This was the first time in Qiao Yu's life that he stood on the podium during class.

The perspective is different, and the feeling is indeed different. It turns out that everything can be seen clearly, and no small movements can be hidden. It seems that many times he is still smart, always thinking that the teacher cannot see some small movements below.

In fact, it's just a matter of whether you want to care or not.

Especially these little...no, they are the brothers and sisters in high school. They don't know how to hide their emotions at all. Whatever they are thinking in their hearts is written directly on their faces, and they look like they are easy to deal with...

Then Qiao Yu glanced at Lan Jie, who was standing on the side of the podium.

Sure enough, the teacher was different. At least just by looking at that emotionless face, one couldn't find out what he was thinking at this time.

After looking around, he finally focused on the topic.

"Let's take a look at this question first. Well, although it is a little simpler, it is a very typical function equation analysis question. You should know at a glance that this question first requires considering the value of the function at a specific point to solve the equation...

…

At the same time, you can also verify and eliminate multiple possibilities by constructing specific functions. With these two points of knowledge, I believe everyone already knows how to solve this problem."

"Ahem... Qiao Yu will explain your thoughts directly." Lan Jie couldn't listen anymore and coughed twice beside him.

"oh."

Qiao Yu nodded and picked up the chalk resignedly: "The first step is to prove that f(x) is strictly monotonically increasing. Let's start with the question f(f(x))=f(x^2)+x

To start, consider the derivation of x on both sides. We can assume that f′(x) represents the derivative of f(x). Obviously when f′(x)=0, then 0=1, which is a contradiction. Therefore, f′(

x)≠0…”

…

Five minutes later, Qiao Yu put down the chalk.

Lan Jie next to him sighed, it turned out that he really couldn't help this kid.

Qiao Yu's attention was focused on the new classmates in the audience. The situation seemed not to be good. There seemed to be only five classmates watching his problem-solving process thoughtfully, including Ma Yufei, who had dealt with him before.

With Lu Jia.

The expressions of most people clearly did not look like they understood, especially many people who were secretly glancing at Lan Jie who was standing on the other side of the podium...

That pure and innocent gaze doesn't seem to be fake.

Not only did they not understand, they didn’t even know whether their interpretation was correct or not!

As expected, Ma Yufei and Lu Jia were not alone in finding him. These guys seemed to have very low levels of skills.

For a moment, Qiao Yu suddenly sympathized with Lan Jie.

It is indeed too difficult to expect these people to participate in competitions after training, and also expect them to win medals and help good people earn some bonuses.

This also made Qiao Yu immediately realize why Lan Jie was so kind to him.

Players like him who can participate in the competition without much training seem to be the kind of scarce resources that are in high demand. As long as the school has a choice, it will not recruit these high school students who are obviously not up to standard in mathematics to participate.

Olympiad training.

"Qiao Yu, you can go down first." Lan Jie nodded at Qiao Yu and then walked up to the podium.

When Qiao Yu stepped off the podium and returned to his position with his back to the blackboard, Teacher Lan's voice also came.

"Student Qiao Yu's problem-solving ideas are very clear, but he briefly mentions some key steps, which you may not fully understand yet. Let's look at the second question analyzed by Qiao Yu. When considering possible functional forms through the construction method, Qiao Yu Yu directly substitutes f(x)=x+1 into the original equation, which actually requires certain calculations...

In the same way, when proving uniqueness, Qiao Yu used the method of contradiction to guide the possibility of analyzing function differences. The steps he omitted here are actually the monotonicity of the derivative derived from the first question. After supplementing, it looks like this...

To sum up, to answer this question, we need to comprehensively use the basics of differential calculus, function property analysis and construction methods, proof by contradiction and other mathematical tools. Today I have prepared a question for you to think about... By the way, Qiao Yu, come with me. office."

…

Qiao Yu was not given any time to stay in the classroom to discuss business with his classmates. After class, he took Qiao Yu back to his office.

However, Qiao Yu didn't feel too sorry. He had realized that helping these students explain the questions clearly was not an easy task.

For example, he just watched Lang Jie push the conclusion of the last example question that was very obvious and could be thought of with a brain again, and most of the blackboard he had vacated was filled to the brim... …

If it were him, by the end of this class, his writing skills would be disabled, which made Qiao Yu deeply feel that the job of a teacher is indeed not easy.

"Come here, move a stool and sit next to me." As soon as he entered the door, Lan Jie waved directly to Qiao Yu.

Qiao Yu honestly brought a stool over and sat down.

Lang Jie had already opened his laptop and opened the Overleaf interface.

"This is an online LaTeX editor. There is no Chinese version, but you should be able to use it depending on your English level. Let me introduce you to the main commands. The first is to create paragraphs..."

Lan Jie didn't talk nonsense. As soon as Qiao Yu sat next to him, he began to teach Qiao Yu how to use the LaTeX editor as promised on the phone yesterday.

The math teacher also witnessed Qiao Yu's talent again.

I didn't see Qiao Yu taking any notes. He only needed to do all the operations once and he understood them clearly.

Commands, formula editing, various lists, commonly used macro package operations and debugging can all be restored perfectly.

At first, Lan Jie thought that Qiao Yu said he knew how to do it, which was a bit reluctant, but when he asked Qiao Yu to start it on his own, his skillful operation even made him wonder if this kid had been exposed to it before and was here to tease him today. To play.

Originally, Lan Jie planned to ask Qiao Yu to come to school again tomorrow to use his rest time to let this extremely talented student consolidate, but now it seems that it is not necessary.

"Okay, there are probably only so many things that need to be mastered. The definition of references is for writing papers, and you won't be able to use it for the time being. Go back and try to do a few questions using the test system on the official website of the Little Libaba Mathematics Competition. , and then you should be able to fully master it by uploading the answers.”

Seeing Qiao Yu finish the demonstration, Lan Jie said with infinite emotion.

After all, it took him at least a week to learn how to use this thing back then, and he was not as proficient as Qiao Yu now.

In the past, Lan Jie felt that the gap between himself and the top students from Huaqing and Yanbei was not that big. At most, it was just that he was a little unlucky. The tutors on the other side were very powerful and could provide more resources. But now he has completely I no longer have this thought.

The world is indeed very unfair. Many students seem to have excellent grades, but most of them have tried their best, and the best among them may just study casually. How can this be compared?

You know, this guy skipped two days of classes two days ago and made money in an Internet cafe...

"It's indeed quite simple. If I had known it, I would have just found a tutorial yesterday and learned it casually." Qiao Yu nodded and stated a fact.

Lan Jie glanced at the little guy wordlessly, and then warned: "By the way, you will start a round of mock exams next Monday. You Class 13 don't have to make up lessons on the weekend. You should prepare first. In addition to mathematics, the questions this time will be... It might be more difficult."
Chapter completed!