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z-0q!*-HpmG#tO}KNDi!Kmp`vt_!nLFeddPpsYH&iaOnJQxk8d>87*XugFhFE3OEqV rImvNqNa)N_Zc#L4cU0nll+yFclyPGiGBh6kw;u;6w6xMHGAREa!+bDI diff --git a/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex b/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex index b35dd70..c58e883 100644 --- a/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex +++ b/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex @@ -48,6 +48,36 @@ \pagestyle{plain} \setcounter{page}{1} +\section{函数中值定理} + +都假定$f(x)$在$[a,b]$上连续。 + +\subsection{有界与最值定理} + +\textcolor{aqua}{\textbf{定理:}}$m\leqslant f(x)\leqslant M$,其中$m$,$M$分别为$f(x)$在$[a,b]$上的最大值和最小值。 + +\subsection{介值定理} + +\textcolor{aqua}{\textbf{定理:}}当$m\leqslant\mu\leqslant M$,存在$\varepsilon\in[a,b]$,使得$f(\varepsilon)=\mu$。 + +\subsection{平均值定理} + +\textcolor{aqua}{\textbf{定理:}}当$a0$,使$\forall x\in U(x_0,\delta)$恒有$f(x)\geqslant f(x_0)$,则$f(x)$在$x_0$处取极小值,恒有$f(x)\leqslant f(x_0)$,则$f(x)$在$x_0$处取极大值。 费马引理\textcolor{violet}{\textbf{定义:}}若$f(x)$在$x_0$处取得极值,且$f(x)$在$x_0$处可导,则$f'(x_0)=0$。 @@ -89,6 +121,15 @@ $\text{罗尔定理}\xrightleftharpoons[\text{特例:}f(a)=f(b)]{\text{泛化 \end{tikzpicture} \end{minipage} +\subsubsection{推广} + +\begin{itemize} + \item 设$f(x)$在$(a,b)$内可导,$\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to b^-}f(x)=A$,则在$(a,b)$内至少存在一点$\varepsilon$,使得$f'(\varepsilon)=0$。 + \item 设$f(x)$在$(a,b)$内可导,$\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to b^-}f(x)=\pm\infty$,则在$(a,b)$内至少存在一点$\varepsilon$,使得$f'(\varepsilon)=0$。 + \item 设$f(x)$在$(a,+\infty)$内可导,$\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to+\infty}f(x)=A$,则在$(a,+\infty)$内至少存在一点$\varepsilon$,使得$f'(\varepsilon)=0$。 + \item 设$f(x)$在$(\infty,+\infty)$内可导,$\lim\limits_{x\to-\infty}f(x)=\lim\limits_{x\to+\infty}f(x)=A$,则在$(-\infty,+\infty)$内至少存在一点$\varepsilon$,使得$f'(\varepsilon)=0$。 +\end{itemize} + \subsection{拉格朗日中值定理} \begin{enumerate} @@ -125,6 +166,18 @@ $\text{罗尔定理}\xrightleftharpoons[\text{特例:}f(a)=f(b)]{\text{泛化 则$\exists\,\xi\in(a,b)$,使得$\dfrac{f(b)-f(a)}{F(b)-F(a)}=\dfrac{f'(\xi)}{F'(\xi)}$。 +\section{积分中值定理} + +\subsection{定理} + +\textcolor{aqua}{\textbf{定理:}}若$f(x)$在$[a,b]$上连续,则存在$\varepsilon\in[a,b]$,使得$\int_a^bf(x)\,\textrm{d}x=f(\varepsilon)(b-a)$。 + +\subsection{证明} + +已知$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$。 + +由介值定理可知$\varepsilon\in[a,b]$,使得$f(\varepsilon)=\dfrac{1}{b-a}\int_a^bf(x)\,\textrm{d}x$。 + \section{洛必达法则} \subsection{定理}