学委真不负责

n+e posted @ 2016年10月16日 18:46 in 日常 with tags 数学 , 2499 阅读

微积分

$1^\infty$ 的处理

令 $$\lim_{x\to x_o} u(x)=1,\ \lim_{x\to x_o} v(x)=\infty$$

则 $$\begin{eqnarray} \lim_{x\to x_o} u(x)^{v(x)}&=&\lim_{x\to x_o} [1+(u-1)]^{\frac{1}{u-1}\cdot (u-1)\cdot v}\\&=&\exp\Big\{\lim_{x\to x_o}(u-1)\cdot v\Big\} \end{eqnarray}$$


例题:$$ \begin{eqnarray} & &\lim_{x\to \infty}(\sin \frac{1}{x}+\cos\frac{1}{x})^x\\ &= &\exp\Big\{\lim_{x\to\infty}x\cdot(\sin\frac{1}{x}+\cos\frac{1}{x}-1)\Big\}\\&=&\exp\Big\{\lim_{x\to\infty}x\cdot (\frac{1}{x}+1-1+o(\frac{1}{x}))\Big\}\\&=&\exp(1)=e\end{eqnarray}$$

还有习题课 函数连续性 第二题 秒出

(可以试一下作业第八题)

各种小量运算

大家都会泰勒展开那我就不说了

$\sin x\sim x-\frac{x^3}{6}+o(x^3)$

$\cos x\sim 1-\frac{x^2}{2}+o(x^2)$

$\tan x\sim x+\frac{x^3}{3}+o(x^3)$

$e^x\sim 1+x+\frac{x^2}{2}+\frac{x^3}{6}+o(x^3)$

$\ln(x+1)\sim x-\frac{x^2}{2}+\frac{x^3}{3}+o(x^3)$

$(1+x)^\alpha\sim 1+\alpha x+\frac{\alpha^2-\alpha}{2}x^2+o(x^2)$

当 $x\to 0$ 时可以直接替代,否则不能用

Stolz定理的应用

我们知道Stolz定理就是离散洛必达,所以洛必达能干的事情Stolz应该也能干,比如 $0\cdot \infty$ 型求极限

方法差不多长这样:

$n\cdot x_n=\frac{n}{\frac{1}{x_n}}\sim \frac{n+1-n}{\frac{1}{x_{n+1}}-\frac{1}{x_n}}=\frac{1}{\frac{1}{x_{n+1}}-\frac{1}{x_n}}$

于是就能把n给消掉了。多用于 $x_{n+1}=f(x_n)$ 型的数列。


例题:设 $x_n\in (0,1), x_{n+1}=x_n(1-x_n),$ 试证明 $\lim_{n\to \infty} nx_n=1$

直接按照上面的来就好了。

练习:设$x_n\in (0,1), x_{n+1}=\sin(x_n),$ 试证明 $\lim_{n\to \infty}\sqrt{\frac{n}{3}}x_n=1$

习题课上的收获

只要是“证明xx收敛”,而没有求值的题目,多半是用柯西;否则只要把要证的东西通过放缩/迭代/奇技淫巧,变成题目里的东西就好

好题推荐

试证明$$\lim_{n\to\infty} n\sin(2\pi e n!)=2\pi$$

线代

“加边法”

线代第二次作业P36/11(1)有无脑做法。对4班那次推送忍了很久了,因此点了几次右下角的投诉,理由是“常识性谣言”。

就是加一行一列不改变行列式:$$D=\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}=\begin{vmatrix}1&bla&bla\\0&a_{11}&a_{12}\\0&a_{21}&a_{22}\end{vmatrix}$$

可以自行选取 bla 的值,然后通过行列变换将原来矩阵里面的东西消掉,比如11(1)

$$\begin{eqnarray} & &\begin{vmatrix}1+a_1+b_1&a_1+b_2&\cdots&a_1+b_n\\a_2+b_1&1+a_2+b_2&\cdots&a_2+b_n\\\vdots&\vdots&\ddots&\vdots\\a_n+b_1&a_n+b_2&\cdots&1+a_n+b_n\end{vmatrix}\\ &=&\begin{vmatrix}1&-b_1&-b_2&\cdots&-b_n\\0&1+a_1+b_1&a_1+b_2&\cdots&a_1+b_n\\0&a_2+b_1&1+a_2+b_2&\cdots&a_2+b_n\\\vdots&\vdots&\vdots&\ddots&\vdots\\0&a_n+b_1&a_n+b_2&\cdots&a_n+b_n\end{vmatrix}\\ &=&\begin{vmatrix}1&-b_1&-b_2&\cdots&-b_n\\1&1+a_1&a_1&\cdots&a_1\\1&a_2&1+a_2&\cdots&a_2\\\vdots&\vdots&\vdots&\ddots&\vdots\\1&a_n&a_n&\cdots&1+a_n\end{vmatrix}\\ &=&\begin{vmatrix}1&0&0&0&\cdots&0\\0&1&-b_1&-b_2&\cdots&-b_n\\-a_1&1&1+a_1&a_1&\cdots&a_1\\\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\-a_n&1&a_n&a_n&\cdots&1+a_n\end{vmatrix}\\ &=&\begin{vmatrix}1&0&1&1&\cdots&1\\0&1&-b_1&-b_2&\cdots&-b_n\\-a_1&1&1&0&\cdots&0\\-a_2&1&0&1&\cdots&0\\\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\-a_n&1&0&0&\cdots&1\end{vmatrix}\\ &=&\begin{vmatrix}1+\sum a_i&0&1&1&\cdots&1\\-\sum a_ib_i&1&-b_1&-b_2&\cdots&-b_n\\0&1&1&0&\cdots&0\\\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\0&1&0&0&\cdots&1\end{vmatrix}\\ &=&\begin{vmatrix}1+\sum a_i&-n&1&\cdots&1\\-\sum a_ib_i&1+\sum b_i&-b_1&\cdots&-b_n\\0&0&1&\cdots&0\\\vdots&\vdots&\vdots&\ddots&\vdots\\0&0&0&\cdots&1\end{vmatrix}\\ &=&\Big(1+\sum_{i=1}^n a_i\Big)\Big(1+\sum_{i=1}^n b_i\Big)-n\sum_{i=1}^n a_ib_i \end{eqnarray}$$

程设

要讲些啥呢?

大家把平时困惑的问题,哪怕是很小很小的问题,哪怕你最后自己解决了,发到公众号里面。我们才能根据大家的需求编写推送哦~

#include <cstdio>
char a[20],len;
int check(int x){
    for(len=0;x;x/=10)a[++len]=x%10;
    for(int i=1;i<=len;i++,len--)
        if(a[i]!=a[len])return 0;
    return 1;
}
int n;
int main(){
    scanf("%d",&n);
    for(int i=1;i*i*i<=n;i++)
        if(check(i)&&check(i*i)&&check(i*i*i))
            printf("%d ",i);
    return 0;
}

我会说上面的代码是我刚睡醒在手机上直接敲的吗233

#include <cstdio>
int n,a[60]={1};
int main(){
    scanf("%d",&n);
    for(int i=1;i<n;i++)
        for(int j=i;j;j--)
            a[j]+=a[j-1];
    for(int i=0;i<n;i++)
        printf("%d ",a[i]);
    puts("");
    return 0;
}

听说周二选李宛洲的课的同学要考试?害怕

思修习题十答案

D;C;ABC;ABCD;

---By n+e
sbi bank credit card 说:
2022年10月03日 16:51

How to Apply for the SBI Credit Card Online? · Visit Finserv MARKETS. Visit us online · Fill in your details. How to Apply for the SBI Credit Card Online? · Visit Finserv MARKETS. Visit us online · Fill in your details. Help us with a few of your details. SBI Credit Card Eligibility. In order to apply for SBI Credit card offers one needs to meet the following eligibility criterias, sbi bank credit card as listed below: Income. Apply online for an SBI Credit Card. Choose from multiple variants to suit your lifestyle. Get offers Shopping. Help us with a few of your details. SBI Credit Card Eligibility. In order to apply for SBI Credit card offers one needs to meet the following eligibility criterias, as listed below: Income. Apply online for an SBI Credit Card. Choose from multiple variants to suit your lifestyle. Get offers Shopping.

apple watch einschal 说:
2023年7月25日 00:01

Wenn Sie versuchen, Ihre Apple Watch aufzuladen, es aber nicht gelingt, müssen Sie möglicherweise verstehen, dass ein Problem mit Ihrem Ladekabel oder Ladeadapter vorliegt. apple watch einschalten beim laden Dies kann daran liegen, dass Ihr Kabel gebrochen ist, dass das Ladekabel defekt ist oder dass das Ladekabel beschädigt ist, oder dass Hardwareprobleme dazu führen, dass sich das Problem mit Ihrer Apple Watch nicht einschaltet.

APSCERT 9th Model P 说:
2023年7月29日 19:10

BSEAP Board Students can Download Their Subject wise PDF Files it will be Beneficial for Study Material Practice and Regular Check Bit Bank and Question Bank, Important Questions, Students who are Attending AP 9th Class Annual Exam can make APSCERT 9th Model Paper 2024 use of this Information which we have Provided here.This Website Available AP Class Last year Exam Question Paper for Students can make a better Performance in Exam 2024.Good News AP Students who are Searching for the AP 9th Class Model Paper 2024 BSEAP Board has been Uploaded the All Subject Wise Download to the Students who are Going to Appear in the Board Examination this year.

como aceptar invita 说:
2023年8月07日 14:56

pueden ser la forma perfecta de compartir recuerdos con personas que están lejos o de trabajar en una actividad con otras personas.Álbumes compartidos es una de las funciones preinstaladas de la aplicación Fotos como aceptar invitacion album compartido iphone de iOS. Esto le permite compartir fotos y videos con una persona en particular y hacer que lo mencionen. Alguien con quien haya intercambiado fotos o videos puede cargar sus propias imágenes y videos.Después de crear un álbum compartido y enviar invitaciones, cada persona que acepte la invitación puede agregar sus propias fotos y videos al álbum.

Tripura 5th Class Q 说:
2023年8月29日 18:43

Tripura 5th Class Model Question Paper 2024 are Prepared as per the updated Board of Education in the Tripura State Syllabus to help Students in their exam Preparations, which Aims to build both the Practical knowledge as well as analytical Skills for High School Every Student, SCERT Tripura Class Question Paper 2024 are Prepared as per the updated Tripura Board Class Syllabus to Tripura 5th Class Question Paper 2024 help Students in their Exam Preparations.Tripura Class Question Paper 2024 will help a lot to the Students to Estimate their writing speed in the Examination 2024, After Completing the Preparation of entire Syllabus.

pavzi.com 说:
2024年1月06日 13:38

Pavzi.com is a startup by passionate webmasters and bloggers who have a passion for providing engaging content that is accurate, interesting, and worthy to read. pavzi.com We are more like a web community where you can find different information, resources, and topics on day-to-day incidents or news. We provide you with the finest web content on every topic possible with the help of the editorial and content team.


登录 *


loading captcha image...
(输入验证码)
or Ctrl+Enter