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更新多元微分

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Didnelpsun
2021-07-17 23:16:30 +08:00
parent bf12e41ca8
commit cbb598ca9e
15 changed files with 878 additions and 476 deletions
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advanced-math/knowledge/1-function-and-limit/function-and-limit.pdf
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advanced-math/knowledge/1-function-and-limit/function-and-limit.tex
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@@ -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$x\in[0,\dfrac{\pi}{2}]$，$\sin x<x<\tan x$。
@@ -957,64 +963,80 @@ $=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\lef
\subsubsection{可去间断点（可补间断点）}
\textcolor{violet}{\textbf{定义：}}若$\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$};
\draw[-latex](0,-0.5) -- (0,3) node[above]{$y$};
\filldraw[black] (0,0) node[below]{$O$};
\draw[black, thick, domain=-0.5:2] plot (\x,{pow(e,\x-1)});
\filldraw[white, draw=black, line width=1pt] (1,1) circle (2pt);
\draw[black, densely dashed](1,2) -- (0,2) node[left]{$B$};
\draw[black, densely dashed](1,1) -- (0,1) node[left]{$A$};
\draw[black, densely dashed](1,2) -- (1,0) node[below]{$x_0$};
\filldraw[black] (1,2) circle (2pt) node[above]{$f(x_0)=B$};
\filldraw[black] (1,1) node[right]{$\lim\limits_{x\to x_0}=A$};
\end{tikzpicture}
\begin{minipage}{0.6\linewidth}
\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0}f(x)=A\neq f(x_0)$（甚至可以没有定义）。
\end{minipage}
\hfill
\begin{minipage}{0.3\linewidth}
\begin{tikzpicture}
\draw[-latex](-0.5,0) -- (3,0) node[below]{$x$};
\draw[-latex](0,-0.5) -- (0,3) node[above]{$y$};
\filldraw[black] (0,0) node[below]{$O$};
\draw[black, thick, domain=-0.5:2] plot (\x,{pow(e,\x-1)});
\filldraw[white, draw=black, line width=1pt] (1,1) circle (2pt);
\draw[black, densely dashed](1,2) -- (0,2) node[left]{$B$};
\draw[black, densely dashed](1,1) -- (0,1) node[left]{$A$};
\draw[black, densely dashed](1,2) -- (1,0) node[below]{$x_0$};
\filldraw[black] (1,2) circle (2pt) node[above]{$f(x_0)=B$};
\filldraw[black] (1,1) node[right]{$\lim\limits_{x\to x_0}=A$};
\end{tikzpicture}
\end{minipage}
\subsubsection{跳跃间断点}
\textcolor{violet}{\textbf{定义：}}若$\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$};
\draw[-latex](0,-0.5) -- (0,3) node[above]{$y$};
\filldraw[black] (0,0) node[below]{$O$};
\draw[black, thick, domain=-0.5:1] plot (\x,{pow(e,\x-1)});
\draw[black, thick, domain=1:1.5] plot (\x,{pow(e,\x-1)+1});
\filldraw[white, draw=black, line width=1pt] (1,1) circle (2pt);
\draw[black, densely dashed](1,2) -- (0,2) node[left]{$B$};
\draw[black, densely dashed](1,1) -- (0,1) node[left]{$A$};
\draw[black, densely dashed](1,2) -- (1,0) node[below]{$x_0$};
\filldraw[black] (1,2) circle (2pt) node[above]{$f(x_0)=B$};
\filldraw[black] (1,1) node[right]{$\lim\limits_{x\to x_0}=A$};
\end{tikzpicture}
\begin{minipage}{0.6\linewidth}
\textcolor{violet}{\textbf{定义：}}若$\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)$。
\end{minipage}
\hfill
\begin{minipage}{0.3\linewidth}
\begin{tikzpicture}
\draw[-latex](-0.5,0) -- (3,0) node[below]{$x$};
\draw[-latex](0,-0.5) -- (0,3) node[above]{$y$};
\filldraw[black] (0,0) node[below]{$O$};
\draw[black, thick, domain=-0.5:1] plot (\x,{pow(e,\x-1)});
\draw[black, thick, domain=1:1.5] plot (\x,{pow(e,\x-1)+1});
\filldraw[white, draw=black, line width=1pt] (1,1) circle (2pt);
\draw[black, densely dashed](1,2) -- (0,2) node[left]{$B$};
\draw[black, densely dashed](1,1) -- (0,1) node[left]{$A$};
\draw[black, densely dashed](1,2) -- (1,0) node[below]{$x_0$};
\filldraw[black] (1,2) circle (2pt) node[above]{$f(x_0)=B$};
\filldraw[black] (1,1) node[right]{$\lim\limits_{x\to x_0}=A$};
\end{tikzpicture}
\end{minipage} \medskip
可去间断点与跳跃间断点的左右极限都存在的间断点都称为第一类间断点。
\subsubsection{无穷间断点}
\textcolor{violet}{\textbf{定义：}}若$\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$};
\draw[-latex](0,-2) -- (0,2) node[above]{$y$};
\filldraw[black] (0,0) node[below]{$O$};
\draw[black, thick, domain=0.5:2] plot (\x,{pow(\x,-1)});
\draw[black, thick, domain=-2:-0.5] plot (\x,{pow(\x,-1)});
\end{tikzpicture}
\begin{minipage}{0.6\linewidth}
\textcolor{violet}{\textbf{定义：}}若$\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$处为无穷间断点。
\end{minipage}
\hfill
\begin{minipage}{0.3\linewidth}
\begin{tikzpicture}
\draw[-latex](-2,0) -- (2,0) node[below]{$x$};
\draw[-latex](0,-2) -- (0,2) node[above]{$y$};
\filldraw[black] (0,0) node[below]{$O$};
\draw[black, thick, domain=0.5:2] plot (\x,{pow(\x,-1)});
\draw[black, thick, domain=-2:-0.5] plot (\x,{pow(\x,-1)});
\end{tikzpicture}
\end{minipage}
\subsubsection{振荡间断点}
\textcolor{violet}{\textbf{定义：}}若$\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$};
\draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$};
\filldraw[black] (0,0) node[below]{$O$};
\draw[black, thick, domain=0.01:2] plot (\x,{sin(pow(\x,-1) r)});
\draw[black, thick, domain=-2:-0.01] plot (\x,{sin(pow(\x,-1) r)});
\end{tikzpicture}
\begin{minipage}{0.6\linewidth}
\textcolor{violet}{\textbf{定义：}}若$\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$就是振荡间断点。
\end{minipage}
\hfill
\begin{minipage}{0.3\linewidth}
\begin{tikzpicture}
\draw[-latex](-2,0) -- (2,0) node[below]{$x$};
\draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$};
\filldraw[black] (0,0) node[below]{$O$};
\draw[black, thick, domain=0.01:2] plot (\x,{sin(pow(\x,-1) r)});
\draw[black, thick, domain=-2:-0.01] plot (\x,{sin(pow(\x,-1) r)});
\end{tikzpicture}
\end{minipage}
无穷间断点与振荡间断点的左右极限都不存在的点都是第二类间断点。