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Chapter 7: Chapter 7: Gasping a James Joule

Lu Yao held the test paper and looked at Li Dong again, sighing softly in her heart.

’This kid doesn’t have the face of a top student; he has the face of someone who’d get into puppy love.’

’Boys like him usually sit in the back of the classroom. They’re good at basketball, but their grades are a complete disaster.’

’If his grades were better, he’d probably have priority in the dating pool at college.’

Lu Yao teased herself inwardly, then picked up her red pen and went straight to the glaring red X’s.

”Look at this multiple-choice question on the concept of sets.”

Lu Yao pointed to the third question.

”The answer should have been A, asking for the intersection. You chose B, which means you obviously missed the case of the empty set.”

”This kind of mistake stems from not having a solid grasp of the basic concepts, and it’s one many students make...”

Just as Lu Yao was about to expand on the three elements of a set for Li Dong, she stopped.

She had noticed the notes written next to the problem, and her voice died in her throat.

[If A∩B=∅, then Δ

The handwriting was a bit messy, but the line of reasoning was perfect. There were no superfluous steps, and the case-by-case analysis neatly avoided every trap in the problem.

”Huh?”

Lu Yao looked up at Li Dong.

”Are these notes you took when your teacher went over the test?”

Li Dong had been thinking about the different descriptions of the definition of a limit in *New Introduction to Mathematical Analysis*. He was startled by the question and took a moment to register it.

”Uh... I guess you could say that.”

Lu Yao nodded.

”Well, you took very good notes. It seems you were paying attention in class after all.”

She continued looking down the page.

The next incorrect problem was on trigonometric functions.

Li Dong’s work was a complete mess; he’d even used the wrong formula.

But Lu Yao had learned her lesson. This time, she looked at the notes beside it first.

Sure enough.

Next to the wrong answer was a simple circle with a few auxiliary lines accurately dividing the angles.

[Using unit circle symmetry, point P(x, y) symmetric about the origin ⇒ P’(-x, -y)]

...

The correct answer had been derived in just three steps.

”This...”

The more Lu Yao looked, the more she felt something was off.

Although she was a student at a second-tier university, she was still a formally trained math major. She had an eye for these things.

This kind of problem-solving approach was definitely not something a student who usually scored in the 80s could come up with on their own.

There was only one explanation—he had an excellent teacher.

”Hey, kiddo, what school do you go to?” Lu Yao couldn’t help but ask.

Li Dong didn’t understand why she was suddenly asking this, but he answered honestly, ”Jiangcheng Seventh Middle School.”

”Seventh Middle School?”

Lu Yao frowned slightly.

She was well aware of Seventh Middle School’s reputation in Jiangcheng. It was known for its hands-off approach, with average students and a faculty that was said to be rather laid-back.

’Are the teachers at Seventh Middle School really this dedicated now?’

Lu Yao muttered to herself.

The notes on this test paper hit the nail on the head every single time. Some of the approaches were even more clever and concise than the standard answers in the official study guides.

This level of teaching would be considered mind-blowing even at Jiangcheng Sixth Middle School, the best in the city.

She couldn’t help but sigh as she spoke to Li Dong.

”You have a great teacher. You’d better not let him down.”

Lu Yao continued to flip through the test. After looking through the first few sections, her impression of Li Dong had improved significantly.

’This kid isn’t a lost cause. At the very least, his attitude toward corrections and the quality of his notes show that he wants to learn.’

That is, until she flipped to the very last page.

The bonus question section.

As expected, the kid hadn’t answered it. But now, in the blank space, was a hand-written variation of the problem.

[Given the function f(x) = e^x - a ln(ex), if f(x) ≥ 0 holds true for all x in (0, +∞), find the maximum value of a.]

Lu Yao’s eyebrows shot up.

”Isn’t this an isomorphic structure? Your teacher actually covers this kind of problem with you?”

The difficulty of this question was already approaching math competition level.

To solve it using conventional methods, you would need to construct two new functions, perform differentiation twice, and in the middle of all that, deal with an extreme value substitution involving an implicit zero. The amount of calculation was staggering.

Even she would need over ten minutes and a full sheet of scratch paper to solve it.

She instinctively looked at the solution process on the paper, preparing to see how this ”note” was written.

However, the very first line of the solution stunned her.

There was no function construction.

No second derivative.

[Using tangent line inequality: e^x ≥ ex...]

[The original equation simplifies to...]

[Therefore, a_max = e]

The entire process was just five short lines!

Lu Yao rubbed her eyes. She wasn’t seeing things. How did it just jump straight to the conclusion?

’What the hell kind of logical leap is this?’

’Is it missing something important?’

She quickly grabbed her scratch paper and traced the logic.

First step, correct.

Second step, substitution with a composite function, correct.

Third step, verification of the equality condition, holds true at x=1, correct!

”Hiss—”

Lu Yao sucked in a sharp breath of James Joules.

This solution method was nothing like high school math. It utilized function convexity and the tangent line inequality to launch a ”dimensional-reduction attack” on the problem.

Forget a regular high school teacher; even her own university professor wouldn’t have thought to use this method to solve this problem.

Lu Yao looked up, her expression complex as she stared at the young man before her.

”Li Dong.”

”Yeah, what is it?” Li Dong, who was taking a sip of water, was startled by her sharp tone.

”This solution...” Lu Yao pointed to the last problem, her voice a bit dry.

”Did your teacher explain how he came up with this method? Or did you just copy what he wrote on the board?”

Li Dong blinked, a little embarrassed.

”Um... Lu Yao, the teacher didn’t actually go over this one.”

”He didn’t?” Lu Yao didn’t quite understand.

”Yeah, it was a variation the teacher wrote down in class. I just tried to work it out myself.”

Li Dong did his best to make his tone sound sincere.

”I figured it’d be too much trouble to brute-force it, so I wondered if I could use an inequality to take a shortcut. I didn’t expect it to actually work.”

Lu Yao was speechless.

Seeing Li Dong’s ”I was just trying to save some effort” expression, Lu Yao couldn’t help but feel a pang of disappointment.

’You’re a slacker who scored an 87, and you’re telling me that, just to be lazy, you casually derived a brilliant solution that even *I* couldn’t think of?’

’That’s just too unbelievable!’

Lu Yao stared into Li Dong’s eyes, trying to discern whether he was lying.

But his gaze was too sincere—so sincere it made her feel like she would be a bad person for doubting him.

Yet, her rational mind told her it was impossible.

’He’s faking it!’

The three words exploded in her mind.

’He must have looked up the answer somewhere, or the teacher explained it, and now he’s claiming he figured it out himself.’

’Boys these days... They’ll say anything for a bit of pathetic vanity.’

She did not like dishonest students.

And when she thought about Aunt Li working so hard, even begging people to find him a tutor, a sour taste filled her mouth.

But she had made a promise, so she had no choice but to bite the bullet and continue.

”Fine. Since you can derive it yourself, that means I don’t need to explain this problem to you.”

”Let’s get back to the fundamentals.”

Lu Yao did her best to drag the kid back onto the ”correct” path.

For the remainder of the session, Lu Yao began to painstakingly explain the basic concepts where students in the 80-90 score range most easily lose points.

From solving quadratic inequalities, to setting up coordinate systems in simple solid geometry, to the summation formulas for sequences.

She explained everything concisely.

Li Dong listened attentively at first, but he soon grew bored of reviewing concepts he had already thoroughly mastered.

As she spoke, his gaze began to drift, and his mind flew back to *New Introduction to Mathematical Analysis*.

”Li Dong, pay attention!” Lu Yao rapped her knuckles on the table.

”Eighty percent of the college entrance exam is basic questions. You have to know this stuff inside and out!”

Li Dong snapped back to attention and asked tentatively,

”Um... Lu Yao, I already know the basics. Could you maybe teach me a little about Calculus?”

Although he felt he could understand it by reading the book himself, he suspected that a formal university education might offer a different perspective than his own unorthodox, self-taught approach.

Hearing this, Lu Yao sighed inwardly.

But in order to raise his score—or at least keep him from failing his next exam—Lu Yao refused him without hesitation.

”Calculus won’t provide any substantial help for your upcoming college entrance exam. Your number one priority right now is to ensure you don’t lose any points on the fundamental questions.”

And so it went. One person gritting her teeth and teaching by the book, the other gritting his teeth and listening while his mind was filled with higher mathematics.

An awkward and very long hour finally crawled by.

”Alright.”

Lu Yao closed the test paper.

”That’s all for today’s tutoring session.”

She glanced at her phone. It had been about an hour, but she had no desire to continue. Besides, judging by Li Dong’s condition, he seemed listless as well.

”Huh? We’re done?” Li Dong was still thinking about *New Introduction to Mathematical Analysis* and didn’t register her words at first.

”Yeah, I have something to do, so I have to head back to campus.”

Lu Yao gathered her things, stood up, and offered a polite but distant smile.

”You can spend the rest of the time reviewing on your own. Don’t bite off more than you can chew. Focus on solidifying your foundation first.”

”Also...”

She paused and said in a meaningful tone,

”It’s not easy for Aunt Li to earn money...”

With that, Lu Yao grabbed her bag and walked out of the room without a backward glance.

Li Dong sat in his chair, utterly bewildered.

’Why do I get the feeling... that she’s angry?’

...

After leaving, Lu Yao felt deeply conflicted.

She wondered, should she say something to Aunt Li?

But if she told her directly, she was afraid she would hurt Aunt Li’s feelings...

Lu Yao was caught in a dilemma.

Meanwhile, the source of her dilemma, Li Dong, was excitedly scribbling integral signs on a piece of scratch paper.

’So this is the Newton-Leibniz formula... It’s so beautiful!’

’If I apply this to calculating the work done by a variable force in physics...’

The young man’s eyes were filled with a pure thirst for knowledge.

He was completely unaware that he had just been labeled as someone who was overly ambitious and out of touch with reality.