On the A4-sized piece of paper, there are three questions.

All three questions have traces of being circled.

Professor Lu naturally did not know in advance that Cheng Nuo was going to come to him to apply for exemption.

So......

He had seemingly pulled out a random question from a stack of materials on his desk. It's not specially prepared for Cheng Nuo.

Judging from the traces of the circle of drawings on the paper, these three questions have been done once.

And that person is likely to be Professor Lu sitting in front of him.

However, figuring out this matter is of no use to Cheng Nuo's current situation.

Regardless of how these three questions came about and who had done them, Cheng Nuo had to make one of these three questions if he wanted Professor Lu to sign the application form.

Choose one of three and get it right!

With Professor Lu's character, being able to put forward such a condition is enough to prove that the three questions on the piece of paper that Cheng Nuo is holding in his hand are by no means idle!

Its power can definitely kill tens of thousands of scumbags in an instant!

Cheng Nuo can't be allowed to be uncautious.

Cheng Nuo looked at Professor Lu, who was sitting at the desk, stepped forward and said, "Teacher, I didn't bring my schoolbag, can you borrow a pen and scratch paper?"

Professor Lu put down the pen, looked up at Cheng Nuo, who had a harmless smile on his face, bent down, opened the drawer of his desk, and handed the pen and scratch paper to Cheng Nuo.

He pointed to a desk next to him, "You can do it over there, call me when you're done." ”

With that, he lowered his head again and continued the work in his hands.

Cheng Nuo was also obedient, took a pen and scratch paper, walked to the desk that Professor Lu pointed out, pulled a chair and sat down.

The A4 paper with three questions listed was also laid flat on the table by Cheng Nuo.

Cheng Nuo looked at the three questions in turn and decided which one to choose as a breakthrough.

Question 1: [Known elliptical cylindrical surface S.

r(u，v)={acosu，bsinu，v}，-π≤u≤π，﹣∞≤v≤＋∞

(1): Find the equation for any geodesic on S.

(2): Let a=b, take p=(a,0,0), Q=r(u,v)={acosu0,bsinu0,v0},-π≤u0≤π,-∞≤v0≤+∞, and write the shortest curve equation connecting P and Q points on S. 】

Question 2: [Derive the calculation format of the conjugate gradient method for solving systems of linear equations, and prove that the format converges after finite step iterations.] 】

Question 3: [Let f(x) be second-order derivable on [0,1], and f(0)=f(1)=0, min(0≤x≤1)f(x)=-1.

Proof that there is a η∈(0,1) such that f(η)" 8. 】

After reading these three questions from beginning to end, Cheng Nuo's brows furrowed.

The first question is a very comprehensive question.

Elliptic equations, trigonometric functions, differential equations, vector operations.

The combination of the four aspects of the content also leads to the super difficulty of this question.

Solving the first question requires knowledge of vectors and trigonometric functions, which is not difficult for Cheng Nuo.

But the second question is that the main thing needed is the knowledge of ordinary differential equations.

Ordinary differential equations are actually covered in the last chapter of the first volume of Advanced Mathematics, which Professor Lu is teaching.

However, it is originally a basic mathematics teaching book, and the content of advanced mathematics is only some of the most basic and simple solutions, just scratching the surface.

Maybe not even the fur.

In the Department of Mathematics, when I was a sophomore, I had a professional course called "Ordinary Differential Equations", which was dedicated to explaining this kind of equation in detail. Cheng Nuo took classes with the mathematics department of his freshman year this year, so naturally he hasn't learned yet.

Judging from the only knowledge that Cheng Nuo has so far, the second question should be solved by using the Pika-Lindelof theorem for solving ordinary differential equations.

But about the pickup-Lindelof theorem, Cheng Nuo has only heard a little. The distance is used flexibly, and Cheng Nuo is still a lot of distance.

In the first question, Cheng Nuo can only give up strategically.

As for the second question, this makes Cheng Nuo even more painful.

The so-called conjugate gradient method of a system of linear equations is to obtain a large system of linear equations by differentially separating the dispersive Laplace equation.

The requirement of the problem is to continuously iterate the general format of this system of equations, and determine the orthogonal system of equations and the convergence value of the approximate equation through the recursive relationship of the residuals.

If you want to talk about the way to solve the differential equation in the first question, it can barely be regarded as something related to high numbers.

The second question has nothing to do with the content explained in high math, and it has nothing to do with half a dime!

What conjugate gradient method, Laplace equation, and residuals recursive relationship are not at all what Cheng Nuo, a freshman, should master.

And indeed, like the previous question, Cheng Nuo has only heard of these contents.

As for solving the problem, I'm sorry, Cheng Nuo really can't do it!

Originally, Cheng Nuo still thought that these three questions would be made for him, which shocked Professor Lu.

But what can I do...... Lack of strength.

However, fortunately for Cheng Nuo, the third question is quite friendly to Cheng Nuo. A special form of Taylor's formula, McLaughlin's expansion, and knowledge of Schlemyrch-Roche's remainder can be used to solve the problem perfectly.

Taylor's formula is the most complex and difficult to understand in the entire high mathematics knowledge. Countless Tianjiao have been buried here.

It is generally used to calculate errors. For general questions about Taylor's formula, you only need a simple formula to substitute.

But the question in front of Cheng Nuo is not like this.

That really needs to be expanded one by one using the Taylor formula.

Workload, quite complicated!

But compared with the first two questions, Cheng Nuo can only choose this question that tests the amount of calculation.

Let's get started!

Cheng Nuo rubbed his hands and brought a stack of scratch paper to him.

Now that you've chosen a topic, do your best to do it.

That exemption application, you must get it!

Close your eyes tightly and your thoughts raced through your head at high speed.

Half a minute later, Cheng Nuo's eyes suddenly opened, and a flash of light flashed. The corners of his mouth curled slightly, and he picked up the pen and calculated as he wrote on scratch paper.

【f(x)=f(t)/0!+f'(t)/1!*(x-a)+f''(t)/2!*(x-a)^2......

............

0=f(0)=-1+f''(t1)/2!x0^2

0=f(1)=......

And because 0≤x≤1, f(η)=max{2/x^2,2/(1-x0)^2}≥8 !]

Get!

It took more than ten minutes for Cheng Nuo to list the formula for a whole piece of A4 paper, and finally calculated the problem.

At that moment, I was full of a sense of accomplishment.

After checking it and confirming that there were no problems, Cheng Nuo put on the cap of his pen, picked up his answer, and got up and walked to Professor Lu.

"Professor, I'm done. Cheng Nuo spoke softly.

Professor Lu looked up at Cheng Nuo first, then raised his wrist to look at the time.

His slightly serious face also showed a slightly surprised expression.

Obviously, Cheng Nuo's speed exceeded his expectations.

He looked up and down seriously, but instead of rushing to receive the answer written by Cheng Nuo, he asked with a smile, "What questions did you do?"

"The third course. Cheng Nuo replied honestly.

"Do you know where I got these three questions from?" Professor Lu began.

Cheng Nuo shook his head.

Professor Lu asked to spit out a sentence, "These are the three questions in the final finale of the third and fourth grade finals of the National College Student Mathematics Competition last year. ”

"That time, not a single student was able to get all three questions right. ”