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%-------------------------------------------------------------------------------------------------
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\title{函数与导数}
\author{贾方正}
\date{2019年五校联考整理}
\begin{document}
\maketitle
\tableofcontents

%正文开始录入
\chapter{导函数符号讨论“界点”如何确定}
在某个区间$(a,b)$内,如果$f'(x)>0$,那么函数$y=f(x)$在这个区间内单调递增;如果$f'(x)<0$,那么函数$y=f(x)$在这个区间内单调递增.

\begin{minipage}[]{0.5\textwidth}
\textbf{例}已知导函数$f'(x)$的下列信息：\\
当$1<x<4$时,$f'(x)>0$;\\
当$x>4$,或$x<1$时,$f'(x)<0$;\\
当$x=4$,或$x=1$时,$f'(x)=0$.\\
试画出函数$f(x)$图象的大致形状.\\
\textcolor{red}{\hspace{2em}解：当$x=4$,或$x=1$时$f'(x)=0$,这两个点比较特殊,我们称为“临界点”,简称为“界点”.}
\end{minipage}%
\begin{minipage}[]{0.5\textwidth}
\centering
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\tikzstyle{every node}=[font=\small,scale=0.65]
%%定义坐标轴大小 %%x正半轴 %%x负半轴
\def\xy{7} \def\xz{1}  %%x正半轴 %%x负半轴
\def\ys{4}  \def\yx{1}  %%y正半轴 %%y负半轴
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\draw[-stealth] (0,-\yx)--(0,\ys) node[right]{$y$} coordinate(y axis);
%\draw[help lines,step=0.5](-\xz,-\yx) grid (\xy,\ys);
%%画函数曲线
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% \draw (-1,4) to [out=-80,in=180] (1,1) to [out=0,in=180] (3,3);
%%绘制坐标轴
\foreach \x/\xtext in {1/1,3.8/4}
\draw[xshift=\x cm] (0pt,4pt) -- (0pt,-4pt) node[below] {$\xtext$};
\end{tikzpicture}
\end{minipage}%

\section{根据二次项系数确定分类“界点”}
\begin{tcolorbox}
已知函数$f(x)=1-x^2+a\ln x(a\in\mathbb{R})$.
\end{tcolorbox}

\begin{marker}
导函数中含有二次三项式,需对最高项的系数分类讨论：\\
(1)根据二次项系数是否为$0$,判断函数是否为二次函数；\\
(2)由二次项系数的正负,判断二次函数图象的开口方向,从而寻找导数的变号零点．\\
\end{marker}


\section{根据判别式确定分类“界点”}
\begin{marker}
求导后，要判断导函数是否有零点(或导函数分子能否分解因式)，若导函数是二次函数
或与二次函数有关，此时涉及二次方程问题，Δ与0 的大小关系往往不确定，所以必须寻找
分界点，进行分类讨论．	
\end{marker}

\section{根据导函数零点的大小确定分类“界点”}
\begin{marker}
(1)根据导函数的“零点”划分定义域时，既要考虑导函数“零点”是否在定义域内，
还要考虑多个“零点”的大小问题，如果多个“零点”的大小关系不确定，也需要分类讨
论．\\
(2)导函数“零点”可求，可根据“零点”之间及“零点”与区间端点之间的大小关系
进行分类讨论.
\end{marker}

\section{根据导函数零点与定义域的关系确定分类“界点”}

\begin{marker}
导函数零点是否分布在定义域内，零点将定义域划分为哪几个区间，若不能确定，则
需要分类讨论．	
\end{marker}


\chapter{到有关$x$与$e^x$，$\ln x$的组合函数}

\section{$x$与 $\ln x$的组合函数问题}


\end{document}