From 00dcdf1f6fe7f0a957e9fa6414d8785e92e0f900 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Mon, 17 Oct 2022 23:11:48 +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 --- ...iation-of-functions-of-single-variable.pdf | Bin 151164 -> 157075 bytes ...iation-of-functions-of-single-variable.tex | 38 ++++++++++++++++ ...ntegal-of-functions-of-single-variable.pdf | Bin 271181 -> 271238 bytes ...ntegal-of-functions-of-single-variable.tex | 2 + .../function-and-limit.pdf | Bin 243666 -> 245769 bytes .../function-and-limit.tex | 43 +++++++++++------- .../applications-of-derivatives.pdf | Bin 155457 -> 157236 bytes .../applications-of-derivatives.tex | 31 ++++++++----- ...efinite-integral-and-definite-integral.pdf | Bin 209922 -> 211799 bytes ...efinite-integral-and-definite-integral.tex | 12 +++++ 10 files changed, 98 insertions(+), 28 deletions(-) diff --git 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+$v'=(x+1)^{-1}$,$v''=(-1)(x+1)^{-2}$,$v^{(n-2)}=\dfrac{(-1)^{n-1}(n-3)!}{(x+1)^{n-2}}$。 + +$v^{(n-2)}(0)=(-1)^{n-1}(n-3)!$,$f^{(n)}(0)=\dfrac{n(n-1)}{2!}2(-1)^{n-1}(n-3)!=\dfrac{(-1)^{n+1}n!}{n-2}$。 + +\subsubsection{归纳法} + +\textbf{例题:}已知$f(x)=\dfrac{x}{x^2-5x+6}$, 求$f^{(n)}(x)$。 + +解:首先式子为多项式形式,所以先因式分解。 + +$f(x)=\dfrac{x}{(x-2)(x-3)}=\dfrac{3}{x-3}-\dfrac{2}{x-2}$。 + +所以根据高阶导数性质$f^{(n)}(x)=\left(\dfrac{3}{x-3}\right)^{(n)}-\left(\dfrac{2}{x-2}\right)^{(n)}$。 + +令$\psi(x)=\dfrac{1}{x-1}=(x-3)^{-1}$,$\psi'(x)=(-1)(x-3)^{-2}$,$\psi''(x)=(-1)(-2)(x-3)^{-3}$,归纳得出$\psi^{(n)}(x)=(-1)^nn!(x-3)^{-(n+1)}$。 + +然后直接将3换成2,得出$f^{(n)}(x)=\dfrac{3(-1)^nn!}{(x-3)^{n+1}}-\dfrac{2(-1)^nn!}{(x-2)^{n+1}}$。 + +\textbf{例题:}已知$f(x)=e^x\sin x$, 求$f^{(n)}(x)$。 + +解:令$f'(x)=e^x\sin x+e^x\cos x=e^x(\sin x+\cos x)=\sqrt{2}e^x\sin\left(x+\dfrac{\pi}{4}\right)$。 + +所以这里就使用三角公式,从而$f^{(n)}(x)=(\sqrt{2})^ne^x\sin\left(x+n\cdot\dfrac{\pi}{4}\right)$。 + +\subsubsection{莱布尼茨公式} + +莱布尼茨公式是基于某个点的,所以题型是求$f(x)$在点$x_0$处的导数,该点的导数就是该函数的泰勒级数的系数。如果题目是直接求$f^{(n)}x$则用不了莱布尼茨公式。 + \subsubsection{高阶导数存在性} \subsubsection{携带未知数的多项式求高阶导} diff --git a/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.pdf b/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.pdf index 8089c5dc7d631a399c9a9dfb5f180bd138fdbc6c..9b7a45315121c4c8c5d89349b4316caaf19bc5a0 100644 GIT binary patch delta 47496 zcmY&5Tu6^OUgfyoeUYLdI!XVcmz36sy=7Mbh@juQ!_gswDsUnkW z72=sC3Be%H1RmxceJ}XL+kV;Q*4;tN4^l#LL}V@qI_+lqe0BSFZ+_Oq86Z#1^t3_s z);_NiFd%9&!T!2`@TuZ<0ltjwxU|oc2-c4aMs{BgftrEqJA8+4{J~)n6!6cf#UfTE zvCP^a6F~3>zDfJm%lU4f(k!=VYJJ|z!{L~Yj*G-(io2~_^Gjp!oFT%$g?a9t1|gnX z9U<*_1_JCN9UA<2Be;J|TNwoQt?t1`MmIHUj(hPLYLxD&S2LJ zkTvy>lrmhiqFTp8?$e5G&1=|nO^2p|nF-5|b&aVTxarDd4RjtYqAUqoI^VfoT}@}> zW?bq5;`tqhwio9(jc8b0Nxs^@rAY^aHfFhpf#r~0d4_nrIf!egOmC8Hn7GL)F7xB^ 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a/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.tex b/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.tex index e6abe62..a9f2205 100644 --- a/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.tex +++ b/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.tex @@ -1265,6 +1265,8 @@ $\rho gA(y)\,\textrm{d}y=\rho g\pi(16-y^2)\,\textrm{d}y$ \section{积分关系式} +基本上要使用到积分中值定理。 + \subsection{积分等式} 包括证明带有积分的等式、方程根、积分中值定理等。 diff --git a/advanced-math/knowledge/1-function-and-limit/function-and-limit.pdf b/advanced-math/knowledge/1-function-and-limit/function-and-limit.pdf index 13734684bb8501b9e6bc7b781229173a3e7aafcf..870e8f8adcd9dbb5ce4209ce1342c924cc9f296a 100644 GIT binary patch delta 184625 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a/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex +++ b/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex @@ -80,7 +80,7 @@ \section{微分中值定理} -三个定理都是建立局部与整体的关系,利用导数控制函数,反之不能使用函数控制导数。 +四个定理都是建立局部与整体的关系,利用导数控制函数,反之不能使用函数控制导数。 $\text{罗尔定理}\xrightleftharpoons[\text{特例:}f(a)=f(b)]{\text{泛化:任意端点值}}\text{拉格朗日中值定理}\xrightleftharpoons[\text{特例:}F(x)=x]{\text{泛化:参数方程}}\text{柯西中值定理}$ @@ -145,7 +145,7 @@ $\text{罗尔定理}\xrightleftharpoons[\text{特例:}f(a)=f(b)]{\text{泛化 有限增量公式\textcolor{violet}{\textbf{定义:}}$\Delta y=f(x_0+\Delta x)-f(x_0)=f'[x_0+\theta\Delta x]\Delta x(0<\theta<1)$。 -有限增量公式中的$\Delta x$不一定很小,这个是一个增量的准确公式。 +有限增量公式中的$\Delta x$不一定很小,这个是一个增量的准确公式。即将增量$\Delta y$用$\Delta x$和该线段上某点的导数来表示。 推论:$f(x)$在$I$上连续且可导,则$I$上$f(x)=C\Leftrightarrow f'(x)\equiv 0$ @@ -166,17 +166,13 @@ $\text{罗尔定理}\xrightleftharpoons[\text{特例:}f(a)=f(b)]{\text{泛化 则$\exists\,\xi\in(a,b)$,使得$\dfrac{f(b)-f(a)}{F(b)-F(a)}=\dfrac{f'(\xi)}{F'(\xi)}$。 -\section{积分中值定理} +\subsection{泰勒中值定理} -\subsection{定理} +即携带拉格朗日余项的泰勒公式。 -\textcolor{aqua}{\textbf{定理:}}若$f(x)$在$[a,b]$上连续,则存在$\xi\in[a,b]$,使得$\int_a^bf(x)\,\textrm{d}x=f(\xi)(b-a)$。 +设$f(x)$在区间$I$上$(n+1)$阶可导,$x_0\in I$,那么对$\forall x\in I$,$\exists\xi$使得$f(x)=f(x_0)+f'(x_0)(x-x_0)+\cdots+\dfrac{f^{(n)}(x_0)}{n!}(x-x_0)^n+R_n(x)$。 -\subsection{证明} - -已知$f(x)$在$[a,b]$上连续,根据有界与最值定理,$m\leqslant f(x)\leqslant M$,$m(b-a)\leqslant\int_a^bf(x)\,\textrm{d}x\leqslant M(b-a)$,所以$m\leqslant\dfrac{1}{b-a}\int_a^bf(x)\,\textrm{d}\leqslant M$。 - -由介值定理可知$\xi\in[a,b]$,使得$f(\xi)=\dfrac{1}{b-a}\int_a^bf(x)\,\textrm{d}x$。 +$R_n(x)=\dfrac{f^{(n+1)(\xi)}}{(n+1)!}(x-x_0)^{n+1}$,$\xi\in I$。 \section{洛必达法则} @@ -420,7 +416,7 @@ $\forall x\in U(x_0,\delta)$恒有$f(x)\leqslant f(x_0)$,则$f(x)$在$x_0$取 \textcolor{aqua}{\textbf{定理:}}(极值第一充分条件) -设$f(x)$在$\mathring{U}(x_0,\delta)$内可导,且$f'(x_0)=0$或在$x_0$连续。 +若$f(x)$在$\mathring{U}(x_0,\delta)$内可导,且$f'(x_0)=0$或在$x_0$连续。 \begin{enumerate} \item 若$xx_0$时$f'(x)\leqslant 0$,则$x_0$取得极大值。 @@ -437,6 +433,15 @@ $\forall x\in U(x_0,\delta)$恒有$f(x)\leqslant f(x_0)$,则$f(x)$在$x_0$取 \item 当$f''(x_0)>0$,则$f(x)$在$x_0$取极小值。 \end{enumerate} +\textcolor{aqua}{\textbf{定理:}}(极值第三充分条件) + +若$f(x)$在$x_0$处$n(n\geqslant2)$阶可导,且$f'(x_0)=f''(x_0)=\cdots=f^{(n-1)}(x_0)=0$,$f^{(n)}(x_0)\neq0$,则: + +\begin{enumerate} + \item 当$n$为偶数时$f(x)$在$x_0$取得极值。当$f^{(n)}(x_0)<0$,则$f(x)$在$x_0$取极大值,当$f^{(n)}(x_0)>0$,则$f(x)$在$x_0$取极小值。 + \item 当$n$为奇数时$f(x)$在$x_0$处无极值。 +\end{enumerate} + \subsection{函数最值} \subsubsection{连续函数闭区间最值} @@ -474,6 +479,8 @@ $\forall x\in U(x_0,\delta)$恒有$f(x)\leqslant f(x_0)$,则$f(x)$在$x_0$取 \item 若$\lim\limits_{x\to\infty}\dfrac{f(x)}{x}=a,b=\lim\limits_{x\to\infty}(f(x)-ax)$,那么$y=ax+b$就是斜渐近线。 \end{itemize} +考的比较多的是斜渐进性,计算较复杂,如果能写成$y=f(x)=ax+b+o(x)$,$o(x)$为$x\to\infty$的高阶无穷小,则能快速得到斜渐进线。 + \section{弧微分与曲率} \subsection{弧微分} @@ -607,6 +614,8 @@ $\therefore\dfrac{\textrm{d}\alpha}{\textrm{d}x}=\dfrac{y''}{1+y'^2}\Rightarrow\ $\therefore 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b/advanced-math/knowledge/1-function-and-limit/function-and-limit.tex index 2176171..c39a2da 100644 --- a/advanced-math/knowledge/1-function-and-limit/function-and-limit.tex +++ b/advanced-math/knowledge/1-function-and-limit/function-and-limit.tex @@ -910,7 +910,9 @@ $=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\lef \section{无穷大与无穷小} -\subsection{无穷定义} +\subsection{无穷} + +\subsubsection{无穷定义} 无穷小\textcolor{violet}{\textbf{定义:}}当$x\to x_0(\infty)$时,函数$f(x)$极限为0,就称$f(x)$为当$x\to x_0(\infty)$时的无穷小,记为:$\lim\limits_{x\to x_0(\infty)}f(x)=0$。 @@ -930,7 +932,7 @@ $=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\lef \textcolor{aqua}{\textbf{定理:}}若同一极限过程中,$f(x)$为无穷大,则$\dfrac{1}{f(x)}$为无穷小,反之若$f(x)$为无穷小且不为0,则$\dfrac{1}{f(x)}$为无穷大。 -\subsection{无穷小的比较} +\subsubsection{无穷小的比较} 设在自变量同一变化过程中,$\lim\alpha(x)=0$,$\lim\beta(x)=0$,且$\beta(x)\neq 0$,则: @@ -954,7 +956,7 @@ $=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\lef 利用二项式展开:$=\lim\limits_{t\to 0}\dfrac{t}{nt+\dfrac{n(n-1)}{2}t^2+\cdots}=\dfrac{1}{n}$。 -\subsection{无穷小运算} +\subsubsection{无穷小运算} 设$m$,$n$为正整数: @@ -982,6 +984,8 @@ $=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\lef 是一个用函数在某点的信息描述其附近取值的公式。如果函数满足一定的条件,泰勒公式可以用函数在某一点的各阶导数值做系数构建一个多项式来近似表达这个函数,即用多项式拟合不规则曲线。 +\subsubsection{麦克劳林公式} + \begin{enumerate} \item $e^x=\sum\limits_{i=0}^n\dfrac{1}{i!}x^i$,$=1+\dfrac{1}{1!}x+\dfrac{1}{2!}x^2+\dfrac{1}{3!}x^3+o(x^3)$。 \item $\ln(1+x)=\sum\limits_{i=1}^n(-1)^{i+1}\dfrac{1}{i}x^i$,$=x-\dfrac{1}{2}x^2+\dfrac{1}{3}x^3+o(x^3)$。 @@ -992,33 +996,38 @@ $=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\lef \item $(1+x)^a=1+\sum\limits_{i=1}^n\dfrac{\prod_{j=1}^i(a-j+1)}{i!}x^i$,$=1+\dfrac{a}{1!}x+\dfrac{a(a-1)}{2!}x^2\\+\dfrac{a(a-1)(a-2)}{3!}x^3+o(x^3)$。 \end{enumerate} -\subsection{常用等价无穷小} +\subsubsection{常用等价无穷小} \textcolor{aqua}{\textbf{定理:}}若$\alpha\sim\alpha_1$,$\beta\sim\beta_1$,则$\lim\dfrac{\alpha}{\beta}=\lim\dfrac{\alpha_1}{\beta}=\lim\dfrac{\alpha}{\beta_1}=\lim\dfrac{\alpha_1}{\beta_1}$。 所以可以使用等价无穷小替换对应式子,这些等价无穷小都是使用泰勒展开得到的。等价无穷小只是泰勒公式在某个固定阶数上(通常为一阶)的特例。 -泰勒一般用于替换整体,等价无穷小一般用于替换部分式子。 - -\textcolor{red}{警告:}一般只有所替换的式子为乘除的整个因子才能替换,加减一般都不能替换,如$x-\sin x\sim\dfrac{1}{6}x^3$。 - 通过麦克劳林公式可以得到当$x\to 0$时的相应等价无穷小: \begin{enumerate} - \item $\sin x\sim x$。 - \item $\tan x\sim x$。 - \item $\arcsin x\sim x$。 - \item $\arctan x\sim x$。 - \item $\ln(1+x)\sim x$。 - \item $e^x-1\sim x$。 + \item $x\sim\sin x\sim\tan x\sim\arcsin x\sim\arctan x\sim\ln(1+x)\sim\ln(x+\sqrt{1+x^2})\sim e^x-1$。 \item $a^x-1\sim x\ln a$。 - \item $1-\cos x\sim\dfrac{1}{2}x^2$。 \item $(1+x)^a-1\sim ax$。 + \item $\log_a(1+x)\sim\dfrac{x}{\ln a}$。 + \item $1-\cos x\sim\dfrac{1}{2}x^2$。 + \item $x-\ln(1+x)\sim\dfrac{1}{2}x^2$。 + \item $x-\sin x\sim\dfrac{1}{6}x^3$。 + \item $\arcsin x-x\sim\dfrac{1}{6}x^3$。 + \item $\tan x-x\sim\dfrac{1}{3}x^3$。 + \item $x-\arctan x\sim\dfrac{1}{3}x^3$。 + \item $\tan x-\sin x\sim\dfrac{1}{2}x^3$ \end{enumerate} -还有$e^{\sin x}-e^x\sim\sin x-x\sim-\dfrac{1}{6}x^3$。 +\subsubsection{等价无穷小适用性} -其中$a\cdot x\ln x$当$x\to 0$的极限必然为0。 +如果是乘除关系可以随便换,但是加减关系需要一定条件: + +\begin{itemize} + \item 若$\alpha\sim\alpha_1$,$\beta\sim\beta_1$,且$\lim\dfrac{\alpha_1}{\beta 1}=A\neq1$,则$\alpha-\beta\sim\alpha_1-\beta_1$。 + \item 若$\alpha\sim\alpha_1$,$\beta\sim\beta_1$,且$\lim\dfrac{\alpha_1}{\beta 1}=A\neq-1$,则$\alpha+\beta\sim\alpha_1+\beta_1$。 +\end{itemize} + +即这两个和不能为0。 \section{函数连续性与间断点} diff --git a/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.pdf b/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.pdf index d206e2887465ecf2c4ac5d2e50c4b22d5e007e73..328a614db42242ad0bbc2c224c0848852da86b36 100644 GIT binary patch delta 146172 zcmZtsQ*fYP(C!V#wvEX|lVoDswr$(a9ox2ziET`5+qNdg`+s(Q2Yc7^RjsPkXMNDs 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