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a/advanced-math/knowledge/1-function-and-limit/function-and-limit.tex b/advanced-math/knowledge/1-function-and-limit/function-and-limit.tex index e0a0060..d43f1e6 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 @@ -830,24 +830,30 @@ $ \subsection{\texorpdfstring{$\lim\limits_{x\to 0}\dfrac{\sin x}{x}=1$}{}} -证明:当$x\to 0$时$x\in[0,\dfrac{\pi}{2}]$。 +证明: -\begin{tikzpicture}[scale=1.5] - \draw (0,0) circle (1); - \filldraw[black] (0,0) node[below]{$O$}; - \draw[black](0,0) -- (1,0) node[right]{$A$}; - \draw[black](0,0) -- (1/2,{sqrt(3)/2}) node[above]{$B$}; - \draw[black](1/2,{sqrt(3)/2}) -- (1/2,0) node[below]{$D$}; - \draw[black](1,0) -- (1,{sqrt(3)}) node[above]{$C$}; - \draw[black](1,0) -- (1/2,{sqrt(3)/2}); - \draw[black](1,{sqrt(3)}) -- (1/2,{sqrt(3)/2}); -\end{tikzpicture} +\begin{minipage}{0.7\linewidth} + 当$x\to 0$时$x\in[0,\dfrac{\pi}{2}]$。 -设$\angle AOB$的弧度为$x$,圆$O$的半径为$1$,则$OD=\sin x$。 + 设$\angle AOB$的弧度为$x$,圆$O$的半径为$1$,则$OD=\sin x$。 -则$S_\vartriangle AOB=\dfrac{\sin x}{2}$。根据扇形面积公式:$S_{\text{扇形}}AOB=\dfrac{x}{2}$。 + 则$S_\vartriangle AOB=\dfrac{\sin x}{2}$。根据扇形面积公式:$S_{\text{扇形}}AOB=\dfrac{x}{2}$。 -又$\because CA=\tan x$,则$S_\vartriangle AOC=\dfrac{\tan x}{2}$。 + 又$\because CA=\tan x$,则$S_\vartriangle AOC=\dfrac{\tan x}{2}$。 +\end{minipage} +\hfill +\begin{minipage}{0.2\linewidth} + \begin{tikzpicture}[scale=1.125] + \draw (0,0) circle (1); + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black](0,0) -- (1,0) node[right]{$A$}; + \draw[black](0,0) -- (1/2,{sqrt(3)/2}) node[above]{$B$}; + \draw[black](1/2,{sqrt(3)/2}) -- (1/2,0) node[below]{$D$}; + \draw[black](1,0) -- (1,{sqrt(3)}) node[above]{$C$}; + \draw[black](1,0) -- (1/2,{sqrt(3)/2}); + \draw[black](1,{sqrt(3)}) -- (1/2,{sqrt(3)/2}); + \end{tikzpicture} +\end{minipage} 根据图,在$x\in[0,\dfrac{\pi}{2}]$,$\sin xSI4Vs8AixVzd`M4qZAPo5-|7ZIan 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zD7x*~LJM)_0tvq3f)+x@A8W|;PPAY<{>Y-|5T}KpdZLNnF-;56^hYK=2VYt+KYySz zr+4EHxyJ=^dIwT>a9v-iB7Sr8?IoMcfBz5HnK^l_%BmFG;?qkfZ8!0JTGSX_6tp%c z>}2iJX?u=`ZCi1Nx9s#^4ejVqxs)AG3%5+o)$*6y^|W%|)VEstOm{zhS~*o#YrXB= zPikjQ`AyyaJ?GxiPkQ%lf7zWll{b}tqnY~SxI67HD;SJF7|0!cTp{G2Jnhu-it_)@ z3$Fjsx!#m?yWVZ-hRwf^N6*T8y2t9{{am}%evkBL>#v$D@^s!$)hnk&OS!af)OgO# zug}@H&M$Du+3>4B1B;ehzWTE!?=x>{m;B%R&mKP3{lqW$ZE^@^?kFxvEGnreN=@T3 RGB7l;u;fxzb@g}S0stRj^Be#G diff --git a/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.tex b/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.tex index 38e668b..b86cda7 100644 --- a/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.tex +++ b/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.tex @@ -487,36 +487,40 @@ $\therefore \sqrt{2}$。 有一个边长为$x$的正方形,变化了$\Delta x$,其面积$\Delta S=(x+\Delta x)^2-x^2=2x\Delta x+(\Delta x)^2$。 -当$\Delta x\to 0$时,将这个变化定义为$2x\cdot\Delta x+o(\Delta x)$,前项为线性主部,后面为误差。这个就是$S$的微分。 +\begin{minipage}{0.45\linewidth} + 当$\Delta x\to 0$时,将这个变化定义为$2x\cdot\Delta x+o(\Delta x)$,前项为线性主部,后面为误差。这个就是$S$的微分。 -增量$\Delta y=f(x_0+\Delta)-f(x_0)=A\Delta x+o(\Delta x)$,这个$A\Delta x$定义为$\textrm{d}y$,叫做$y$的微分。 + 增量$\Delta y=f(x_0+\Delta)-f(x_0)=A\Delta x+o(\Delta x)$,这个$A\Delta x$定义为$\textrm{d}y$,叫做$y$的微分。 -$\therefore \textrm{d}y\vert_{x=x_0}=A\Delta x=y'(x_0)\cdot\Delta x=y'(x_0)\cdot\textrm{d}x$ + $\therefore \textrm{d}y\vert_{x=x_0}=A\Delta x=y'(x_0)\cdot\Delta x=y'(x_0)\cdot\textrm{d}x$。 -由此,可导必可微,可微必可导。 - -\begin{tikzpicture}[scale=0.9] - \draw[-latex](-0.5,0) -- (4.5,0) node[below]{$x$}; - \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$}; - \draw[black, thick, domain=-0.5:3] plot (\x,{pow(\x-1,2)/2+1}) node[above]{$y(x)$}; - \filldraw[black] (0,0) node[below]{$O$}; - \draw[black, densely dashed](1.5,1.125) -- (1.5,0) node[below]{$x_0$}; - \draw[black, densely dashed](1.5,1.125) -- (0,1.125) node[left]{$y_0$}; - \draw[black, densely dashed](3,3) -- (3,0) node[below]{$x_0+\Delta x$}; - \draw[black, densely dashed](3,3) -- (0,3) node[left]{$y_0+\Delta x$}; - \draw[black, densely dashed](3,1.875) -- (0,0.375) node[left]{$\textrm{d}y\cdot x+b$}; - \draw[<->, black](1.5,1.125) -- (3,1.125); - \draw[<->, black](4,1.125) -- (4,3); - \draw[<->, black](3.25,1.125) -- (3.25,1.875); - \draw[<->, black](3.25,3) -- (3.25,1.875); - \draw[black](3,3) -- (4.5,3); - \draw[black](3,1.125) -- (4.5,1.125); - \draw[black](3,1.875) -- (3.75,1.875); - \filldraw[black] (2.25,0.75) node{$\Delta x$}; - \filldraw[black] (4.3,2) node{$\Delta y$}; - \filldraw[black] (3.5,1.5) node{\scriptsize{$\textrm{d}y$}}; - \filldraw[black] (3.5,2.5) node{\scriptsize{$o(\Delta x)$}}; -\end{tikzpicture} + 由此,可导必可微,可微必可导。 +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \begin{tikzpicture}[scale=0.9] + \draw[-latex](-0.5,0) -- (4.5,0) node[below]{$x$}; + \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$}; + \draw[black, thick, domain=-0.5:3] plot (\x,{pow(\x-1,2)/2+1}) node[above]{$y(x)$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, densely dashed](1.5,1.125) -- (1.5,0) node[below]{$x_0$}; + \draw[black, densely dashed](1.5,1.125) -- (0,1.125) node[left]{$y_0$}; + \draw[black, densely dashed](3,3) -- (3,0) node[below]{$x_0+\Delta x$}; + \draw[black, densely dashed](3,3) -- (0,3) node[left]{$y_0+\Delta x$}; + \draw[black, densely dashed](3,1.875) -- (0,0.375) node[left]{$\textrm{d}y\cdot x+b$}; + \draw[<->, black](1.5,1.125) -- (3,1.125); + \draw[<->, black](4,1.125) -- (4,3); + \draw[<->, black](3.25,1.125) -- (3.25,1.875); + \draw[<->, black](3.25,3) -- (3.25,1.875); + \draw[black](3,3) -- (4.5,3); + \draw[black](3,1.125) -- (4.5,1.125); + \draw[black](3,1.875) -- (3.75,1.875); + \filldraw[black] (2.25,0.75) node{$\Delta x$}; + \filldraw[black] (4.3,2) node{$\Delta y$}; + \filldraw[black] (3.5,1.5) node{\scriptsize{$\textrm{d}y$}}; + \filldraw[black] (3.5,2.5) node{\scriptsize{$o(\Delta x)$}}; + \end{tikzpicture} +\end{minipage} \medskip 所以可微就是用简单线性取代复杂线性,如图用直线取替代曲线。微分就是瞬时改变量,而导数就是瞬时改变速率。 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 773bb15bae813d7953bc301765885ae2c66dcec7..36f6ead19f0dc40a6f241429cf0187d0a77219ba 100644 GIT binary patch delta 143708 zcmY(qQ;;T1xUAc@ZQHhO+s3r*{@S)}+qP|E+P2-Z{}r*%iM{SBqAu#9Dl(qTd~3c5 zKIj)JffK;NE+7Eo;_7T}YzO1Hb)!99x5ELq`$pR`qpugL$R9u=1j+{71?%l$hw1OW zOxJXIm!9_^Cy|%VF0aJK3=za0MyTfFP(a5V4LBlrRM z`g|Sj^$(952p76EkO-INBkew$$Sc$RaS$X?4hPtQQspXDU(6CR_GjSnwpfxckYV~R zoac%Anruk}F8~pltuNN`18wV}1i%P2c%7Gjb+7vSZ*CVM6jQ2Yu8~O@MC`CqJbU%i zypUB?@=KxfMk3q@t8d+n6lXNlo~oO*Nn5uWL|YcmTX0;rR|g9nB9JkKNJa1~4e$D; ziU0;-@6mH*l?BjG_KBmUA6o!CGFwuV8b_6I;*8vL8qH-9WI-G37Iq!=OGfMq?{b#j 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a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex +++ b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex @@ -315,6 +315,10 @@ $\dfrac{P}{x^n}=\dfrac{A}{x}+\dfrac{B_0x+B_1}{x^2}+\cdots+\dfrac{N_0x^{n-1}+\cdo \subsection{性质} +\textcolor{aqua}{\textbf{定理:}}定积分由积分上下限与函数关系确定,与积分变量无关。$\int_a^xf(x)\,\textrm{d}x=\int_a^xf(t)\,\textrm{d}t=\varPhi(x)$。 + +\subsubsection{计算性质} + 设函数$f(x)$在区间$[a,b]$上连续,则得到定积分基本计算性质: \begin{enumerate} @@ -329,8 +333,15 @@ $\dfrac{P}{x^n}=\dfrac{A}{x}+\dfrac{B_0x+B_1}{x^2}+\cdots+\dfrac{N_0x^{n-1}+\cdo \item 已知$f(x)\in[m,M]$在$[a,b]$上成立,则$m(b-a)\leqslant\int_a^bf(x)\,\textrm{d}x\leqslant M(a-b)$。 \item 估值定理:当$M$,$m$分别为$f(x)$在$[a,b]$上的最大值和最小值,则$m(b-a)\leqslant\int_a^bf(x)\,\textrm{d}x\leqslant M(b-a)$。 \item 积分中值定理:$\exists\,\xi\in[a,b]$,使得$\int_a^bf(x)\,\textrm{d}x=f(\xi)(b-a)$。 + \item 积分中值定理推广:设$f(x)\in[a,b]$,$\exists\,\xi\in(a,b)$,使得$\int_a^bf(x)\,\textrm{d}x=f(\xi)(b-a)$。 \end{enumerate} +证明第八条:已知$-\vert f(x)\vert\leqslant f(x)\leqslant\vert f(x)\vert$。 + +即得到$-\int_a^b\vert f(x)\vert\,\textrm{d}x\leqslant\int_a^bf(x)\,\textrm{d}x\leqslant\int_a^b\vert f(x)\vert\,\textrm{d}x$。 + +从而得证。 + 证明第十一条积分中值定理: 设函数$f(x)$在区间$[a,b]$上连续,因为闭区间上连续函数必然有最大最小值,所以设最大值为$M$,最小值为$m$,$M\geqslant m$。 @@ -349,7 +360,11 @@ $\dfrac{P}{x^n}=\dfrac{A}{x}+\dfrac{B_0x+B_1}{x^2}+\cdots+\dfrac{N_0x^{n-1}+\cdo 对$F(x)$使用拉格朗日中值定理:$F(b)-F(a)=F'(\xi)(b-a)$,即$\int_a^bf(x)\,\textrm{d}x=f(\xi)(b-a)$,其中$\xi\in(a,b)\subset[a,b]$。 -定积分的存在性性质: +证明第十二条积分中值定理的推广。令$F(x)=\int_a^xf(t)\,\textrm{d}t$,$F'(x)=f(x)$。 + +$\int_a^bf(x)\,\textrm{d}x=F(b)-F(a)=F'(\xi)(b-a)=f(\xi)(b-a)$($a<\xio(62>Ipx@BlXINdYSWGW53@_Wdtz)6dk$TLv*8R8Qp~l@N+fA(6wrx2R!B zDk{6%sPw4W&f%;2IzlK5JdE#XD<($COQXNEc(8@t+7Dg|69K`AbGkyruw@`M5&m(v zU6%Q5r_c(Z*iSxrFKbKlFU7WdY=;>o;Lz_Cwa^HYA>d7p_1Gar>d@$I$mr*hoK2?! 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zPnQ~g*cmUe)pKO5`h7noNyyYwW|2}FQEbtv-f^ppDMNLwd1@5WEmfk4YLv>6-?r|J zH!74WtQ|@GX-nj!I4rDETKLmiuJAgZScXRVtW?ISRSKJNLXp@8l3uQSCTLZqc>60$,则平面上以$M_0$为圆心,$\delta$为半径的圆的内部称为$M_0$的\textbf{$\delta$领域},记为$U(M_0,\delta)$。 + +若领域中去掉圆心$M_0$,称为$M_0$的\textbf{$\delta$去心邻域},记为$\mathring{U}(M_0,\delta)$。 + +\subsubsection{点的分类} + +\textcolor{violet}{\textbf{定义:}}设$M$为平面上一个点,若存在$\delta>0$,使得$U(M,\delta)\subset E$,则$M$为点集$E$的\textbf{内点}。 + +\textcolor{violet}{\textbf{定义:}}若存在$\delta>0$,使得$U(M,\delta)\cap E=\varnothing$,则$M$为点集$E$的的\textbf{外点}。 + +\textcolor{violet}{\textbf{定义:}}若对任意$\delta>0$,$U(M,\delta)$即有$E$内的点也有外的点,则$M$为点集$E$的\textbf{边界点}。 + +\textcolor{violet}{\textbf{定义:}}$E$所有边界点的集合称为$E$的\textbf{边界},记为$\partial E$。对于任意一个点集$E$与其余集$E^C$有公共边界,即$\partial E=\partial E^C$。 + +\subsubsection{集合} + +\textcolor{violet}{\textbf{定义:}}设$E$为一个平面点集,若存在常数$\delta>0$,使得$E\subset U(O,\delta)$,则$E$为\textbf{有界集},否则为\textbf{无界集}。 + +\textcolor{violet}{\textbf{定义:}}若$E$中的每个点都是$E$的内点,则$E$为\textbf{开集},若$E$的边界点都是$E$的点,则$E$为\textbf{闭集}。若一个点集是开集,则其余集为闭集,若一个点集为闭集,则其余集为开集。 + +\textcolor{violet}{\textbf{定义:}}若$E$中任意两点,都可用一条完全属于$E$的曲线将其两点连接,则$E$为\textbf{(道路)连通集},连通的开集为\textbf{开区域},一个开区域和其边界点的并集为\textbf{闭区域},统称\textbf{区域}。 + +\textcolor{violet}{\textbf{定义:}}若$E$内任意一条\textbf{简单闭曲线}的内部还在$E$内,则$E$为\textbf{单连通区域},否则为\textbf{多连通区域}。 + +\subsubsection{聚点} + +\textcolor{violet}{\textbf{定义:}}对一个平面点集$E$,$M_0$为平面上一点,若对任意$\delta>0$,总有$\mathring{U}(M_0,\delta)\cap E\neq\varnothing$,即$M_0$的任意邻域中都含有异于$M_0$的$E$中的点,则$M_0$为$E$的\textbf{聚点}。 + +\textcolor{aqua}{\textbf{定理:}}非空开集的内点余边界点都是这个点集的聚点,闭区域的任意一点都是其聚点。 + +\textcolor{violet}{\textbf{定义:}}若存在$\delta>0$,使得$U(M_0,\delta)\cap E=\{M_0\}$,即如果$M_0$的某一邻域与点集$E$的交集是一个孤立的点$M_0$,则称$M_0$为$E$的\textbf{孤立点}。边界点要么是聚点要么是孤立点。 + +\subsection{极限} + +对于一元函数的极限可用列举法,从两端逼近该点取极限,但是对于多元函数所处的邻域,逼近方向为无穷,所以不可能再通过取两个方向逼近的方式求极限。 + +从点集来看\textcolor{violet}{\textbf{定义:}}设二元函数$f(P)=f(x,y)$的定义域为$D$,$P_0(x_0,y_0)$为$D$聚点。若存在常数$A$,对于任意给定正数$\epsilon$,总存在正数$\delta$,使得当$P(x,y)\in D\cap\mathring{U}(P_0,\delta)$时,都有$\vert f(x,y)-A\vert<\epsilon$成立,则常数$A$为$f(x,y)$当$(x,y)\rightarrow(x_0,y_0)$时的极限,记为$\lim\limits_{(x,y)\to(x_0,y_0)}f(x,y)=A$或$f(x,y)\to A((x,y)\to(x_0,y_0))$。 + +如$\because xy\neq0$排除$xy$轴:$\lim\limits_{(x,y)\to(0,0)}\dfrac{\sqrt{xy+1}-1}{xy}=\lim\limits_{(x,y)\to(0,0)}\dfrac{xy+1-1}{xy(\sqrt{xy+1}+1)}$\\$=\lim\limits_{(x,y)\to(0,0)}\dfrac{1}{\sqrt{xy+1}+1}=\dfrac{1}{2}$。\medskip + +从邻域来看\textcolor{violet}{\textbf{定义:}}若二元函数$f(x,y)$在$(x_0,y_0)$的去心邻域内有定义,且$(x,y)$以任意方式趋向$(x_0,y_0)$时,$f(x,y)$均趋向于$A$,则$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f(x,y)=A$。 + +根据邻域的定义,由于函数$\lim\limits_{(x,y)\to(0,0)}\dfrac{\sqrt{xy+1}-1}{xy}$在坐标轴上无定义,则极限不存在。 + +此时两种定义就会有两种结论,所以为了避免这种定义不同的矛盾,就只会出现哪种定义下极限存在或都不存在的函数,如$\lim\limits_{\substack{x\to0\\y\to0}}(x^2+y^2)\sin\dfrac{1}{x^2+y^2}=0$。 + +从现实角度来看,点集定义是更合理的,若要求一根弯曲铁丝在某点的导数,第二种定义无法求,所以不合理。而第二种定义是从一元极限定义直接升级过来,所以有一定局限性。 + +\subsection{连续} + +\textcolor{violet}{\textbf{定义:}}若$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f(x,y)=f(x_0,y_0)$则称$f(x,y)$在点$(x_0,y_0)$处连续。 + +若不连续,则不讨论间断类型。 + +\subsection{偏导数} + +当含有两个以及三个变量时,若求一个极限,则有多个变量同时趋向,所以多个变量同时在变。为了运算简单,就假定只有一个变量在变,其他变量固定,从而直接降低成一元变量,只对一个变量求导,从而就是偏导数。 + +\textcolor{violet}{\textbf{定义:}}设函数$z=f(x,y)$在点$(x_0,y_0)$的某邻域内有定义,若极限\\$\lim\limits_{\Delta x\to0}\dfrac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}$存在,则称此极限为函数$z=f(x,y)$在点$(x_0,y_0)$处对$x$的\textbf{偏导数},记为$\dfrac{\partial z}{\partial x}\bigg|_{\substack{x=x_0\\y=y_0}}$,$\dfrac{\partial f}{\partial x}\bigg|_{\substack{x=x_0\\y=y_0}}$,$z'\bigg|_{\substack{x=x_0\\y=y_0}}$或$f'_x(x_0,y_0)$。\medskip + +$f'_x(x_0,y_0)=\lim\limits_{\Delta x\to0}\dfrac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}=\lim\limits_{x\to x_0}\dfrac{f(x,y_0)-f(x_0,y_0)}{x-x_0}$。 + +$f'_y(x_0,y_0)=\lim\limits_{\Delta y\to0}\dfrac{f(x_0,y_0+\Delta y)-f(x_0,y_0)}{\Delta y}=\lim\limits_{y\to y_0}\dfrac{f(x_0,y)-f(x_0,y_0)}{y-y_0}$。 + +\textcolor{violet}{\textbf{定义:}}若函数$z=f(x,y)$在区域$D$内的偏导数$f_x'(x,y)$、$f_y'(x,y)$仍具有偏导数,则其偏导数为函数$z=f(x,y)$的\textbf{二阶偏导数}。按照求导次序不同,有如下四个二阶偏导数。 + +$\dfrac{\partial}{\partial x}\left(\dfrac{\partial z}{\partial x}\right)=\dfrac{\partial^2z}{\partial x^2}=f''_{xx}(x,y)$,$\dfrac{\partial}{\partial y}\left(\dfrac{\partial z}{\partial y}\right)=\dfrac{\partial^2z}{\partial y^2}=f''_{yy}(x,y)$, + +$\dfrac{\partial}{\partial y}\left(\dfrac{\partial z}{\partial x}\right)=\dfrac{\partial^2z}{\partial x\partial y}=f''_{xy}(x,y)$,$\dfrac{\partial}{\partial x}\left(\dfrac{\partial z}{\partial y}\right)=\dfrac{\partial^2z}{\partial y\partial x}=f''_{yx}(x,y)$。 + +其中$f''_{xy}(x,y)$和$f''_{yx}(x,y)$为\textbf{混合偏导数}。二阶以及以上的偏导数均为\textbf{高阶偏导数}。 + +\subsection{全微分} + +\textcolor{violet}{\textbf{定义:}}若函数$z=f(x,y)$在点$(x,y)$的全增量$\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y)$可表示为$\Delta z=A\Delta x+B\Delta y+o(\rho)$,其中$\rho=\sqrt{(\Delta x)^2+(\Delta y)^2}$,$AB$不依赖$\Delta x$,$\Delta y$而仅与$x,y$相关,则称函数$z=f(x,y)$在点$(x,y)$可微,而称$A\Delta x+B\Delta y$为函数$z=f(x,y)$在点$(x,y)$的\textbf{全微分},记为$\textrm{d}z$。 + +$\textrm{d}z=A\Delta x+B\Delta y=\dfrac{\partial z}{\partial x}\Delta x+\dfrac{\partial z}{\partial y}\Delta y=\dfrac{\partial z}{\partial x}\textrm{d}x+\dfrac{\partial z}{\partial y}\textrm{d}y$。 + +判断可微的步骤: + +\begin{enumerate} + \item 写出全增量$\Delta z=f(x_0+\Delta x,y_0+\Delta y)-f(x_0,y_0)$。 + \item 写出线性增量$A\Delta x+B\Delta y$,$A=f_x'(x_0,y_0)$,$B=f_y'(x_0,y_0)$。 + \item 写出极限$\lim\limits_{\substack{\Delta x\to0\\\Delta y\to0}}\dfrac{\Delta z-(A\Delta x+B\Delta y)}{\sqrt{(\Delta x)^2+(\Delta y)^2}}$,若极限等于0,则$z=f(x,y)$在点$(x_0,y_0)$可微,否则不可微。 +\end{enumerate} + +% \textbf{例题:}已知函数$z=f(x,y)$的全微分$\textrm{d}z=2x\,\textrm{d}x+\sin y\,\textrm{d}y$且$f(1,0)=2$,求$f(x,y)$。 + +% 解: + +\subsection{偏导数连续性} + +对$z=f(x,y)$,讨论其在某特殊点$(x_0,y_0)$处偏导数是否连续的步骤: + +\begin{enumerate} + \item 用定义法求$f_x'(x_0,y_0)$,$f_y'(x_0,y_0)$。(求某点偏导数) + \item 用公式法求$f_x'(x,y)$,$f_y'(x,y)$。(求偏导函数) + \item 计算$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f_x'(x,y)$,$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f_y'(x,y)$。(偏导函数求极限) + \item 若$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f_x'(x,y)=f_x'(x_0,y_0)$,$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f_y'(x,y)=f_y'(x_0,y_0)$若成立则连续,否则不连续。 +\end{enumerate} + +\textbf{例题:}设$z=f(x,y)=\left\{\begin{array}{ll} + (x^2+y^2)\sin\dfrac{1}{\sqrt{x^2+y^2}}, & x^2+y^2\neq0 \\ + 0, & x^2+y^2=0 +\end{array}\right.$,则四个结论中正确的个数为()。 + +\ding{172}$f(x,y)$在$(0,0)$处连续。\qquad\ding{173}$f'_x(0,0)$,$f'_y(0,0)$存在。 + +\ding{174}$f_x'(x,y)$,$f_y'(x,y)$在$(0,0)$处连续。\qquad\ding{174}$f(x,y)$在$(0,0)$可微。 + +$A.1$\qquad$B.2$\qquad$C.3$\qquad$D.4$ + +解:$\lim\limits_{\substack{x\to0\\y\to0}}(x^2+y^2)\sin\dfrac{1}{\sqrt{x^2+y^2}}=0=f(0,0)$。所以$A$正确。 + +$f_x'(0,0)=\lim\limits_{\Delta x\to0}\dfrac{f(0+\Delta x,0)-f(0,0)}{\Delta x}=\lim\limits_{\Delta x\to0}\dfrac{(\Delta x)^2\sin\dfrac{1}{\sqrt{(\Delta x)^2}}-0}{\Delta x}=$\\$\lim\limits_{\Delta x\to0}(\Delta x)\sin\dfrac{1}{\vert\Delta x\vert}=0$。同理$f'_y(0,0)=0$。 + +判断连续性,首先计算偏导数值,之前计算过:$f_x'(0,0)=f_y'(0,0)=0$;然后求偏导函数$f_x'(x,y)=2x\sin\dfrac{1}{\sqrt{x^2+y^2}}+(x^2+y^2)\cos\dfrac{1}{\sqrt{x^2+y^2}}\left(-\dfrac{1}{2}\right)\dfrac{2x}{\sqrt{(x^2+y^2)^3}}$\\$=2x\sin\dfrac{1}{\sqrt{x^2+y^2}}-\dfrac{x}{\sqrt{x^2+y^2}}\cos\dfrac{1}{\sqrt{x^2+y^2}}$,同理得$f_y'(x,y)=2y\sin\dfrac{1}{\sqrt{x^2+y^2}}$\\$-\dfrac{y}{\sqrt{x^2+y^2}}\cos\dfrac{1}{\sqrt{x^2+y^2}}$;最后一步查看偏导函数值与偏导数值是否相等,$\because\lim\limits_{\substack{x\to0\\y\to0}}2x\sin\dfrac{1}{\sqrt{x^2+y^2}}=0$,且$\lim\limits_{\substack{x\to0\\y\to0}}\dfrac{x}{\sqrt{x^2+y^2}}\cos\dfrac{1}{\sqrt{x^2+y^2}}$震荡,所以总的来说极限值不存在,就不会等于偏导数值,同理可得函数的偏导数在该点不连续。 + +要求一个函数在某点可微,首先$\Delta z=f(0+\Delta x,0+\Delta y)-f(0,0)=[(\Delta x)^2+(\Delta y)^2]\sin\dfrac{1}{\sqrt{(\Delta x)^2+(\Delta y)^2}}$。然后$A\Delta x+B\Delta y=f_x'(0,0)\Delta x+f_y'(0,0)\Delta y=0$。最后求极限$\lim\limits_{\substack{\Delta x\to0\\\Delta y\to0}}\dfrac{\Delta z-(A\Delta x+B\Delta y)}{\sqrt{(\Delta x)^2+(\Delta y)^2}}=\lim\limits_{\substack{\Delta x\to0\\\Delta y\to0}}\sqrt{(\Delta x)^2+(\Delta y)^2}\sin\dfrac{1}{\sqrt{(\Delta x)^2+(\Delta y)^2}}$\\$=0$,所以在此点可微。 + +综上正确的结论有\ding{172}\ding{173}\ding{175}三个,所以选$C$。 + +\section{多元函数微分法则} + +\subsection{链式求导法则} + +主要对显函数的微分。 + +多元函数链式求导法则与一元函数的求导法则类似。都是从因变量从中间变量走到自变量。一条路径是一个加项,多少条从因变量到所有自变量的路就有多少个加项。每条路上由不同的路段组成,若有$n$层中间变量,则有$n+1$路段,路段之间项是乘积形式,若变量只与一个变量有一条路,则是导数$\textrm{d}$,若一个变量到多个变量有多条路,则是偏导数$\partial$。 + +\begin{minipage}{0.65\linewidth} + 因变量$z$到$x$一共有两条路,所以两个和项。每条路都有两端,所以和项中有两个乘项。$z$到$uv$两个中间变量,所以是两个偏导$\dfrac{\partial z}{\partial u}$和$\dfrac{\partial z}{\partial v}$。$uv$都只有一条路直接连通$x$,所以都是导数$\dfrac{\textrm{d}u}{\textrm{d}x}$和$\dfrac{\textrm{d}v}{\textrm{d}x}$。一条路的每个路段的项相乘:$\dfrac{\partial z}{\partial u}\dfrac{\textrm{d}u}{\textrm{d}x}$和$\dfrac{\partial z}{\partial v}\dfrac{\textrm{d}v}{\textrm{d}x}$。最后将每条路段相加:$\dfrac{\textrm{d}z}{\textrm{d}x}=\dfrac{\partial z}{\partial v}\dfrac{\textrm{d}v}{\textrm{d}x}$。 +\end{minipage} +\hfill +\begin{minipage}{0.25\linewidth} + \begin{tikzpicture}[scale=1] + \filldraw[black] (-0.25,0) node{$z$}; + \draw[black](0,0) -- (1,1) node[right]{$u$}; + \draw[black](0,0) -- (1,-1) node[right]{$v$}; + \draw[black](1.5,1) -- (2.5,0) node[right]{$x$}; + \draw[black](1.5,-1) -- (2.5,0); + \end{tikzpicture} +\end{minipage} \medskip + +\begin{minipage}{0.25\linewidth} + \begin{tikzpicture}[scale=1] + \filldraw[black] (-0.25,0) node{$z$}; + \draw[black](0,0) -- (1,1) node[right]{$u$}; + \draw[black](0,0) -- (1,0) node[right]{$v$}; + \draw[black](0,0) -- (1,-1) node[right]{$w$}; + \draw[black](1.5,1) -- (2.5,1) node[right]{$x$}; + \draw[black](1.5,0) -- (2.5,1); + \draw[black](1.5,-1) -- (2.5,-1) node[right]{$y$}; + \draw[black](1.5,1) -- (2.5,-1); + \end{tikzpicture} +\end{minipage} +\hfill +\begin{minipage}{0.65\linewidth} + 因为因变量$z$到自变量$x,y$有较多条路径,所以分开分析。 + + 对于$x$,有$z-u-x$,所以这条路为$\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial x}$,还有一条$z-v-x$,由于$v$只与$x$连通,所以是导数,该路为$\dfrac{\partial z}{\partial v}\dfrac{\textrm{d}v}{\textrm{d}x}$,所以$\dfrac{\partial z}{\partial x}=\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial x}+\dfrac{\partial z}{\partial v}\dfrac{\textrm{d}v}{\textrm{d}x}$。、 + + 同理对于$y$,有$z-u-y$和$z-w-y$,且$u$有两条出路,$w$只有一条,所以$u$偏导,$v$导数,$\dfrac{\partial z}{\partial y}=\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial y}+\dfrac{\partial z}{\partial w}\dfrac{\textrm{d}w}{\textrm{d}y}$。 +\end{minipage} \medskip + +无论$z$对谁求导也无论求了几阶到,求导过后的新函数仍具有与原函数完全相同的复合结构。 + +\textbf{例题:}设$z=f(e^x\sin y,x^2+y^2)$,其中$f$具有二阶连续偏导数,求$\dfrac{\partial^2z}{\partial x\partial y}$。 + +解:$\because\dfrac{\partial^2z}{\partial x\partial y}=\dfrac{\partial}{\partial y}\left(\dfrac{\partial z}{\partial x}\right)$,$\therefore\dfrac{\partial z}{\partial x}=f_1'\cdot e^x\sin y+f_2'\cdot2x$。 + +在求偏导时,将第一个中间变量记为$f_1$即之前的$u$,第二个中间变量记为$f_2$即之前的$v$。记$f$对$u$求偏导为$f_1'$,对$v$求偏导为$f_2'$同理二阶导也如此,下标为求导顺序。 + +$\dfrac{\partial^2z}{\partial x\partial y}=\dfrac{\partial(f_1'\cdot e^x\sin y)}{\partial y}+\dfrac{\partial(f_2'\cdot2x)}{\partial y}$。 + +其中$\dfrac{\partial(f_1'\cdot e^x\sin y)}{\partial y}=\dfrac{\partial f_1'}{\partial y}\cdot e^x\sin y+f_1'\cdot e^x\cos y$。所以难点就是$\dfrac{\partial f_1'}{\partial y}$。 + +求导路径$f_1'-1-y$和$f_1'-2-y$:$=(f_{11}''e^x\cos y+f_{12}''2y)\cdot e^x\sin y+f_1'\cdot e^x\cos y$。 + +\subsection{隐函数存在定理} + +主要对隐函数的微分。 + +\section{多元函数极值最值} + +\subsection{概念} + +\subsection{无条件极值} + +\subsubsection{隐函数} + +\subsubsection{显函数} + +\subsection{条件极值与拉格朗日乘数法} + +\subsubsection{闭区域边界最值} + +\subsubsection{闭区域上最值} \end{document} diff --git a/probability-theory-and-mathematical-statistics/knowledge/5-mathematical-statistics/mathematical-statistics.pdf b/probability-theory-and-mathematical-statistics/knowledge/5-mathematical-statistics/mathematical-statistics.pdf index 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X7Y`>>GXOIeD<>-}000$,$y=x^{\mu}$都有定义: +\begin{minipage}{0.4\linewidth} + $y=x^{\mu}$,$\mu$为实数,当$x>0$,$y=x^{\mu}$都有定义: -\begin{tikzpicture}[scale=0.9] - \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; - \draw[-latex](0,-2) -- (0,4) node[above]{$y$}; - \draw[black, thick, smooth, domain=0.3:2] plot (\x,1/\x) node[right]{$\mu =-1$}; - \draw[black, thick, smooth, domain=-2:-0.5] plot (\x,1/\x) node[right]{$\mu =-1$}; - \draw[black, thick, smooth, domain=0.01:2] plot (\x, {sqrt(\x)}) node[right]{$\mu =\dfrac{1}{2}$}; - \draw[black, thick, smooth, domain=-2:2] plot (\x,\x) node[right]{$\mu =1$}; - \draw[black, thick, smooth, domain=-2:2] plot (\x, {\x*\x}) node[right]{$\mu =2$}; - \filldraw[black] (0,0) node[below]{$O$}; - \filldraw[black] (1,1) circle (2pt) node at(1.25,0.5){$(1,1)$}; -\end{tikzpicture} + \begin{tikzpicture}[scale=0.9] + \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,4) node[above]{$y$}; + \draw[black, thick, smooth, domain=0.3:2] plot (\x,1/\x) node[right]{$\mu =-1$}; + \draw[black, thick, smooth, domain=-2:-0.5] plot (\x,1/\x) node[right]{$\mu =-1$}; + \draw[black, thick, smooth, domain=0.01:2] plot (\x, {sqrt(\x)}); + \filldraw[black] (2.75,1.25) node {$\mu =\dfrac{1}{2}$}; + \draw[black, thick, smooth, domain=-2:2] plot (\x,\x) node[right]{$\mu =1$}; + \draw[black, thick, smooth, domain=-2:2] plot (\x, {\x*\x}) node[right]{$\mu =2$}; + \filldraw[black] (0,0) node[below]{$O$}; + \filldraw[black] (1,1) circle (2pt) node at(1.25,0.5){$(1,1)$}; + \end{tikzpicture} +\end{minipage} +\hfill +\begin{minipage}{0.5\linewidth} + 对于幂函数可以根据不同幂下相同单调性来研究最值: -所以对于幂函数,可以根据不同幂下相同单调性来研究最值: - -\begin{enumerate} - \item $\sqrt{u},\sqrt[3]{u}$可以使用$u$来研究。 - \item $\vert u\vert$可以使用$u^2$来研究。 - \item $\dfrac{1}{u},u>0$可以使用$u$来研究,但是最值相反。 - \item $u_1u_2...u_n$可以使用$\sum_{i=1}^{n}\ln u_i$来研究。 -\end{enumerate} + \begin{enumerate} + \item $\sqrt{u},\sqrt[3]{u}$可以使用$u$来研究。 + \item $\vert u\vert$可以使用$u^2$来研究。 + \item $\dfrac{1}{u},u>0$可以使用$u$来研究,但是最值相反。 + \item $u_1u_2...u_n$可以使用$\sum_{i=1}^{n}\ln u_i$来研究。 + \end{enumerate} +\end{minipage} \subparagraph{指数函数} \leavevmode \medskip -$y=a^x(a>0,a\neq 1)$: +\begin{minipage}{0.4\linewidth} + $y=a^x(a>0,a\neq 1)$: -\begin{tikzpicture}[scale=0.9] - \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; - \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$}; - \draw[black, thick, domain=-2:2] plot (\x,{pow(1/2,\x)}) node[right]{$01$}; - \filldraw[black] (0,0) node[below]{$O$}; -\end{tikzpicture} + \begin{tikzpicture}[scale=0.9] + \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$}; + \draw[black, thick, domain=-2:2] plot (\x,{pow(1/2,\x)}) node[right]{$01$}; + \filldraw[black] (0,0) node[below]{$O$}; + \end{tikzpicture} +\end{minipage} +\hfill +\begin{minipage}{0.5\linewidth} + 指数函数具有如下性质: -指数函数具有如下性质: - -\begin{enumerate} - \item 特殊函数值:$a^0=1$。 - \item 定义域:$(-\infty, +\infty)$,值域:$(0,+\infty)$。 - \item 单调性:$a>1$,$y=a^x$单调增,$01$,$y=a^x$单调增,$00,a\neq 1)$为$y=a^x$的反函数: +\begin{minipage}{0.45\linewidth} + $y=log_ax(a>0,a\neq 1)$为$y=a^x$的反函数: -\begin{tikzpicture}[scale=0.9] - \draw[-latex](-0.5,0) -- (4,0) node[below]{$x$}; - \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; - \draw[black, thick, domain=0.2:4] plot (\x,{ln(1/\x)}) node[right]{$01$}; - \filldraw[black] (0,0) node[below]{$O$}; -\end{tikzpicture} + 常用公式:$x=e^{\ln x}$,$u^v=e^{\ln u^v}=e^{v\ln u}(x>0,u>0)$。 +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \begin{tikzpicture}[scale=0.9] + \draw[-latex](-0.5,0) -- (4,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \draw[black, thick, domain=0.2:4] plot (\x,{ln(1/\x)}) node[right]{$01$}; + \filldraw[black] (0,0) node[below]{$O$}; + \end{tikzpicture} +\end{minipage} 对数函数具有如下性质: @@ -130,9 +151,9 @@ $y=log_ax(a>0,a\neq 1)$为$y=a^x$的反函数: \item 单调性:$a>1$,$y=\log_ax$单调增,$00,u>0)$ \end{enumerate} + \subparagraph{三角函数} \leavevmode \medskip 正弦函数: @@ -177,40 +198,44 @@ $y=log_ax(a>0,a\neq 1)$为$y=a^x$的反函数: \item 有界性:$\vert\sin x\vert\leqslant 1$,$\vert\cos x\vert\leqslant 1$。 \end{enumerate} -正切函数: +\begin{minipage}{0.5\linewidth} + 正切函数: -\begin{tikzpicture}[scale=0.7] - \draw[-latex](-6,0) -- (6,0) node[below]{$x$}; - \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; - \draw[black, thick, domain=-pi/2+0.5:pi/2-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$}; - \draw[black, densely dashed](pi/2,2) -- (pi/2,-2); - \draw[black, densely dashed](-pi/2,2) -- (-pi/2,-2); - \draw[black, thick, domain=-pi/2*3+0.5:-pi/2-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$}; - \draw[black, densely dashed](pi/2*3,2) -- (pi/2*3,-2); - \draw[black, thick, domain=pi/2+0.5:pi/2*3-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$}; - \draw[black, densely dashed](-pi/2*3,2) -- (-pi/2*3,-2); - \filldraw[black] (0,0) node[below]{$O$}; - \filldraw[black] (pi/2+0.5,-0.75) node{$\dfrac{\pi}{2}$}; - \filldraw[black] (-pi/2-0.75,-0.75) node{$-\dfrac{\pi}{2}$}; - \filldraw[black] (pi/2*3+0.5,-0.75) node{$\dfrac{3\pi}{2}$}; - \filldraw[black] (-pi/2*3-0.75,-0.75) node{$-\dfrac{3\pi}{2}$}; -\end{tikzpicture} + \begin{tikzpicture}[scale=0.55] + \draw[-latex](-6,0) -- (6,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \draw[black, thick, domain=-pi/2+0.5:pi/2-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$}; + \draw[black, densely dashed](pi/2,2) -- (pi/2,-2); + \draw[black, densely dashed](-pi/2,2) -- (-pi/2,-2); + \draw[black, thick, domain=-pi/2*3+0.5:-pi/2-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$}; + \draw[black, densely dashed](pi/2*3,2) -- (pi/2*3,-2); + \draw[black, thick, domain=pi/2+0.5:pi/2*3-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$}; + \draw[black, densely dashed](-pi/2*3,2) -- (-pi/2*3,-2); + \filldraw[black] (0,0) node[below]{$O$}; + \filldraw[black] (pi/2+0.5,-0.75) node{$\dfrac{\pi}{2}$}; + \filldraw[black] (-pi/2-0.75,-0.75) node{$-\dfrac{\pi}{2}$}; + \filldraw[black] (pi/2*3+0.5,-0.75) node{$\dfrac{3\pi}{2}$}; + \filldraw[black] (-pi/2*3-0.75,-0.75) node{$-\dfrac{3\pi}{2}$}; + \end{tikzpicture} +\end{minipage} +\hfill +\begin{minipage}{0.4\linewidth} + 余切函数: -余切函数: - -\begin{tikzpicture}[scale=0.7] - \draw[-latex](-4,0) -- (4,0) node[below]{$x$}; - \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; - \draw[black, thick, domain=0.5:pi-0.5] plot (\x,{cot(\x r)}) node at(pi-1,2){$\cot(x)$}; - \draw[black, densely dashed](pi,2) -- (pi,-2); - \draw[black, thick, domain=-0.5:-pi+0.5] plot (\x,{cot(\x r)}) node at(-1,2){$\cot(x)$}; - \draw[black, densely dashed](-pi,2) -- (-pi,-2); - \filldraw[black] (0,0) node[below]{$O$}; - \filldraw[black] (pi/2,0) node[below]{$\dfrac{\pi}{2}$}; - \filldraw[black] (pi+0.5,-0.5) node{$\pi$}; - \filldraw[black] (-pi/2-0.25,0) node[below]{$-\dfrac{\pi}{2}$}; - \filldraw[black] (-pi-0.5,-0.5) node{$-\pi$}; -\end{tikzpicture} + \begin{tikzpicture}[scale=0.65] + \draw[-latex](-4,0) -- (4,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \draw[black, thick, domain=0.5:pi-0.5] plot (\x,{cot(\x r)}) node at(pi-1,2){$\cot(x)$}; + \draw[black, densely dashed](pi,2) -- (pi,-2); + \draw[black, thick, domain=-0.5:-pi+0.5] plot (\x,{cot(\x r)}) node at(-1,2){$\cot(x)$}; + \draw[black, densely dashed](-pi,2) -- (-pi,-2); + \filldraw[black] (0,0) node[below]{$O$}; + \filldraw[black] (pi/2,0) node[below]{$\dfrac{\pi}{2}$}; + \filldraw[black] (pi+0.5,-0.5) node{$\pi$}; + \filldraw[black] (-pi/2-0.25,0) node[below]{$-\dfrac{\pi}{2}$}; + \filldraw[black] (-pi-0.5,-0.5) node{$-\pi$}; + \end{tikzpicture} +\end{minipage} 切函数有如下特征: @@ -221,9 +246,7 @@ $y=log_ax(a>0,a\neq 1)$为$y=a^x$的反函数: \item 周期性:最小正周期为$\pi$。 \end{enumerate} -$$ - \sec x=\dfrac{1}{\cos x},\csc x=\dfrac{1}{\sin x} -$$ +$\sec x=\dfrac{1}{\cos x},\csc x=\dfrac{1}{\sin x}$: 正割函数: @@ -284,31 +307,35 @@ $$ \subparagraph{反三角函数} \leavevmode \medskip -反正弦函数: +\begin{minipage}{0.45\linewidth} + 反正弦函数: -\begin{tikzpicture} - \draw[-latex](-1.5,0) -- (1.5,0) node[below]{$x$}; - \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; - \draw[black, thick, domain=-1:1] plot (\x,{rad(asin(\x))}) node[right]{$\arcsin(x)$}; - \draw[black, densely dashed](1,pi/2) -- (0,pi/2) node[left]{$\dfrac{\pi}{2}$}; - \filldraw[black] (0,0) node[below]{$O$}; - \draw[black, densely dashed](1,pi/2) -- (1,0) node[below]{$1$}; - \draw[black, densely dashed](-1,-pi/2) -- (0,-pi/2) node[right]{$-\dfrac{\pi}{2}$}; - \draw[black, densely dashed](-1,-pi/2) -- (-1,0) node[above]{$-1$}; -\end{tikzpicture} + \begin{tikzpicture} + \draw[-latex](-1.5,0) -- (1.5,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \draw[black, thick, domain=-1:1] plot (\x,{rad(asin(\x))}) node[right]{$\arcsin(x)$}; + \draw[black, densely dashed](1,pi/2) -- (0,pi/2) node[left]{$\dfrac{\pi}{2}$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, densely dashed](1,pi/2) -- (1,0) node[below]{$1$}; + \draw[black, densely dashed](-1,-pi/2) -- (0,-pi/2) node[right]{$-\dfrac{\pi}{2}$}; + \draw[black, densely dashed](-1,-pi/2) -- (-1,0) node[above]{$-1$}; + \end{tikzpicture} +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + 反余弦函数: -反余弦函数: - -\begin{tikzpicture} - \draw[-latex](-1.5,0) -- (1.5,0) node[below]{$x$}; - \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$}; - \draw[black, thick, domain=-1:1] plot (\x,{rad(acos(\x)}) node at (-2, pi){$\arccos(x)$}; - \filldraw[black] (0,pi/2+0.5) node[right]{$\dfrac{\pi}{2}$}; - \draw[black](1,0) -- (1,0) node[below]{$1$}; - \filldraw[black] (0,0) node[below]{$O$}; - \draw[black, densely dashed](-1,pi) -- (0,pi) node[right]{$\pi$}; - \draw[black, densely dashed](-1,pi) -- (-1,0) node[below]{$-1$}; -\end{tikzpicture} + \begin{tikzpicture} + \draw[-latex](-1.5,0) -- (1.5,0) node[below]{$x$}; + \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$}; + \draw[black, thick, domain=-1:1] plot (\x,{rad(acos(\x)}) node at (-2, pi){$\arccos(x)$}; + \filldraw[black] (0,pi/2+0.5) node[right]{$\dfrac{\pi}{2}$}; + \draw[black](1,0) -- (1,0) node[below]{$1$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, densely dashed](-1,pi) -- (0,pi) node[right]{$\pi$}; + \draw[black, densely dashed](-1,pi) -- (-1,0) node[below]{$-1$}; + \end{tikzpicture} +\end{minipage} 反弦函数有如下特征: @@ -327,29 +354,33 @@ $$ 又$f(0)=\dfrac{\pi}{2}$,所以该函数等于$\dfrac{\pi}{2}$。 -反正切函数: +\begin{minipage}{0.45\linewidth} + 反正切函数: -\begin{tikzpicture}[scale=0.9] - \draw[-latex](-3,0) -- (3,0) node[below]{$x$}; - \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; - \draw[black, thick, domain=-3:3] plot (\x,{rad(atan(\x))}) node[right]{$\arctan(x)$}; - \filldraw[black] (0,0) node[below]{$O$}; - \draw[black, densely dashed](-3,pi/2) -- (3,pi/2); - \draw[black, densely dashed](-3,-pi/2) -- (3,-pi/2); - \filldraw[black] (0.5,pi/2-0.5) node{$\dfrac{\pi}{2}$}; - \filldraw[black] (0.5,-pi/2-0.5) node{$-\dfrac{\pi}{2}$}; -\end{tikzpicture} + \begin{tikzpicture}[scale=0.75] + \draw[-latex](-3,0) -- (3,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \draw[black, thick, domain=-3:3] plot (\x,{rad(atan(\x))}) node[right]{$\arctan(x)$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, densely dashed](-3,pi/2) -- (3,pi/2); + \draw[black, densely dashed](-3,-pi/2) -- (3,-pi/2); + \filldraw[black] (0.5,pi/2-0.5) node{$\dfrac{\pi}{2}$}; + \filldraw[black] (0.5,-pi/2-0.5) node{$-\dfrac{\pi}{2}$}; + \end{tikzpicture} +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + 反余切函数: -反余切函数: - -\begin{tikzpicture}[scale=0.9] - \draw[-latex](-3,0) -- (3,0) node[below]{$x$}; - \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$}; - \draw[black, thick, domain=-3:3] plot (\x,{pi/2-rad(atan(\x))}) node[right]{$\textrm{arccot}(x)$}; - \filldraw[black] (0,0) node[below]{$O$}; - \draw[black, densely dashed](-3,pi) -- (3,pi); - \filldraw[black] (-0.5,pi/2-0.5) node{$\dfrac{\pi}{2}$}; -\end{tikzpicture} + \begin{tikzpicture}[scale=0.75] + \draw[-latex](-3,0) -- (3,0) node[below]{$x$}; + \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$}; + \draw[black, thick, domain=-3:3] plot (\x,{pi/2-rad(atan(\x))}) node[right]{$\textrm{arccot}(x)$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, densely dashed](-3,pi) -- (3,pi); + \filldraw[black] (-0.5,pi/2-0.5) node{$\dfrac{\pi}{2}$}; + \end{tikzpicture} +\end{minipage} 反切函数有如下特征: @@ -388,27 +419,32 @@ x的不同范围对应不同的法则,经典形式如下: \subparagraph{绝对值函数} \leavevmode \medskip -$ +\begin{minipage}{0.45\linewidth} + $ y=\vert x\vert=\left\{ \begin{array}{lcl} x, & & x\geqslant 0 \\ -x, & & x<0 \end{array} \right. -$ - -\begin{tikzpicture} - \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; - \draw[-latex](0,-0.5) -- (0,2.5) node[above]{$y$}; - \draw[black, thick, domain=0:2] plot (\x,\x); - \draw[black, thick, domain=-2:0] plot (\x,-\x); - \filldraw[black] (0.5,1.5) node{$\vert x\vert$}; - \filldraw[black] (0,0) node[below]{$O$}; -\end{tikzpicture} + $ +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \begin{tikzpicture} + \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-0.5) -- (0,2.5) node[above]{$y$}; + \draw[black, thick, domain=0:2] plot (\x,\x); + \draw[black, thick, domain=-2:0] plot (\x,-\x); + \filldraw[black] (0.5,1.5) node{$\vert x\vert$}; + \filldraw[black] (0,0) node[below]{$O$}; + \end{tikzpicture} +\end{minipage} \subparagraph{符号函数} \leavevmode \medskip -$ +\begin{minipage}{0.45\linewidth} + $ y=\textrm{sgn}\,x=\left\{ \begin{array}{lcl} 1, & & x>0 \\ @@ -416,228 +452,280 @@ $ -1, & & x<0 \end{array} \right. -$ - -\begin{tikzpicture} - \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; - \draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$}; - \draw[black, thick, domain=0:2] plot (\x,1); - \draw[black, thick, domain=-2:0] plot (\x,-1); - \filldraw[black] (-1.5,1) node{$\textrm{sgn}\,x$}; - \filldraw[black] circle (2pt) (0,0) node[below]{$O$}; - \filldraw[white, draw=black, line width=1pt] (0,1) circle (2pt); - \filldraw[black] (0,1) node[left]{$1$}; - \filldraw[white, draw=black, line width=1pt] (0,-1) circle (2pt); - \filldraw[black] (0,-1) node[right]{$-1$}; -\end{tikzpicture} + $ +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \begin{tikzpicture} + \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$}; + \draw[black, thick, domain=0:2] plot (\x,1); + \draw[black, thick, domain=-2:0] plot (\x,-1); + \filldraw[black] (-1.5,1) node{$\textrm{sgn}\,x$}; + \filldraw[black] circle (2pt) (0,0) node[below]{$O$}; + \filldraw[white, draw=black, line width=1pt] (0,1) circle (2pt); + \filldraw[black] (0,1) node[left]{$1$}; + \filldraw[white, draw=black, line width=1pt] (0,-1) circle (2pt); + \filldraw[black] (0,-1) node[right]{$-1$}; + \end{tikzpicture} +\end{minipage} \subparagraph{取整函数} \leavevmode \medskip $x$为实数,不超过$x$的最大整数称为其整数部分$[x]$,其定义域为$R$,值域为$Z$。 -\begin{enumerate} - \item $x-1<[x]\leqslant x$。 - \item $\lim\limits_{x\to 0^+}[x]=0$。 - \item $\lim\limits_{x\to 0^-}[x]=-1$。 -\end{enumerate} - -\begin{tikzpicture}[scale=0.6] - \draw[-latex](-3.5,0) -- (4.5,0) node[below]{$x$}; - \draw[-latex](0,-3.5) -- (0,3.5) node[above]{$y$}; - \draw[black, thick, domain=1:2] plot (\x,1); - \draw[black, thick, domain=2:3] plot (\x,2); - \draw[black, thick, domain=3:4] plot (\x,3); - \draw[black, thick, domain=-1:0] plot (\x,-1); - \draw[black, thick, domain=-2:-1] plot (\x,-2); - \draw[black, thick, domain=-3:-2] plot (\x,-3); - \filldraw[black] (-2,2) node{$[x]$}; - \filldraw[black] circle (2pt) (0,0) node[below]{$O$}; - \foreach \x in {-2,...,4} - \filldraw[white, draw=black, line width=1pt] (\x,\x-1) circle (2pt); - \foreach \x in {3,...,-3} - \filldraw[black] (\x,\x) circle (2pt); - \foreach \x/\xtext in {-3,...,-1} - \filldraw[black] (\x,0) node[below]{\xtext} -- ++(0, 3pt); - \foreach \x/\xtext in {1,...,4} - \filldraw[black] (\x,0) node[below]{\xtext} -- ++(0, 3pt); - \foreach \x/\xtext in {1,...,3} - \filldraw[black] (0,\x) node[left]{\xtext} -- +(3pt, 0); - \foreach \x/\xtext in {-3,...,-1} - \filldraw[black] (0,\x) node[right]{\xtext} -- +(3pt, 0); -\end{tikzpicture} +\begin{minipage}{0.45\linewidth} + \begin{enumerate} + \item $x-1<[x]\leqslant x$。 + \item $\lim\limits_{x\to 0^+}[x]=0$。 + \item $\lim\limits_{x\to 0^-}[x]=-1$。 + \end{enumerate} +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \begin{tikzpicture}[scale=0.6] + \draw[-latex](-3.5,0) -- (4.5,0) node[below]{$x$}; + \draw[-latex](0,-3.5) -- (0,3.5) node[above]{$y$}; + \draw[black, thick, domain=1:2] plot (\x,1); + \draw[black, thick, domain=2:3] plot (\x,2); + \draw[black, thick, domain=3:4] plot (\x,3); + \draw[black, thick, domain=-1:0] plot (\x,-1); + \draw[black, thick, domain=-2:-1] plot (\x,-2); + \draw[black, thick, domain=-3:-2] plot (\x,-3); + \filldraw[black] (-2,2) node{$[x]$}; + \filldraw[black] circle (2pt) (0,0) node[below]{$O$}; + \foreach \x in {-2,...,4} + \filldraw[white, draw=black, line width=1pt] (\x,\x-1) circle (2pt); + \foreach \x in {3,...,-3} + \filldraw[black] (\x,\x) circle (2pt); + \foreach \x/\xtext in {-3,...,-1} + \filldraw[black] (\x,0) node[below]{\xtext} -- ++(0, 3pt); + \foreach \x/\xtext in {1,...,4} + \filldraw[black] (\x,0) node[below]{\xtext} -- ++(0, 3pt); + \foreach \x/\xtext in {1,...,3} + \filldraw[black] (0,\x) node[left]{\xtext} -- +(3pt, 0); + \foreach \x/\xtext in {-3,...,-1} + \filldraw[black] (0,\x) node[right]{\xtext} -- +(3pt, 0); + \end{tikzpicture} +\end{minipage} \subsubsection{图像变换} \paragraph{平移变换} \subparagraph{左右平移} \leavevmode \medskip -$f(x)$沿$x$轴左移$x_0$个单位长度得到$f(x+x_0)$,向右移动$x_0$个单位则得到$f(x-x_0)$: - -\begin{tikzpicture}[scale=0.9] - \draw[-latex](-4,0) -- (4,0) node[below]{$x$}; - \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; - \filldraw[black] (0,0) node[below]{$O$}; - \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1); - \filldraw[black] (0,1.5) node{$-x^2+1$}; - \draw[black, thick, domain=0.5:3.5] plot (\x,{-pow((\x-2),2)+1}); - \filldraw[black] (2.5,1.5) node{$-(x-2)^2+1$}; - \draw[black, thick, domain=-3.5:-0.5] plot (\x,{-pow((\x+2),2)+1}); - \filldraw[black] (-2.5,1.5) node{$-(x+2)^2+1$}; - \filldraw[black] (1,0.5) node{$\rightarrow$}; - \filldraw[black] (-1,0.5) node{$\leftarrow$}; -\end{tikzpicture} +\begin{minipage}{0.35\linewidth} + $f(x)$沿$x$轴左移$x_0$个单位长度得到$f(x+x_0)$,向右移动$x_0$个单位则得到$f(x-x_0)$: +\end{minipage} +\hfill +\begin{minipage}{0.55\linewidth} + \begin{tikzpicture}[scale=0.9] + \draw[-latex](-4,0) -- (4,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1); + \filldraw[black] (0,1.5) node{$-x^2+1$}; + \draw[black, thick, domain=0.5:3.5] plot (\x,{-pow((\x-2),2)+1}); + \filldraw[black] (2.5,1.5) node{$-(x-2)^2+1$}; + \draw[black, thick, domain=-3.5:-0.5] plot (\x,{-pow((\x+2),2)+1}); + \filldraw[black] (-2.5,1.5) node{$-(x+2)^2+1$}; + \filldraw[black] (1,0.5) node{$\rightarrow$}; + \filldraw[black] (-1,0.5) node{$\leftarrow$}; + \end{tikzpicture} +\end{minipage} \subparagraph{上下平移} \leavevmode \medskip -$f(x)$沿$y$轴上移$y_0$个单位长度得到$f(x)+y_0$,向下移动$y_0$个单位则得到$f(x)-y_0$: - -\begin{tikzpicture}[scale=0.9] - \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; - \draw[-latex](0,-4) -- (0,4) node[above]{$y$}; - \filldraw[black] (0,0) node[below]{$O$}; - \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1); - \filldraw[black] (0,-0.75) node{$-x^2+1$}; - \draw[black, thick, domain=-1.5:1.5] plot (\x,{-\x*\x+3}); - \filldraw[black] (0,1.5) node{$-x^2+3$}; - \draw[black, thick, domain=-1.5:1.5] plot (\x,{-\x*\x+-1}); - \filldraw[black] (0,-2.5) node{$-x^2-1$}; - \filldraw[black] (-2,2.5) node{$\uparrow $}; - \filldraw[black] (-2,-2.5) node{$\downarrow $}; -\end{tikzpicture} +\begin{minipage}{0.45\linewidth} + $f(x)$沿$y$轴上移$y_0$个单位长度得到$f(x)+y_0$,向下移动$y_0$个单位则得到$f(x)-y_0$: +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \begin{tikzpicture}[scale=0.9] + \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-4) -- (0,4) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1); + \filldraw[black] (0,-0.75) node{$-x^2+1$}; + \draw[black, thick, domain=-1.5:1.5] plot (\x,{-\x*\x+3}); + \filldraw[black] (0,1.5) node{$-x^2+3$}; + \draw[black, thick, domain=-1.5:1.5] plot (\x,{-\x*\x+-1}); + \filldraw[black] (0,-2.5) node{$-x^2-1$}; + \filldraw[black] (-2,2.5) node{$\uparrow $}; + \filldraw[black] (-2,-2.5) node{$\downarrow $}; + \end{tikzpicture} +\end{minipage} \paragraph{对称变换} \subparagraph{上下对称} \leavevmode \medskip -将$f(x)$关于$x$轴对称得到$-f(x)$: - -\begin{tikzpicture} - \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; - \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; - \filldraw[black] (0,0) node[below]{$O$}; - \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1); - \filldraw[black] (0,1.5) node{$-x^2+1$}; - \draw[black, thick, domain=-1.5:1.5] plot (\x,\x*\x-1); - \filldraw[black] (0,-1.5) node{$x^2-1$}; -\end{tikzpicture} +\begin{minipage}{0.5\linewidth} + 将$f(x)$关于$x$轴对称得到$-f(x)$: +\end{minipage} +\hfill +\begin{minipage}{0.4\linewidth} + \begin{tikzpicture} + \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1); + \filldraw[black] (0,1.5) node{$-x^2+1$}; + \draw[black, thick, domain=-1.5:1.5] plot (\x,\x*\x-1); + \filldraw[black] (0,-1.5) node{$x^2-1$}; + \end{tikzpicture} +\end{minipage} \subparagraph{左右对称} \leavevmode \medskip -将$f(x)$关于$y$轴对称得到$f(-x)$: - -\begin{tikzpicture}[scale=0.8] - \draw[-latex](-4,0) -- (4,0) node[below]{$x$}; - \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; - \filldraw[black] (0,0) node[below]{$O$}; - \draw[black, thick, domain=0.25:3.5] plot (\x,{ln(\x)}); - \filldraw[black] (1.5,1.5) node{$\ln x$}; - \draw[black, thick, domain=-0.25:-3.5] plot (\x,{ln(-\x)}); - \filldraw[black] (-1.5,1.5) node{$\ln -x$}; -\end{tikzpicture} +\begin{minipage}{0.4\linewidth} + 将$f(x)$关于$y$轴对称得到\\$f(-x)$: +\end{minipage} +\hfill +\begin{minipage}{0.5\linewidth} + \begin{tikzpicture}[scale=0.8] + \draw[-latex](-4,0) -- (4,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=0.25:3.5] plot (\x,{ln(\x)}); + \filldraw[black] (1.5,1.5) node{$\ln x$}; + \draw[black, thick, domain=-0.25:-3.5] plot (\x,{ln(-\x)}); + \filldraw[black] (-1.5,1.5) node{$\ln -x$}; + \end{tikzpicture} +\end{minipage} \subparagraph{原点对称} \leavevmode \medskip -将$f(x)$关于$x$轴$y$轴即关于原点对称得到$-f(-x)$: - -\begin{tikzpicture}[scale=0.8] - \draw[-latex](-4,0) -- (4,0) node[below]{$x$}; - \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; - \filldraw[black] (0,0) node[below]{$O$}; - \draw[black, thick, domain=0.25:3.5] plot (\x,{ln(\x)}); - \filldraw[black] (1.5,1.5) node{$\ln x$}; - \draw[black, thick, domain=-0.25:-3.5] plot (\x,{-ln(-\x)}); - \filldraw[black] (-1.5,-1.5) node{$-\ln -x$}; -\end{tikzpicture} +\begin{minipage}{0.4\linewidth} + 将$f(x)$关于$x$轴$y$轴即关于原点对称得到$-f(-x)$: +\end{minipage} +\hfill +\begin{minipage}{0.5\linewidth} + \begin{tikzpicture}[scale=0.8] + \draw[-latex](-4,0) -- (4,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=0.25:3.5] plot (\x,{ln(\x)}); + \filldraw[black] (1.5,1.5) node{$\ln x$}; + \draw[black, thick, domain=-0.25:-3.5] plot (\x,{-ln(-\x)}); + \filldraw[black] (-1.5,-1.5) node{$-\ln -x$}; + \end{tikzpicture} +\end{minipage} \subparagraph{反函数对称} \leavevmode \medskip -将$f(x)$关于$y=x$轴对称得到$f^{-1}(x)$: - -\begin{tikzpicture}[scale=0.8] - \draw[-latex](-2,0) -- (e,0) node[below]{$x$}; - \draw[-latex](0,-2) -- (0,e) node[above]{$y$}; - \filldraw[black] (0,0) node[below]{$O$}; - \draw[black, thick, domain=0.25:e] plot (\x,{ln(\x)}); - \filldraw[black] (1.5,-1.5) node{$\ln x$}; - \draw[black, thick, domain=-1:1] plot (\x,{exp(\x)}); - \filldraw[black] (-1.5,1.5) node{$e^x$}; - \draw[black, densely dashed] (-2,-2) -- (e-0.5,e-0.5) node[above]{$y=x$}; -\end{tikzpicture} +\begin{minipage}{0.55\linewidth} + 将$f(x)$关于$y=x$轴对称得到$f^{-1}(x)$: +\end{minipage} +\hfill +\begin{minipage}{0.35\linewidth} + \begin{tikzpicture}[scale=0.8] + \draw[-latex](-2,0) -- (e,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,e) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=0.25:e] plot (\x,{ln(\x)}); + \filldraw[black] (1.5,-1.5) node{$\ln x$}; + \draw[black, thick, domain=-1:1] plot (\x,{exp(\x)}); + \filldraw[black] (-1.5,1.5) node{$e^x$}; + \draw[black, densely dashed] (-2,-2) -- (e-0.5,e-0.5) node[above]{$y=x$}; + \end{tikzpicture} +\end{minipage} \subparagraph{函数绝对值} \leavevmode \medskip -保留$f(x)$函数值在$[0,\infty]$的部分,并对$[-\infty,0]$部分进行上下对称: - -\begin{tikzpicture} - \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; - \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; - \filldraw[black] (0,0) node[below]{$O$}; - \draw[black, thick, domain=1:1.5] plot (\x,\x*\x-1); - \draw[black, thick, densely dashed, domain=-1:1] plot (\x,\x*\x-1); - \draw[black, thick, domain=-1:1] plot (\x,-\x*\x+1); - \draw[black, thick, domain=-1.5:-1] plot (\x,\x*\x-1); - \filldraw[black] (0,1.5) node{$\vert x^2-1\vert$}; -\end{tikzpicture} +\begin{minipage}{0.5\linewidth} + 保留$f(x)$函数值在$[0,\infty]$的部分,并对$[-\infty,0]$部分进行上下对称: +\end{minipage} +\hfill +\begin{minipage}{0.4\linewidth} + \begin{tikzpicture} + \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=1:1.5] plot (\x,\x*\x-1); + \draw[black, thick, densely dashed, domain=-1:1] plot (\x,\x*\x-1); + \draw[black, thick, domain=-1:1] plot (\x,-\x*\x+1); + \draw[black, thick, domain=-1.5:-1] plot (\x,\x*\x-1); + \filldraw[black] (0,1.5) node{$\vert x^2-1\vert$}; + \end{tikzpicture} +\end{minipage} \subparagraph{自变量绝对值} \leavevmode \medskip -先只保留$f(x)$定义域在$[0,\infty]$的部分,然后在$[-\infty,0]$部分使用$[0,\infty]$的部分进行左右对称: - -\begin{tikzpicture} - \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; - \draw[-latex](0,-1) -- (0,3) node[above]{$y$}; - \filldraw[black] (0,0) node[below]{$O$}; - \draw[black, thick, domain=0:1.25] plot (\x,{-pow(\x,3)+1}); - \draw[black, thick, densely dashed, domain=-1.25:0] plot (\x,{-pow(\x,3)+1}); - \draw[black, thick, domain=-1.25:0] plot (\x,{-pow(-\x,3)+1}); - \filldraw[black] (1,2) node{$-\vert x\vert^3+1$}; -\end{tikzpicture} +\begin{minipage}{0.5\linewidth} + 先只保留$f(x)$定义域在$[0,\infty]$的部分,然后在$[-\infty,0]$部分使用$[0,\infty]$的部分进行左右对称: +\end{minipage} +\hfill +\begin{minipage}{0.4\linewidth} + \begin{tikzpicture} + \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-1) -- (0,3) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=0:1.25] plot (\x,{-pow(\x,3)+1}); + \draw[black, thick, densely dashed, domain=-1.25:0] plot (\x,{-pow(\x,3)+1}); + \draw[black, thick, domain=-1.25:0] plot (\x,{-pow(-\x,3)+1}); + \filldraw[black] (1,2) node{$-\vert x\vert^3+1$}; + \end{tikzpicture} +\end{minipage} \paragraph{伸缩变换} \subparagraph{水平伸缩} \leavevmode \medskip -纵坐标不变,当$k>1$时,$y=f(kx)$是$y=f(x)$缩短k倍得到,当$01$时,$y=f(kx)$是$y=f(x)$缩短k倍得到,当$00$,周期为$2\pi$。 + 其中$r$为线的极径,$\theta$为极角,$a$为形状参数且$a>0$,周期为$2\pi$。 +\end{minipage} +\hfill +\begin{minipage}{0.35\linewidth} + \begin{tikzpicture}[scale=0.8] + \draw[-latex](-5,0) -- (1,0) node[below]{$x$}; + \draw[-latex](0,-3) -- (0,3) node[above]{$y$}; + \draw[black, thick, domain=0:360,smooth,variable=\t, samples=300] plot ({\t}:{2*(1-cos(\t))}); + \filldraw[black] (0,0) node[below]{$O$}; + \draw (-1,0) circle [radius=1]; + \draw (-3,0) circle [radius=1]; + \end{tikzpicture} +\end{minipage} 在直角坐标系下表达式:$x^2+y^2+a\cdot x=a\cdot\sqrt{x^2+y^2}$和$x^2+y^2-a\cdot x=a\cdot\sqrt{x^2+y^2}$。 -参数方程:$x=a\cdot(2\cdot\cos(t)-cos(2\cdot t))$与$y=a\cdot(2\cdot\sin(t)-sin(2\cdot t))$ - -\begin{tikzpicture}[scale=0.8] - \draw[-latex](-5,0) -- (1,0) node[below]{$x$}; - \draw[-latex](0,-3) -- (0,3) node[above]{$y$}; - \draw[black, thick, domain=0:360,smooth,variable=\t, samples=300] plot ({\t}:{2*(1-cos(\t))}); - \filldraw[black] (0,0) node[below]{$O$}; -\end{tikzpicture} +参数方程:$x=a\cdot(2\cdot\cos(t)-cos(2\cdot t))$与$y=a\cdot(2\cdot\sin(t)-sin(2\cdot t))$。 水平心形线对应参数: \leavevmode \medskip @@ -650,18 +738,22 @@ $f(x)$沿$y$轴上移$y_0$个单位长度得到$f(x)+y_0$,向下移动$y_0$个 \paragraph{玫瑰线} \leavevmode \medskip -表达式:$r=a\sin(n\theta)$,周期为$\dfrac{2\pi}{n}$。 +\begin{minipage}{0.55\linewidth} + 表达式:$r=a\sin(n\theta)$,周期为$\dfrac{2\pi}{n}$。 -当$n$为3时为三叶,2时为四叶,$\dfrac{3}{2}$为六叶。三叶时周期为$\dfrac{2\pi}{3}$。 + 当$n$为3时为三叶,2时为四叶,$\dfrac{3}{2}$为六叶。三叶时周期为$\dfrac{2\pi}{3}$。 -直角坐标系下表达式:$x=a\cdot\sin(n\cdot\theta)\cdot\cos(\theta)$与$y=a\cdot\sin(n\cdot)\cdot\sin(\theta)$ - -\begin{tikzpicture}[scale=0.8] - \draw[-latex](-3,0) -- (3,0) node[below]{$x$}; - \draw[-latex](0,-pi) -- (0,pi/2) node[above]{$y$}; - \draw[domain=0:180,samples=100] plot (\x:{3*sin(\x*3)}); - \filldraw[black] (0,0) node[below]{$O$}; -\end{tikzpicture} + 直角坐标系下表达式:$x=a\cdot\sin(n\cdot\theta)\cdot\cos(\theta)$与$y=a\cdot\sin(n\cdot)\cdot\sin(\theta)$ +\end{minipage} +\hfill +\begin{minipage}{0.35\linewidth} + \begin{tikzpicture}[scale=0.8] + \draw[-latex](-3,0) -- (3,0) node[below]{$x$}; + \draw[-latex](0,-pi) -- (0,pi/2) node[above]{$y$}; + \draw[domain=0:180,samples=100] plot (\x:{3*sin(\x*3)}); + \filldraw[black] (0,0) node[below]{$O$}; + \end{tikzpicture} +\end{minipage} 三叶玫瑰线对应参数: \leavevmode \medskip @@ -674,14 +766,18 @@ $f(x)$沿$y$轴上移$y_0$个单位长度得到$f(x)+y_0$,向下移动$y_0$个 \paragraph{阿基米德螺线} \leavevmode \medskip -表达式:$r=a\theta$,其中$a>0$,$\theta\geqslant 0$由0开始增大时$r$也在不断增大。 - -\begin{tikzpicture}[scale=0.2] - \draw[-latex](-10,0) -- (15,0) node[below]{$x$}; - \draw[-latex](0,-15) -- (0,10) node[above]{$y$}; - \draw[domain=0:720,samples=100] plot (\x:{rad(\x)}); - \filldraw[black] (0,0) node[below]{$O$}; -\end{tikzpicture} +\begin{minipage}{0.5\linewidth} + 表达式:$r=a\theta$,其中$a>0$,$\theta\geqslant 0$由0开始增大时$r$也在不断增大。 +\end{minipage} +\hfill +\begin{minipage}{0.4\linewidth} + \begin{tikzpicture}[scale=0.2] + \draw[-latex](-10,0) -- (15,0) node[below]{$x$}; + \draw[-latex](0,-15) -- (0,10) node[above]{$y$}; + \draw[domain=0:720,samples=100] plot (\x:{rad(\x)}); + \filldraw[black] (0,0) node[below]{$O$}; + \end{tikzpicture} +\end{minipage} \paragraph{伯努利双扭线} \leavevmode \medskip @@ -711,16 +807,20 @@ $f(x)$沿$y$轴上移$y_0$个单位长度得到$f(x)+y_0$,向下移动$y_0$个 \subsubsection{直角坐标系下画极坐标图像} -令$\theta$为$x$,令$r$为$y$。如心形线$r=2(1-\cos\theta)$: +\begin{minipage}{0.5\linewidth} + 令$\theta$为$x$,令$r$为$y$。如心形线$r=2(1-\cos\theta)$: -\begin{tikzpicture}[scale=0.5] - \draw[-latex](-5,0) -- (5,0) node[below]{$x$}; - \draw[-latex](0,-0.5) -- (0,5) node[above]{$y$}; - \draw[black, thick, smooth, domain=-5:5] plot (\x,{2*(1-cos(\x r))}) node at (0,4){$2(1-\cos(\theta))$}; - \filldraw[black] (0,0) node[below]{$O$}; -\end{tikzpicture} - -按直角坐标系的图就可以计算出对应的$r$从而能画出对应的图像。 + 按直角坐标系的图就可以计算出对应的$r$从而能画出对应的图像。 +\end{minipage} +\hfill +\begin{minipage}{0.4\linewidth} + \begin{tikzpicture}[scale=0.5] + \draw[-latex](-5,0) -- (5,0) node[below]{$x$}; + \draw[-latex](0,-0.5) -- (0,5) node[above]{$y$}; + \draw[black, thick, smooth, domain=-5:5] plot (\x,{2*(1-cos(\x r))}) node at (0,4){$2(1-\cos(\theta))$}; + \filldraw[black] (0,0) node[below]{$O$}; + \end{tikzpicture} +\end{minipage} \subsection{参数法} @@ -737,33 +837,56 @@ $ 摆线,又称旋轮线、圆滚线,是一个圆沿一条直线滚动时,圆边界上一定点所形成的轨迹。 -令圆半径为$r$,摆点与圆心所成直线所转动夹角对应弧度为$t$,其中$t\in[0,2\pi]$,所对应参数方程为: - -$$ +令圆半径为$r$,摆点与圆心所成直线所转动夹角对应弧度为$t$,其中$t\in[0,2\pi]$,所对应参数方程为:$ \left\{ \begin{array}{lcl} x=r(t-\sin t) \\ y=r(1-\cos t) \end{array} \right. -$$ +$ + +\begin{tikzpicture}[scale=1.5] + \draw[-latex](-1.5,0) -- (5,0) node[below]{$x$}; + \draw[-latex](0,-0.5) -- (0,2) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black,scale=0.35, domain=-1.7:2*4.0, smooth, variable=\t ] + plot ( {2*(\t-sin(\t r))}, {2*(1-cos(\t r))}); + \draw (0.7,0.7) circle [radius=0.7]; + \draw[black](0.7,0) -- (0.7,1.4); + \draw[black, densely dashed](2.25,0) -- (2.25,1.4); + \filldraw[black] (2.5,0.625) node{$2a$}; +\end{tikzpicture} \subsubsection{星形线(内摆线)} -与半径为$r$的定圆内切的半径为$\dfrac{r}{4}$的动圆沿定圆无滑动地滚动,动圆上一点的轨迹称为星形线。 +\begin{minipage}{0.5\linewidth} + 与半径为$r$的定圆内切的半径为$\dfrac{r}{4}$的动圆沿定圆无滑动地滚动,动圆上一点的轨迹称为星形线。 -令$t$表示摆点与圆心的连线所构成夹角的弧度,其中$t\in[0,2\pi]$,得对应参数方程: - -$$ + 令$t$表示摆点与圆心的连线所构成夹角的弧度,其中$t\in[0,2\pi]$,得对应参数方程:$ \left\{ \begin{array}{lcl} x=r\cos^3t \\ y=r\sin^3t \end{array} \right. -$$ + $ -由$\cos^2t+\sin^2t=1$得到直角坐标方程:$x^{\frac{2}{3}}+y^{\frac{2}{3}}=r^{\frac{2}{3}}$ + 由$\cos^2t+\sin^2t=1$得到直角坐标方程:$x^{\frac{2}{3}}+y^{\frac{2}{3}}=r^{\frac{2}{3}}$。 +\end{minipage} +\hfill +\begin{minipage}{0.4\linewidth} + \begin{tikzpicture}[scale=2] + \draw[-latex](-1.25,0) -- (1.25,0) node[below]{$x$}; + \draw[-latex](0,-1.25) -- (0,1.25) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black,scale=1, domain=-1.7:2*4.0, smooth, variable=\t ] + plot ( {(pow(cos(\t r),3))}, {(pow(sin(\t r),3))}); + \filldraw[black] (-0.25,0.125) node{$a$}; + \draw (0,0) circle [radius=1]; + \draw (0.75,0) circle [radius=0.25]; + \end{tikzpicture} +\end{minipage} \section{常用基础知识} \subsection{数列} 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 f5e1a553a0f7c66d6abf090106d1bec9946b6d7a..b60af38b08673fcb195b7feb97bb4f2cc563720c 100644 GIT binary patch delta 219947 zcmY(KLwGJ+u%(j|+qP}n=80|FPQLiWwr$(CZQC~czulAD(_Mo-soKxGYOUQ6jhYn+ zAIA=0W##9Gc6MU2cxhp5jSCXk;5v@` z1xR_8@HJiCHyvRPAhTe4N6FM`-5x5HFHgsUW7`AfL~V>D<&_r!tvIo6KArfn9KXGQ zpQoj-Zv@seT}aC5#Wg|FRMboeoAJ zeZO$CC!*ujIoZl;5e<>DLp{= z)#JZUD;w&obK7=P(rs}i2mS^yCzptGwgvd^bH=R^1DC9W=0RNg?B2Kg9u=e1z^bCB zLy%IJZsgn9%5QadK7&ViV~x*l*lj!FxJ1a6K+^!IATp-gc##W5GpDQO7t*m;Z#Yo- z(xbXL-gz$6@Q{3-I7x$b3L8SCjz0j>jvn-Iok!6FMN;(86cVOc8ql*OlYpU3N|VP4 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