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Didnelpsun
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advanced-math/1-perpare/perpare.tex
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@@ -531,7 +531,7 @@ $$
\begin{tikzpicture}[scale=0.9]
\draw[-latex](-3,0) -- (3,0) node[below]{$x$};
\draw[-latex](0,-0.5) -- (0,4) node[above]{$y$};
\draw[black, thick, domain=-3:3] plot (\x,{pi/2-rad(atan(\x))}) node[right]{$\rm{arccot}(x)$};
\draw[black, thick, domain=-3:3] plot (\x,{pi/2-rad(atan(\x))}) node[right]{$\rm{arccot}(\textit{x})$};
\filldraw[black] (0,0) node[below]{$O$};
\draw[black, densely dashed](-3,pi) -- (3,pi);
\filldraw[black] (-0.5,pi/2-0.5) node{$\dfrac{\pi}{2}$};
@@ -541,11 +541,11 @@ $$
\begin{enumerate}
\item 特殊函数值：$\arctan 0=0$，$\arctan\dfrac{\pi}{6}=\dfrac{\sqrt{3}}{3}=$，$\arctan 1=\dfrac{\pi}{4}$，$\arctan\sqrt{3}=\dfrac{\pi}{3}$，$\rm{arccot}0=\dfrac{\pi}{2}$，$\rm{arccot}\sqrt{3}=\dfrac{\pi}{6}$，$\rm{arccot}1=\dfrac{\pi}{4}$，$\rm{arccot}\dfrac{\sqrt{3}}{3}=\dfrac{\pi}{3}$。
\item 定义域：$(-\infty, +\infty)$，值域：$\arctan x:[-\dfrac{\pi}{2},+\dfrac{\pi}{2}]$，$\rm{arccot}x:[0,\pi]$。
\item 定义域：$(-\infty, +\infty)$，值域：$\arctan x:[-\dfrac{\pi}{2},+\dfrac{\pi}{2}]$，$\rm{arccot}\,\textit{x}:[0,\pi]$。
\item 单调性：$y=\arctan x$单调增，$y=\rm{arccot}x$单调减。
\item 奇偶性：$y=\arctan x$为奇函数。
\item 有界性：$\vert\arctan x\vert\leqslant\dfrac{\pi}{2}$，$0\leqslant\rm{arccot}x\leqslant\pi$。
\item 性质：$\arctan x+\rm{arccot}x=\dfrac{\pi}{2}(-\infty<x<+\infty)$
\item 有界性：$\vert\arctan x\vert\leqslant\dfrac{\pi}{2}$，$0\leqslant\rm{arccot}\,\textit{x}\leqslant\pi$。
\item 性质：$\arctan x+\rm{arccot}\,\textit{x}=\dfrac{\pi}{2}(-\infty<x<+\infty)$
\end{enumerate}
\subparagraph{初等函数} \leavevmode \bigskip

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advanced-math/3-function-and-limit/function-and-limit.synctex(busy) Normal file
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advanced-math/4-single-variable-function-differential-calculus/single-variable-function-differential-calculus.tex
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@@ -533,14 +533,57 @@ $\therefore \dfrac{y^{(6)}(0)}{6!}=-\dfrac{1}{6}\Rightarrow y^{(6)}(0)=-5!=-120$
\subsection{基本求导公式}
\subsubsection{对幂指函数}
\begin{center}
\begin{tabular}{|c|c|}
\begin{tabular}{|c|c|c|c|}
\hline
原函数 & 导函数 \\ \hline
$C$ & $0$ \\ \hline
$n^x$ & $n^x\ln n$ \\ \hline
$\log_ax$ & $\dfrac{1}{x\ln a}$ \\ \hline
$\ln x(\ln\vert x\vert)$ & $\dfrac{1}{x}$ \\
原函数 & 导函数 & 原函数 & 导函数\\ \hline
$C$ & $0$ & $n^x$ & $n^x\ln n$ \\ \hline
$\log_ax$ & $\dfrac{1}{x\ln a}$ & $\ln x=\ln\vert x\vert$ & $\dfrac{1}{x}$ \\ \hline
$x^n$ & $nx^{n-1}$ & $\sqrt[n]{x}$ & $\dfrac{x^{-\frac{n-1}{n}}}{n}$ \\ \hline
$\dfrac{1}{x^n}$ & $-\dfrac{n}{x^{n+1}}$ & & \\
\hline
\end{tabular}
\end{center}
\subsubsection{三角与反三角函数}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
原函数 & 导函数 & 原函数 & 导函数\\ \hline
$\sin x$ & $\cos x$ & $\cos x$ & $-\sin x$ \\ \hline
$\tan x$ & $\dfrac{1}{\cos^2x}=\sec^2x$ & $\cot x$ & $\dfrac{1}{\sin^2x}=\csc^2x$ \\ \hline
$\sec x$ & $\sec x\tan x$ & $\csc x$ & $-\csc x\cot x$ \\ \hline
$\arcsin x$ & $\dfrac{1}{1-x^2}$ & $\arccos x$ & $-\dfrac{1}{1-x^2}$ \\ \hline
$\arctan x$ & $\dfrac{1}{1+x^2}$ & $\rm{arccot}\,\textit{x}$ & $-\dfrac{1}{1+x^2}$ \\ \hline
$\rm{arcsec}\,\textit{x}$ & $\dfrac{1}{x\sqrt{x^2-1}}$ & $\rm{arccsc}\,\textit{x}$ & $-\dfrac{1}{x\sqrt{x^2-1}}$ \\ \hline
\hline
\end{tabular}
\end{center}
\subsubsection{双曲与反双曲函数}
\begin{itemize}
\item 双曲正弦：$\rm{sinh}\,\textit{x}=\rm{sh}\,\textit{x}=\dfrac{e^x-e^{-x}}{2}$。
\item 双曲余弦：$\rm{cosh}\,\textit{x}=\rm{ch}\,\textit{x}=\dfrac{e^x+e^{-x}}{2}$。
\item 双曲正切：$\rm{tanh}\,\textit{x}=\rm{th}\,\textit{x}=\dfrac{\rm{sinh}\,\textit{x}}{\rm{cosh}\,\textit{x}}=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}$。
\item 双曲余切：$\rm{coth}\,\textit{x}=\dfrac{\rm{cosh}\,\textit{x}}{\rm{sinh}\,\textit{x}}=\dfrac{e^x+e^{-x}}{e^x-e^{-x}}$。
\item 双曲正割：$\rm{sech}\,\textit{x}=\dfrac{1}{\rm{cosh}\,\textit{x}}=\dfrac{2}{e^x+e^{-x}}$。
\item 双曲余割：$\rm{csch}\,\textit{x}=\dfrac{1}{\rm{sinh}\,\textit{x}}=\dfrac{2}{e^x-e^{-x}}$。
\item 反双曲正弦：$\rm{arcsinh}\,\textit{x}=\ln\left(x+\sqrt{x^2+1}\right)$。
\item 反双曲余弦：$\rm{arccosh}\,\textit{x}=\ln\left(x+\sqrt{x^2-1}\right)$。
\item 反双曲正切：$\rm{arctanh}\,\textit{x}=\dfrac{1}{2}\ln\left(\dfrac{1+x}{1-x}\right)$。
\end{itemize}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
原函数 & 导函数 & 原函数 & 导函数\\ \hline
$\rm{sinh}\,\textit{x}$ & $\rm{cosh}\,\textit{x}$ & $\rm{cosh}\,\textit{x}$ & $\rm{sinh}\,\textit{x}$ \\ \hline
$\rm{tanh}\,\textit{x}$ & $\dfrac{1}{\rm{cosh}\,\textit{x}^2}$ & $\rm{arcsinh}\,\textit{x}$ & $\dfrac{1}{\sqrt{x^2+1}}$ \\ \hline
$\rm{arccosh}\,\textit{x}$ & $\dfrac{1}{\sqrt{x^2-1}}$ & $\rm{arctan}\,\textit{x}$ & $\dfrac{1}{1-x^2}$ \\
\hline
\end{tabular}
\end{center}