her eyes lighting up quietly when she saw the annotations with the little diagrams, though she still didn't dare to look up.

"Don't mention it." Uesugi Hara didn't linger. As he turned to leave, he caught sight of the dog-eared page in her textbook—the very section Kamuro Masumi had been stuck on—and added casually, "For the second derivative part, I drew a diagram of mountain climbing. If you don't understand it, just ask me directly."

Yamamura Miki's fingers tightened around the notebook, and she gave a soft "Mm." By the time she looked up, Uesugi Hara had already returned to his seat and was busy organizing the newly distributed test papers. She stared at the simple little mountain-climbing diagram in the notebook, the blush on her cheeks slowly spreading to the tips of her ears. Her pen hovered over her scratch paper for a long time before she carefully wrote down the first formula.

The sound of Morishita Ran and Kamuro Masumi discussing problems drifted from the front, blending with the wind outside the window, lending a warm, gentle movement to the classroom during the break.