Xueba starts with change

Chapter 52 It's Right Now
On the first day of the Lunar New Year, Chen Zhou spent this efficient problem solving.

There are still 4 and a half cans of psychotropic potion left.

On the second day of the new year, Chen Zhou needs to go to his grandparents' house to pay New Year's greetings.

However, after receiving the red envelopes, having lunch, and chatting with grandma and grandpa for a while, Chen Zhou went home by himself.

After simply tidying up the messy desks, Chen Zhou thought about it, and it seems that there is no need to go out these two days.

So, now is the best time.

Chen Zhou drank the remaining half can of spirit potion.

Then, he began to search for more knowledge of Lagrange's median value theorem, preparing to figure out the ins and outs of this theorem.

Let's start with the proof method.

"It is possible to prove Lagrange's mean value theorem by means of auxiliary functions:
It is known that f(x) is continuous on the closed interval [a, b] and can be derived in the open interval (a, b);
Then, construct the auxiliary function g(x)=f(x)-f(a)-[f(b)-f(a)](xa)/(ba);

It can be obtained that g(a)=g(b);

And because g(x) is continuous on [a, b], it is derivable in the open interval (a, b);
Therefore, according to Rolle's mean value theorem, there must be a point ε∈(a, b) such that g'(ε)=0;

From this we can get g'(ε)=f'(ε))-[f(b)-f(a)]/(ba)=0;

Deformed to f(b)-f(a)=f'(ε)(ba);

The theorem is proved. "

The process is very simple, and Chen Zhou understands it, but why he constructed such an auxiliary function and what Rolle's mean value theorem is, he is at a loss.

Chen Zhou thought about it and immediately searched for the related concepts of Rolle's theorem.

"Rohr's mean value theorem is an important theorem in differential calculus. It is one of the three major differential mean value theorems. The other two are: Lagrange's mean value theorem and Cauchy's mean value theorem."

"It turns out that this guy also belongs to differential calculus."

Chen Zhou continued to look at the description of Rolle's mean value theorem and the proof process.

This, the more he looked at it, the bigger the head, Chen Zhou found that he didn't understand anything, and he didn't know anything. Seeing a new theorem or lemma was a brand new knowledge.

Sure enough, the 12-year basic education is the real foundation
Chen Zhou raised a desire, and he strongly wanted to understand these theorems.

His thirst for knowledge was opened, instead of just studying for the college entrance examination.

At this time, Chen Zhou felt that this hidden mission seemed to become interesting.

He did not just focus on the Lagrange mean value theorem and Cauchy mean value theorem mentioned in the task.

He started with the differential mean value theorem, a branch that intrigued him a lot, starting with Rolle's mean value theorem.

The proof process, geometric meaning, and several special cases are all understood.

For the Fermat's Lemma and the Limit Existence Theorem mentioned in it, he put them aside for the incomprehensible ones, and simply looked at the Rolle's mean value theorem.

One afternoon is definitely not enough. After Chen Zhou hastily settled his dinner, he began to indulge again.

In order not to break this thirst for knowledge, Chen Zhou took out a can of new spiritual potion and drank it.

Dry mouth like this, and only before the start of school, the most suitable time, he dares to do it.

This is no joke, Xiuxian needs the right posture, the right time, and the right place.

It has to be said that with the powerful assistance of the spiritual potion, he went from Rolle's median theorem to the already familiar Lagrange median theorem to the only Cauchy median theorem mentioned in the mission. , and then to Taylor's formula, Darbe's theorem, and L'Hopital's law, which he had never heard of, he actually read them all.

Some are understood and learned, some are half-knowing, no matter how bad it is, they are familiar.

Chen Zhou also finally understood why he had to pick out Lagrange's mean value theorem and Cauchy's mean value theorem for hidden tasks.

Not only because of their wide application in the college entrance examination, but more importantly, the Lagrangian mean value theorem is the core of the differential mean value theorem, and other mean value theorems are special cases and generalizations of the Lagrange mean value theorem. , which is a bridge for differential calculus applications and has extremely high research value in theory and practice.

Lagrange's mean value theorem is also a special case of Cauchy's mean value theorem.

It was not until dawn in the morning, when Chen Zhou was called out for breakfast by Chen Jianguo, that he briefly escaped from the ocean of knowledge.

Chen Jianguo looked at his two deep bags under his eyes and was a little puzzled: "Xiaozhou, didn't you sleep well last night?"

Chen Zhou replied belatedly: "Well, I didn't sleep well."

After eating, Chen Zhou hurried back to the house to continue.

Although the strength of the potion has not yet passed, Chen Zhou was afraid that he would be sleepy, so he made another can.

On the third day of the new year, all day long, Chen Zhou didn't take half a step out of the room except out of the house to eat.

For some physiological needs, he is incidentally solved during the meal time.

After another sleepless night, Chen Zhou fell deeply into this thirst for knowledge.

When he went to the toilet to wash his hands before meals, he found himself in the mirror, except that the dark circles under his eyes were a little heavier and the bags under his eyes were a little bigger, but he didn't feel anything.

As for the spirit, it is very full!
Then go ahead.
On the fourth day of the Lunar New Year, after breakfast, I drank another can of psychotropic medicine.

Chen Zhou's mind is now full of these incomprehensible things such as the differential median theorem, the limit theorem, the θ of finite increment formulas, inequalities, functions, derivatives and so on.

On the fifth day of the new year, Chen Zhou looked at the last pot of medicine left, and he hesitated: "Will it be sudden death? Is it really worth it for a hidden mission?"

After thinking about it, Chen Zhou calculated the time left for the previous medicine stacking, and it seemed that there was not much.

If he can't overcome this hidden task during the winter vacation, Chen Zhou feels that it is unlikely that he will be able to complete it with only three months.

As the new semester begins, it will definitely be bombarded, and the time that can be allocated to hidden tasks will become less and less.

Taking another look at the formula on the scratch paper, he guessed that Lagrange, Cauchy, Rolle and the others could come up with these things.

Thinking of this, Chen Zhou no longer hesitated, he raised his head and drank the last can.

Whether the cultivation of immortality can succeed this time depends on the last wave.

The pen in his hand records his thoughts on the scratch paper almost all the time, and then goes to correspond to these theorems to verify his thoughts.

Thoroughly understand every proof process, and know every application example by heart.

Go back and sort out the inner connection of these theorems.

"Lagrange's mean-value theorem is a generalization of Rolle's mean-value theorem and a special case of Cauchy's mean-value theorem, a weak form (first-order expansion) of Taylor's formula."

In winter, the days are short and the nights are long.

But for Chen Zhou, there is no day or night, he just feels that one day passes too fast.

He even thought he had just had breakfast, why did he eat breakfast again?
After another night of fighting, at 7 o'clock in the morning on the sixth day of the Lunar New Year, Chen Zhou finished his breakfast and went back to the house to sit down.

He sorted out the draft paper that he wrote these days.

Chen Zhou has already learned about these differential median theorems.

He even knew a lot about advanced mathematics.

But he wondered why the system hadn't judged him to complete the hidden mission.

While tidying up, Chen Zhou watched the proof process of the Lagrangian mean value theorem written on the first day, and couldn't help smiling.

The logical sequence here, he has now fully understood.

It is because Rolle’s median theorem needs to be applied when proving Lagrange’s median value theorem, so a constructor is required to satisfy the conditions of Rolle’s median value theorem, and the constructed function is not unique, as long as it can satisfy the conditions of Rolle’s theorem can.

Thinking of this, Chen Zhou picked up a pen and began to try to construct a new function to prove Lagrange's median value theorem.

"Let F(x)=f(x)-[f(b)-f(a)]x/(ba), because the function."

".So F(x) is in."

".And F(a)=f(a)-[f(b)-f(a)]a/(ba)"

". Then F(a)=F(b), so F(x) satisfies the three conditions of Rolle's theorem.."

"Therefore, proof."

The moment Chen Zhou finished writing, the voice of the system sounded in his mind.

"Congratulations to the host! Done"

Chen Zhou didn't hear the next words, the spirit of the potion was over, he slumped, fell on the bed, and fell asleep.

(End of this chapter)