For the 10,000 yuan transferred by Fang Yuanyang, Fang Chao accepted it, and at the same time praised his father.

Ate and was sleepy.

Full of warm thoughts**.

He stayed up all night for seven days and seven nights in a row, and the sleep time at night was basically no more than three hours, but whenever an idea came out, Fang Chao got up directly to work, the light in his room was almost not turned off, it was on every night, and it was also on during the day, unless he was particularly tired, he would go to bed and lie down for a while.

But more often than not, he slept on his stomach on the table in his room.

Now the waveguide function is over for the time being, and the rest is handed over to Shen Lang, and he has nothing to do, at this time, he feels that the upper and lower eyelids are fighting, and no one is convinced of anyone, and finally he does it directly, which is difficult to solve......

This sleep was so dark that the sun and the moon were dark, Fang Chao didn't wake up for a day and a night, and Fang Chao woke up at four o'clock in the afternoon the next day, he was woken up by the ringtone of his mobile phone.

He's still a student, and not many people know his mobile phone number, and it's still class time, so it's basically not the school that looks for him.

Shen Lang!

The remark on the caller ID is Shen Lang.

Got news?

Fang Chao suddenly came to his senses, and he didn't feel sleepy at all.

He connected the phone.

"Brother Lang, how is it? Do I have any problems with the waveguide function equation I wrote? Is there anything that needs to be corrected? ”

Shen Lang quickly said, "I have taken a general look at your waveguide function, the problem is not very big, I didn't expect that you were only seventeen years old and would be able to come up with this kind of function equation." ”

Fang Chao said, "I also had a whim. ”

Shen Lang said again, "Regarding the function equation you uploaded to me this time, I have read it, whether it is an example or an idea, etc., there is no big problem, I personally only have a few questions to ask you briefly." ”

Hearing Shen Lang say this, Fang Chao immediately breathed a sigh of relief.

There is only one reason why Shen Lang said this, that is, the waveguide function he calculated is true, which is a new idea, although it is not as great as Kaijiang, but it can provide high school students with a better idea of how to solve functions.

Seven days of continuous fighting, not in vain, Fang Chao's heart is definitely surging!

"Brother Lang, if you have any questions, just ask."

Shen Lang: "First of all, the first point, that is, the problem of the definition domain of f(x), I did not see it in the materials you sent me, if it were not for the fact that I knew what you wanted to express, I rushed to this question, and your waveguide function would not be valid." ”

When Fang Chao heard this, he directly patted his head, "Dizzy, I forgot about this......"

When he was thinking about this question, he naturally had a definition domain in his mind, but he didn't express this definition domain, because he knew it, but if he knew, it didn't mean that other people knew it, which was a huge problem, and it was very serious.

Shen Lang said that it was just a small problem, that was from his side, because he was also a genius, and it was not too difficult to understand the idea of a genius, but what about other people?

Even the wise can make mistakes!

Fang Chao said helplessly, "I didn't expect me to make such a huge mistake in this aspect, I really shouldn't be, I should reflect on it." ”

Shen Lang said, "It's not a big problem, don't be nervous, it's just some routine problems and mistakes, which can be avoided." ”

Fang Chao, "En." ”

Shen Lang, "Within the definition domain interval you mentioned, how to determine the symbol of f(x) at the endpoint?" ”

Fang Chao continued to answer, "If there is a different sign, then f(x) has only one zero point in the interval, if it is the same sign, then f(x) has no zero point in the interval, and if the value of f(x) at a certain endpoint is zero, then the endpoint is a zero point of f(x). ”

Shen Lang: "Okay, so what do you think about the approximation?" ”

Fang Chao: "The main idea of using derivatives for approximation is to replace the curve with a tangent when x is close enough to x0. ”

Shen Lang, "Well, then how to calculate it?" ”

Fang Chao: "When |x|is very small, we have to approximate the equation, △y=f(xo+△x)-f(xo)≈f'(xo)△x,f(xo+△x)≈f(xo)+f'(xo)△x, so that x=xo+△x,f(x)≈f(xo)+f'(xo)(xo)(xo)(x-xo)

In particular, when xo=o, |x|is small, f(x)=f(o)+f'(o)x......"

The topic is what Fang Chao thinks, and the entire solution idea is also done by Fang Chao, and Fang Chao has answered all kinds of questions raised by Shen Lang, and he has never had any questions.

He can be said to know every step by heart.

The two chatted on the phone for more than an hour, and then Shen Lang said with a smile, "The questions are just now, it's not a big problem, it just needs to be supplemented, because you have to write it out a lot of things, mathematics is like this, even if you know, but the process and definition of the calculation are also very important." ”

Fang Chao said, "Thank you, Brother Lang." ”

Shen Lang, "It's not a big deal, it's just how I think about what I said?" Do you want to come to Qingbei? If you come to Qingbei, there is absolutely no problem in mathematics, and even if you want to go to Princeton in the future, it is not impossible to start from Qingbei, can it be said that my Qingbei is really worse than Peking University, there is no reason, and if you come to Qingbei, Professor Lin will also provide you with convenience, and write a letter of recommendation to Princeton in the future, you will be a student of Princeton. ”

Shen Lang was still thinking of a way to pull Fang Chao to Qingbei, so that the two swords of the junior brothers would be invincible in the world.

When Fang Chao heard this, he was a little moved in his heart, he wanted to go to Peking University before, it was because the road from Peking University to Princeton was a shortcut, and there were indeed many seniors from Peking University in Princeton, so Fang Chao wanted to go to Princeton through Peking University.

But because of this, he also fell into a misunderstanding, that is, if he wants to go to Princeton, is there really only one way to go to Peking University? No, Qingbei is okay too!

Besides, is Princeton's recruitment of students based on your Peking University's face?

Apparently not!

As long as you have the ability, then you can get through Princeton's admissions anywhere.

Iron striking still needs its own hardness, not to mention, Fang Chao's own level has no problems, so what is the difference between going to Peking University or Qingbei?

Fang Chao said, "Brother Lang, I will think carefully about what you say, maybe Qingbei is indeed a good choice." ”

()