From b9c83265a678b4249d0b6cefd2177a9648b87380 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Wed, 16 Jun 2021 23:35:11 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../function-and-limit.pdf | Bin 230347 -> 329138 bytes .../function-and-limit.tex | 60 ++--- .../derivatives-and-differential.pdf | Bin 142834 -> 202285 bytes .../derivatives-and-differential.tex | 28 ++- ...heorem-and-applications-of-derivatives.pdf | Bin 147302 -> 219466 bytes ...heorem-and-applications-of-derivatives.tex | 10 +- ...efinite-integral-and-definite-integral.pdf | Bin 171112 -> 254142 bytes ...efinite-integral-and-definite-integral.tex | 16 +- .../differential-equation.pdf | Bin 100184 -> 145286 bytes .../differential-equation.tex | 12 +- .../knowledge/1-determinant/determinant.pdf | Bin 77862 -> 115890 bytes .../knowledge/1-determinant/determinant.tex | 53 ++--- 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b/advanced-math/knowledge/3-differential-mean-value-theorem-and-applications-of-derivatives/differential-mean-value-theorem-and-applications-of-derivatives.tex @@ -258,9 +258,9 @@ $\therefore \dfrac{y^{(6)}(0)}{6!}=-\dfrac{1}{6}\Rightarrow y^{(6)}(0)=-5!=-120$ $\lim\limits_{x\to 0}\dfrac{x-\sin x}{x^3}=\lim\limits_{x\to 0}\dfrac{x-\left[x-\dfrac{1}{6}x^3+o(x^3)\right]}{x^3}=\lim\limits_{x\to 0}\dfrac{\dfrac{1}{6}x^3+o(x^3)}{x^3}=\dfrac{1}{6}$ -\subsubsection{\texorpdfstring{$A-B$}\ 型,幂次最低} +\subsubsection{\texorpdfstring{$A-B$}\ 型,幂次最高} -将$A$,$B$分别展到他们系数不相等的$x$的最低次幂为止。 +将$A$,$B$分别展到他们系数不相等的$x$的最高次幂为止。 如已知当$x\to 0$时,$\cos x-e^{-\frac{x^2}{2}}$与$ax^k$为等价无穷小,求$a$,$b$。 @@ -484,7 +484,7 @@ $\forall x\in U(x_0,\delta)$恒有$f(x)\leqslant f(x_0)$,则$f(x)$在$x_0$取 曲率\textcolor{violet}{\textbf{定义:}}表明曲线在某一点的弯曲程度的数值,针对曲线上某个点的切线方向角对弧长的转动率,通过微分来定义,表明曲线偏离直线的程度。曲率越大,表示曲线的弯曲程度越大。 -曲率的倒数就是曲率半径。\medskip +曲率的倒数就是\textbf{曲率半径}。\medskip \begin{minipage}{0.5\linewidth} 两点切线改变角相同时,弯曲程度与两点之间的弧长度成反比。 @@ -548,12 +548,12 @@ $\forall x\in U(x_0,\delta)$恒有$f(x)\leqslant f(x_0)$,则$f(x)$在$x_0$取 $\therefore\dfrac{\textrm{d}\alpha}{\textrm{d}x}=\dfrac{y''}{1+y'^2}\Rightarrow\textrm{d}\alpha=\dfrac{y''}{1+y\,'^2}\textrm{d}x$。 -$\therefore k=\left\lvert\dfrac{\textrm{d}\alpha}{\textrm{d}s}\right\rvert=\dfrac{\vert y''\vert}{(1+y'^2)^{\frac{3}{2}}}$。 +$\therefore \textcolor{aqua}{\textbf{定理:}}k=\left\lvert\dfrac{\textrm{d}\alpha}{\textrm{d}s}\right\rvert=\dfrac{\vert y''\vert}{(1+y'^2)^{\frac{3}{2}}}$。 \subsection{曲率半径} \begin{minipage}{0.5\linewidth} - $\bigcirc O$为函数$L$在点$X$处的曲率圆,该圆与$L$在$X$处相切,切线为$T$。 + $\odot\,O$为函数$L$在点$X$处的曲率圆,该圆与$L$在$X$处相切,切线为$T$。 该点的曲率半径为$R$,其中$R=\dfrac{1}{K}$。 \end{minipage} diff --git a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf index 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D$,如果存在某正数$M$,对于任一$x\in I$,有$\vert f(x)\vert\leqslant M$,则$f(x)$在$I$上有界,否则无界。 +函数指明定义域区间才能讨论函数是否有界。 如果$f(x)\geqslant M$有下界,$f(x)\leqslant M$则有上界。 \subsubsection{单调性} -$y=f(x)\,x\in D$,如果$\forall x_1,x_2\in D$且$x_1\dfrac{1}{\varepsilon}$。 @@ -303,9 +301,7 @@ $\therefore$证明完毕。 \textbf{例题:}用定义证明$\lim\limits_{n\to\infty}q^n=0$($q$为常数且$\vert q\vert<1$)。 -证明: - -\ding{172}$\vert q^n-0\vert<\varepsilon$。 +证明:\ding{172}$\vert q^n-0\vert<\varepsilon$。 \ding{173}$\vert q^n\vert<\varepsilon$,取对数进行反解$n\ln\vert q\vert<\ln\varepsilon$,又因为$\vert q\vert<1$,所以$\ln\vert q\vert<0$,所以得到$n>\dfrac{\ln\varepsilon}{\ln\vert q\vert}$。(若$\varepsilon>1$则$n$就是负数,这样条件必然成立) @@ -317,9 +313,9 @@ $\therefore$当$n>N$时,必然$n>\dfrac{\ln\varepsilon}{\ln\vert q\vert}$, \subsubsection{数列绝对值} -\textbf{例题:}证明若$\lim\limits_{x\to\infty}a_n=A$,则$\lim\limits_{x\to\infty}\vert a_n\vert=\vert A\vert$。 +\textcolor{aqua}{\textbf{定理:}}若$\lim\limits_{x\to\infty}a_n=A$,则$\lim\limits_{x\to\infty}\vert a_n\vert=\vert A\vert$。 -因为$\lim\limits_{n\to\infty}a_n=A\Leftrightarrow\forall\varepsilon>0,\exists N>0,\text{当}n>N$,恒有$\vert a_n-A\vert<\varepsilon$。 +证明:$\because\lim\limits_{n\to\infty}a_n=A\Leftrightarrow\forall\varepsilon>0,\exists N>0,\text{当}n>N$,恒有$\vert a_n-A\vert<\varepsilon$。 又由重要不等式$\vert\vert a\vert-\vert b\vert\vert\leqslant\vert a-b\vert$,所以$\vert\vert a_n-\vert A\vert\vert\leqslant\varepsilon$。 @@ -350,9 +346,7 @@ $\therefore$当$n>N$时,必然$n>\dfrac{\ln\varepsilon}{\ln\vert q\vert}$, \textcolor{violet}{\textbf{定义:}}若数列$\{x_n\}$收敛于$a$,则$a$是唯一的。 -证明: - -设$\lim\limits_{n\to\infty}a_n=A$且$\lim\limits_{n\to\infty}a_n=B$且$A\neq B$。 +证明:设$\lim\limits_{n\to\infty}a_n=A$且$\lim\limits_{n\to\infty}a_n=B$且$A\neq B$。 不如设$A>B$。任意取$\varepsilon=\dfrac{A-B}{2}>0$。 @@ -378,9 +372,7 @@ $\therefore\exists N_2>0$,当$n>N_2$时,$\vert a_n-B\vert<\dfrac{A-B}{2}$。 即$\lim\limits_{n\to\infty}a_n=A$,则存在$M>0$,使得$\vert a_n\vert\leqslant M$。 -证明: - -由极限定义,取$\varepsilon=1$。 +证明:由极限定义,取$\varepsilon=1$。 $\because\lim\limits_{n\to\infty}a_n=A$ @@ -408,9 +400,7 @@ $\forall n$,有$\vert a_n\vert\leqslant M$ 推论,戴帽法:若数列$\{a_n\}$从某项开始$a_n\geqslant b$,且$\lim\limits_{n\to\infty}a_n=a$,则$a\geqslant b$。这里一定要带等号。 -证明: - -设$A>0$,取$\varepsilon=\dfrac{A}{2}>0$。 +证明:设$A>0$,取$\varepsilon=\dfrac{A}{2}>0$。 $\because\lim\limits_{n\to\infty}a_n=A$ @@ -420,7 +410,7 @@ $\therefore\exists N>0$,当$n>N$时,$\vert a_n-A\vert<\dfrac{A}{2}\Rightarro \subsection{海涅定理(归结原则)} -设$f(x)$在$\mathring{U}(x_0,\delta)$内有定义,则$\lim\limits_{x\to x_0}f(x)=A$存在$\Leftrightarrow$对任何$\mathring{U}(x_0,\delta)$内以$x_0$为极限的数列$\{x_n\}(x_n\neq x_0)$,极限$\lim\limits_{n\to\infty}f(x_n)=A$存在。 +\textcolor{aqua}{\textbf{定理:}}设$f(x)$在$\mathring{U}(x_0,\delta)$内有定义,则$\lim\limits_{x\to x_0}f(x)=A$存在$\Leftrightarrow$对任何$\mathring{U}(x_0,\delta)$内以$x_0$为极限的数列$\{x_n\}(x_n\neq x_0)$,极限$\lim\limits_{n\to\infty}f(x_n)=A$存在。 海涅定理用来连接数列极限与函数极限。在极限存在下他们可以相互转换。 @@ -452,7 +442,7 @@ $\therefore e^{\lim\limits_{x\to 0}\frac{1}{x^2}\ln\frac{\tan x}{x}}=e^{\lim\lim \subsubsection{极限定义} -设函数$f(x)$在点$x_0$的某一个去心邻域有定义,若存在常数$A$,对于任意给定的$\varepsilon>0$,总存在正数$\delta$,使得当$0<\vert x-x_0\vert<\delta$式,对应的函数值$f(x)$都满足不等式$\vert f(x)-A\vert <\varepsilon$,则$A$就是函数$f(x)$当$x\to x_0$时的极限,记作$\lim\limits_{x\to x_0}f(x)=A$或$f(x)\rightarrow A(x\rightarrow x_0)$。 +\textcolor{violet}{\textbf{定义:}}设函数$f(x)$在点$x_0$的某一个去心邻域有定义,若存在常数$A$,对于任意给定的$\varepsilon>0$,总存在正数$\delta$,使得当$0<\vert x-x_0\vert<\delta$式,对应的函数值$f(x)$都满足不等式$\vert f(x)-A\vert <\varepsilon$,则$A$就是函数$f(x)$当$x\to x_0$时的极限,记作$\lim\limits_{x\to x_0}f(x)=A$或$f(x)\rightarrow A(x\rightarrow x_0)$。 写成$\varepsilon-\delta$语言:$\lim\limits_{x\to x_0}f(x)=A\Leftrightarrow\forall\varepsilon>0,\exists\delta>0,\text{当}0<\vert x-x_0\vert<\delta$时,有$\vert f(x)-A\vert\varepsilon$。 @@ -503,11 +493,11 @@ $\therefore e^{\lim\limits_{x\to 0}\frac{1}{x^2}\ln\frac{\tan x}{x}}=e^{\lim\lim \subsubsection{唯一性} -若极限存在,则极限唯一。 +\textcolor{violet}{\textbf{定义:}}若极限存在,则极限唯一。 \subsubsection{局部有界性} -若极限存在且为$A$,则存在正常数$M$和$\delta$,使得当$0<\vert x-x_0\vert<\delta$时,有$\vert f(x)\vert\leqslant M$。 +\textcolor{violet}{\textbf{定义:}}若极限存在且为$A$,则存在正常数$M$和$\delta$,使得当$0<\vert x-x_0\vert<\delta$时,有$\vert f(x)\vert\leqslant M$。 \begin{enumerate} \item 极限存在是函数局部有界性的充分不必要条件。 @@ -571,13 +561,11 @@ $\lim\limits_{x\to -1}\dfrac{\vert x\vert\sin(x-2)}{x(x-1)(x-2)^2}=-\dfrac{\sin \subsubsection{局部保号性} -若极限存在,则存在常数$\delta>0$,使得当$0<\vert x-x_0\vert<\delta$时,$f(x)$与$A$同号。 +\textcolor{violet}{\textbf{定义:}}若极限存在,则存在常数$\delta>0$,使得当$0<\vert x-x_0\vert<\delta$时,$f(x)$与$A$同号。 简单来说,函数值在$x\to x_0$时函数值与极限值同号。 -证明局部保号性: - -首先根据极限存在定义:$\forall\varepsilon>0,\exists\delta>0,0<\vert x-x_0\vert<\delta$时,恒有$\vert f(x)-A\vert<\varepsilon$。 +证明:首先根据极限存在定义:$\forall\varepsilon>0,\exists\delta>0,0<\vert x-x_0\vert<\delta$时,恒有$\vert f(x)-A\vert<\varepsilon$。 $\Rightarrow -\varepsilon0$,$\exists N$,当$n>N$时,$\vert y_n<\varepsilon$,$\vert z_n<\varepsilon$。 @@ -877,9 +863,7 @@ $\therefore\lim\limits_{x\to 0}\dfrac{\sin x}{x}=1$。 书上通过数列进行单调有界定理证明极限存在性。 -证明: - -$\lim\limits_{x\to\infty}\left(1+\dfrac{1}{x}\right)^x=\lim\limits_{x\to\infty}e^{\ln(1+\frac{1}{x})^x}=\lim\limits_{x\to\infty}e^{x\ln(1+\frac{1}{x})}=e^{\lim\limits_{x\to\infty}x\ln(1+\frac{1}{x})}=e^{\lim\limits_{x\to\infty}\frac{\ln(1+\frac{1}{x})}{\frac{1}{x}}}$ +证明:$\lim\limits_{x\to\infty}\left(1+\dfrac{1}{x}\right)^x=\lim\limits_{x\to\infty}e^{\ln(1+\frac{1}{x})^x}=\lim\limits_{x\to\infty}e^{x\ln(1+\frac{1}{x})}=e^{\lim\limits_{x\to\infty}x\ln(1+\frac{1}{x})}=e^{\lim\limits_{x\to\infty}\frac{\ln(1+\frac{1}{x})}{\frac{1}{x}}}$ $=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\left(-\frac{1}{x^2}\right)}{-\frac{1}{x^2}}}=e^{\lim\limits_{x\to\infty}\frac{1}{1+x}}=e$\medskip @@ -971,7 +955,7 @@ $=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\lef \subsubsection{可去间断点(可补间断点)} -若$\lim\limits_{x\to x_0}f(x)=A\neq f(x_0)$(甚至可以没有定义)。 +\textcolor{violet}{\textbf{定义:}}若$\lim\limits_{x\to x_0}f(x)=A\neq f(x_0)$(甚至可以没有定义)。 \begin{tikzpicture} \draw[-latex](-0.5,0) -- (3,0) node[below]{$x$}; @@ -988,7 +972,7 @@ $=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\lef \subsubsection{跳跃间断点} -若$\lim\limits_{x\to x_0^-}f(x)$与$\lim\limits_{x\to x_0^+}f(x)$都存在,但是$\lim\limits_{x\to x_0^-}f(x)\neq\lim\limits_{x\to x_0^+}f(x)$。 +\textcolor{violet}{\textbf{定义:}}若$\lim\limits_{x\to x_0^-}f(x)$与$\lim\limits_{x\to x_0^+}f(x)$都存在,但是$\lim\limits_{x\to x_0^-}f(x)\neq\lim\limits_{x\to x_0^+}f(x)$。 \begin{tikzpicture} \draw[-latex](-0.5,0) -- (3,0) node[below]{$x$}; @@ -1008,7 +992,7 @@ $=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\lef \subsubsection{无穷间断点} -若$\lim\limits_{x\to x_0}f(x)=\infty$,或至少一个方向为无穷大(定义分歧)。如$y=\dfrac{1}{x}$在$x=0$处为无穷间断点。 +\textcolor{violet}{\textbf{定义:}}若$\lim\limits_{x\to x_0}f(x)=\infty$,或至少一个方向为无穷大(定义分歧)。如$y=\dfrac{1}{x}$在$x=0$处为无穷间断点。 \begin{tikzpicture} \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; @@ -1020,7 +1004,7 @@ $=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\lef \subsubsection{振荡间断点} -若$\lim\limits_{x\to x_0}f(x)$为振荡不存在。如$\lim\limits_{x\to 0}\sin\dfrac{1}{x}$的$x=0$就是振荡间断点。 +\textcolor{violet}{\textbf{定义:}}若$\lim\limits_{x\to x_0}f(x)$为振荡不存在。如$\lim\limits_{x\to 0}\sin\dfrac{1}{x}$的$x=0$就是振荡间断点。 \begin{tikzpicture} \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; diff --git a/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.pdf b/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.pdf index 50b5f8e4f4a1862000867bbbab06b498b48aad01..3c9979ec4d10cb4d3ddc918ca66c7d94ad245cb4 100644 GIT binary patch literal 202285 zcma&NL#!^)wyis@wWe*`wr$(CZQHhO{nNH>+qTzy`zEh&PVO!2WE4haRC-!VzSh$Q zse*_YEh8N(4C(Rx%{vS;3jqUxy^$3R4-dVVrHzZJ6TO&?p^K@Asj}u-xmDd_0!em`=XDZ9iQLG zXz-r_>8EZ>g^AMhY)I^%*Yba3HHEv9V@yY-dmee-Ui(K5y=Ztn~A;%Y9``0y*Lbw_JHe#&(Schv8J+rQLUh zvCmfk?KE}>x=0(LXt+(a`#qgd91!Gw-G3+lsp}HiOXKQuhoB0OYn;XTI{bYO>vKN+ 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