Volume 1: Student Life Chapter 52: This is the right time
Chen Zhou spent the first day of the Lunar New Year doing exercises in this efficient way.
There are 4 and a half bottles of mental potion left.
On the second day of the Lunar New Year, Chen Zhou needs to go to his grandparents' house to pay New Year's greetings.
However, after receiving the red envelopes, eating lunch, and chatting with his grandparents for a while, Chen Zhou went home by himself.
After tidying up the messy desk, Chen Zhou thought that he didn't need to go out for the next two days.
Then, now is the most suitable time.
Chen Zhou drank the remaining half bottle of mental potion.
Then, he began to search for more knowledge about Lagrange's mean value theorem, preparing to figure out the ins and outs of this theorem.
Let’s start with the proof method.
“The Lagrange mean value theorem can be proved by using auxiliary functions:
It is known that f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b);
Then, construct the auxiliary function g(x) = f(x) - f(a) - [f(b) - f(a)] (x - a) / (b - a);
It can be obtained that g(a)=g(b);
And because g(x) is continuous on [a, b], it is differentiable on the open interval (a, b);
Therefore, according to Rolle's middle 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)]/(b-a)=0;
After deformation, we get f(b)-f(a)=f'(ε)(b-a);
The theorem is proved.”
The process was very simple and Chen Zhou understood it, but he was confused as to why such an auxiliary function should be constructed and what Rolle's Mean Value Theorem was.
Chen Zhou thought about it and immediately searched for the relevant concepts of Rolle's theorem.
"Rohr's mean value theorem is an important theorem in differential calculus. It is one of the three major mean value theorems of differential calculus. The other two are: Lagrange's mean value theorem and Cauchy's mean value theorem..."
"So this guy is also a differential calculus student..."
Chen Zhou continued to look at the description and proof of Rolle's Middle Value Theorem.
The more Chen Zhou read on, the more confused he became. He found that he didn't understand anything and didn't know anything. Seeing a new theorem or lemma was a completely new knowledge.
It turns out that the twelve-year basic education is the real foundation...
Chen Zhou felt a desire, he strongly wanted to understand these theorems.
His curiosity for knowledge was aroused and he no longer studied just for the college entrance examination.
At this time, Chen Zhou felt that this hidden mission seemed to become interesting.
He was not only concerned with the Lagrange mean value theorem and the Cauchy mean value theorem mentioned in the task.
He started with the mean value theorem of differential calculus, a branch that aroused his great interest, and began with Rolle's mean value theorem.
I understood the proof process, geometric meaning, and several special cases.
As for the Fermat's lemma and the limit existence theorem mentioned in it, which he didn't understand, he put them aside for the time being and just looked at Rolle's mean value theorem.
One afternoon was definitely not enough, so after a quick dinner, Chen Zhou began to indulge himself again.
In order to keep his thirst for knowledge burning, Chen Zhou took out a new can of mental potion and drank it all.
He would only dare to do it in one gulp like this before school starts, which is the most suitable time.
This is no joke. Cultivating immortals requires the correct posture, the right time, and the right place.
It has to be said that with the help of the powerful mental potion, he actually went through all of them in one night, from Rolle's mean value theorem, to the already familiar Lagrange mean value theorem, to the only two Cauchy mean value theorems mentioned in the task, to Taylor's formula, Darboux's theorem, and L'Hôpital's rule that he had never heard of.
Some of them I understood and learned, some of them I just had a vague understanding, or at worst just became familiar with them.
Chen Zhou finally understood why the hidden task picked out Lagrange's mean value theorem and Cauchy's mean value theorem.
It is not only because they are widely used in the college entrance examination, but more importantly, the Lagrange mean value theorem is the core of the mean value theorem of differential calculus, and other mean value theorems are special cases and generalizations of the Lagrange mean value theorem. It is a bridge for the application of differential calculus and has extremely high research value in theory and practice.
The Lagrange mean value theorem is just a special case of the Cauchy mean value theorem.
It was not until dawn that Chen Zhou was called out by Chen Jianguo to have breakfast, and he was able to briefly escape from the ocean of knowledge.
Chen Jianguo looked at his two deep eye bags and asked in confusion: "Xiao Zhou, did you not sleep well last night?"
Chen Zhou realized belatedly and replied: "Yeah, I didn't sleep... OK..."
After finishing his meal, Chen Zhou hurried back to the house to continue.
Although the effect of the potion had not yet worn off, Chen Zhou was afraid of feeling sleepy, so he drank another can.
On the third day of the New Year, Chen Zhou did not step out of his room for the entire day except for going out to eat.
He would take care of some of his physiological needs during meal times.
Another sleepless night passed, and Chen Zhou was deeply immersed in this desire for knowledge.
When he went to the toilet to wash his hands before dinner, he found himself in the mirror. Apart from the dark circles under his eyes becoming darker and the bags under his eyes becoming bigger, there seemed to be nothing special about him.
As for my spirit, it is extremely abundant!
Then let’s continue…
On the fourth day of the Lunar New Year, after breakfast, I drank another can of mental medicine.
Chen Zhou's mind is now full of these difficult things such as the mean value theorem of differential calculus, theorem for finding limits, finite increment formula of θ, inequalities, functions, derivatives, etc.
On the fifth day of the Lunar New Year, Chen Zhou looked at the last remaining can of medicine and hesitated: "Will I die suddenly? Is it really worth it for a hidden mission?"
After thinking about it, Chen Zhou calculated the time left for the previous potions to stack, and it seemed that there was not much time left.
If he couldn't complete this hidden mission during the winter vacation, Chen Zhou felt that it would be unlikely to be completed in the limited three months.
With the start of the new semester, you will definitely be bombarded with time, and the time available for hidden tasks will become less and less.
Taking another look at the formulas on the draft paper, he guessed that Lagrange, Cauchy, and Rolle must have put in a lot of effort to come up with these things, and they might even have become immortals.
Thinking of this, Chen Zhou no longer hesitated. He raised his head and drank the last can.
Whether or not you can succeed in cultivating immortality this time depends on the last wave.
With the pen in hand, I almost kept writing down my thoughts on the draft paper, and then compared them with these theorems to verify my ideas.
Understand every proof process thoroughly and know every application example by heart.
Let’s go back and sort out the internal connections between these theorems.
"The Lagrange mean value theorem is a generalization of Rolle's mean value theorem, and is also a special case of the Cauchy mean value theorem, and is 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 a day passes too quickly.
He even felt that he had just had breakfast, so why was he having breakfast again?
After another night of fierce fighting, at 7 o'clock in the morning on the sixth day of the New Year, Chen Zhou finished his breakfast and went back to sit down in the house.
He sorted out the draft papers he had written these days.
Chen Zhou has almost learned all these mean value theorems of differential calculus.
He even learned a lot about advanced mathematics.
But he was very confused as to why the system had not yet determined that he had completed the hidden task.
While cleaning up, Chen Zhou looked at the proof of Lagrange's mean value theorem that he wrote on the first day and couldn't help but smile.
He now understands the logical order here.
This is because Rolle's Middle Value Theorem needs to be applied when proving Lagrange's Middle Value Theorem, so a constructor is needed to satisfy the conditions of Rolle's Middle Value Theorem. The constructed function is not unique, as long as it satisfies the conditions of Rolle's Middle Value Theorem.
Thinking of this, Chen Zhou picked up a pen and began to try to construct a new function to prove Lagrange's mean value theorem.
"Let F(x) = f(x) - [f(b) - f(a)]x/(b - a), since the function ..."
"…so F(x) is…"
"...and F(a)=f(a)-[f(b)-f(a)]a/(b-a)..."
"…then F(a)=F(b), so F(x) satisfies the three conditions of Rolle's theorem…"
"So, it's proven."
The moment Chen Zhou finished writing, the system's voice rang in his mind.
"Congratulations host ! Completed..."
Chen Zhou did not hear what was said next. After the effect of the psychiatric drug wore off, he fell on the bed and fell asleep.
