pjbooks Chapter 16: Cao Wen: I think you can give it a try!
Chapter 16: Cao Wen: I think you can give it a try!
It must be said that Mr. Zhou Peiyuan's teaching methods and Mr. Ge Jun's teaching methods are almost completely opposite.
The latter's teaching style combines sharp wit and satire with teaching "how to think," using the harshest words to explain the most profound principles.
A sharp tongue but a soft heart; the fiercest criticisms are precisely when they are paving the way for the truth.
But Mr. Zhou was completely different. He was like a refined and gentle teacher from ancient times, speaking slowly and calmly, with a composure that made people unconsciously relax.
When his handwriting appears on the pages of a book, the strokes are always so calm and gentle, without a trace of anger.
Even if Han Chuan asked the same question three times, Old Zhou would only gently rephrase it and guide him to think about it patiently.
Two different teachers, with completely different styles.
If Ge Jun taught him techniques, then what Mr. Zhou Peiyuan taught him was more like internal strength.
As night fell and the bright moon rose outside the window, Han Chuan finally finished all the physics exam papers in his hands.
A line of handwriting appeared on the textbook "Theoretical Mechanics," accompanied by a gentle smile.
"Xiao Han, get some rest. Learning takes time and effort. If you don't understand anything, feel free to ask me anytime."
Han Chuan tidied up the test papers and draft paper on the table, nodded, and said, "Okay!"
......
After spending two days at home, Han Chuan took a ride back to school with two jars of meat and vegetables prepared for him by his mother, Li Xiumei, and resumed his normal high school life.
With guidance from Ge Jun and Zhou Peiyuan, he revised his study plan.
After all, he now has two book spirits from different disciplines guiding his studies, so he naturally needs to focus his time and energy on mathematics and physics.
After some thought, Han Chuan wrote down a new plan in his notebook.
The daytime school curriculum remains unchanged, but an hour is set aside during morning self-study and lunch break to study mathematics.
The two evening study sessions were used to make up for the lack of physics knowledge, and occasionally to go to the office to ask math teacher Cao Wen questions.
Although he was taught by two great teachers, Ge Jun and Zhou Peiyuan, he hardly needed to ask his teachers about math and physics problems.
But Mr. Cao Wen was different. He paid attention to and took care of him when he first started working hard. He even made it clear that he could go to Mr. Cao Wen's evening self-study sessions on Mondays, Wednesdays and Fridays to tutor him in junior high and high school math.
Han Chuan definitely remembers this kindness.
Of course, he also wanted Cao Wen to witness his growth process.
After all, returning home in wealth and honor is like wearing fine clothes at night.
He was never a top student before, but now that he is, shouldn't we let him experience it?
Therefore, Han Chuan would spend one or two days each week going to the math office of the second-year high school group to ask Teacher Cao one or two questions.
Days passed by like this, and in mid-to-late April, in the office of the second-year high school group.
Looking at the student in front of him, Cao Wen couldn't help but smile.
"You're here?"
Han Chuan nodded, walked over and placed the exercise book in front of him on the desk, saying, "Teacher Cao, I don't quite understand this question."
"Which one? Let me see."
Reaching out to take the exercise book, Cao Wen looked at the question Han Chuan was pointing to.
[Given the function f(x) = cos5x + acosx, where a is a real number.]
【(1) When a=1, find the maximum value of f(x) on the interval [0, π];】
【(2) If for any real number x, f(x)≥0 always holds, find the range of values for the real number a;】
【(3) Does there exist a real number φ such that the function.....】
After glancing at the problem in the exercise book, Cao Wen smiled and said, "This is a problem combining derivatives and trigonometric functions. The core difficulty lies in finding the derivatives and performing isotropic transformations of trigonometric functions."
He paused for a moment, then looked at Han Chuan, pointed to the chair next to him, and said with a smile, "Come on, sit down, and I'll explain it to you in detail."
Han Chuan obediently pulled over a chair and sat down beside him.
Cao Wen pulled an A4 sheet of paper from his desk drawer, picked up a pen, and began to instruct.
"The core difficulty in combining derivatives and trigonometric functions lies in the periodicity of trigonometric functions; no matter how you differentiate them, the trigonometric functions still exist."
"Therefore, it is difficult to reduce it to a simple polynomial function by taking multiple derivatives, which requires a change in thinking."
"For example, in the first question, when a=1, find the maximum value of f(x) on the interval [0, π].
"This is a classic derivative problem involving trigonometric functions. It requires finding the derivative, simplifying the derivative, and analyzing the sign of the derivative to determine the monotonicity and extrema of the function."
As he spoke, he wrote on the manuscript paper: "First, differentiate with respect to x, f(x) = cos5x + cosx, then f′(x) = −5sin5x − sinx...."
In the office, Cao Wen explained step by step, and after writing down this series of derivatives, he paused, turned his head to look at Han Chuan.
"After finding the derivative, how should I solve it?"
Han Chuan stared at the equation on the draft paper, pondered for a moment, and then asked, "Teacher Cao, can we use the sum-to-product method here?"
"Um?"
Cao Wen glanced at him and handed him the pen.
Han Chuan took the pen and continued writing on the manuscript paper: "f′(x)=−5sin5x−sinx"
=−5[sin3xcos2x+cos3xsin2x]−[sin3xcos2x−cos3xsin2x]
=−5sin3xcos2x−5cos3xsin2x−sin3xcos2x+cos3xsin2x
=−6sin3xcos2x−4cos3xsin2x。」
After writing down his thoughts, he stopped and looked at Cao Wen with some helplessness, saying, "I feel that it should be able to be transformed into a single product form, but I don't seem to be able to write it down."
Hearing this, Cao Wen looked at Han Chuan with a meaningful glint in his eyes and asked curiously, "How did you come up with this format?"
Han Chuan thought for a moment and said, "Just a gut feeling?"
停顿了一下，他补充道：「或者说从一个预算的结果倒推，这些零点都能用sin3xcos2x=0覆盖，所以我猜它可能是一个类似于−8sin3xcos2x这样的单一的乘积形式？」
"But I can't seem to push it out."
After listening, Cao Wen remained silent for a few seconds before speaking, "It's normal that you can't deduce it. Have you checked the coefficients?"
Han Chuan shook his head: "No, I was just guessing."
"Then calculate it for me."
Han Chuan nodded and began to write on the paper.
sin3x=sin·π/2 =1，cos2x=cos·π/3=0.5，sin3xcos2x=0.5。
The original derivative function is −5sin.....
As he deduced, an answer came into view.
[2cos3xcos2x?]
Looking at the equation in front of him, Han Chuan himself could hardly believe that he had calculated the result so easily.
Cao Wen, standing to the side, smiled and said, "You're stuck on the concept of 'the final, most challenging problem in calculus'!"
As he spoke, he took the pen from Han Chuan's hand, moved the manuscript paper over, and wrote on it: "When a=1, f(x)=cos5x+cosx=2cos3xcos2x. This is a product form, so we don't actually need to find the derivative to analyze the maximum value!"
Han Chuan stared at the equation on the draft paper, stunned. He stared at it for a few seconds, then his eyes lit up: "Sum and difference to product!"
Cao Wen smiled and nodded, saying, "That's right!"
After a slight pause, he continued, "Your first reaction to this problem is, 'This is a derivative problem, so I should find the derivative,' rather than, 'This is a problem of finding the maximum or minimum value, so I should first see if the function itself can be simplified.'"
"It's not your fault, because even many senior high school students can't break free from this kind of mindset."
"But your performance just now surprised me, because when you found that the simplification after differentiation was very complicated, you did not give up, but tried a variety of decomposition methods."
"In the end, we even used special points to deduce the form of the function."
"This kind of flexibility of thought and spirit of exploration is more valuable than being able to solve the problem itself."
At this point, he looked at Han Chuan and said with some emotion, "I have been teaching for so many years and have seen one class of students after another. Among them are geniuses who have been admitted to Tsinghua and Peking Universities, as well as students who have participated in competitions."
"But you are the first person to be able to catch up to a score of 130 or 140 on the college entrance exam in just over a month, starting from a foundation that was not even strong enough in junior high school."
"The question you just asked is one of the most difficult questions in the college entrance examination."
"Although you can't completely solve it yet, you're only lacking a little foundation. If you continue learning at this pace, it won't be long before you'll be able to solve these kinds of difficult problems on your own."
In the office, upon hearing Cao Wen praise him like this, Han Chuan smiled shyly and flattered him, saying, "It's all thanks to Teacher Cao's excellent teaching. If it weren't for your tutoring me every Monday, Wednesday, and Friday, how could I have improved so quickly?"
Cao Wen shook his head and said, "I know very well how much knowledge I've given you. The most important thing is still your own hard work and talent."
He paused for a moment, then looked at Han Chuan and asked, "Han Chuan, have you ever thought about participating in a math competition?"
"A math competition?"
Han Chuan paused for a moment, then asked, "The Olympiad? I'm still far from that level."
He did know about the competition; it was a place where true academic geniuses and top talents gathered.
Cao Wen smiled and said, "Don't underestimate yourself. You are qualified now."
"And with your learning speed, it's only the end of April now. By the time September starts, you'll definitely be able to break through the encirclement."
As he spoke, he took out a blue-covered document from the drawer and handed it over: "The Olympiad is also known as the National High School Mathematics League, which is the name of the domestic CMO competition under the IMO competition."
"Our county's No. 1 High School has a quota every year, but in previous years very few students from our grade could win awards. The last time a student entered the national competition was several years ago."
"But this year..."
He looked at Han Chuan with a serious expression in his eyes.
"I think you can give it a try."
.....