<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[cankaodaan]{hfutexam}
\RequirePackage{extarrows} % ç”¨äºŽç­‰å·ä¸Šé¢åŠ&nbsp;æ–‡å­—
\newcommand{\diff}{\,\mathrm{d}} % å®šä¹‰å¾®åˆ†ç¬¦å·
% \renewcommand{\score}[1]{}
% \renewcommand{\Score}[1]{}

\begin{document}
\BiaoTi{åˆè‚¥å·¥ä¸šå¤§å­¦è¯•å·å‚è€ƒç­”æ¡ˆï¼ˆAï¼‰}
\XueNian{2021}{2022}
\XueQi{äºŒ}
\KeChengDaiMa{034Y01}
\KeChengMingCheng{æ•°å­¦ï¼ˆä¸‹ï¼‰}
\XueFen{5}
\KeChengXingZhi{å¿…ä¿®}
\KaoShiXingShi{é—­å·}
\ZhuanYeBanJi{å°‘æ•°æ°‘æ—é¢„ç§‘ç­}
\KaoShiRiQi{2022å¹´6æœˆ18æ—¥8:00-10:00}
\MingTiJiaoShi{é›†ä½“}
% \XiZhuRenQianMing{dengbing.png}

\tigan{ä¸€ã€å¡«ç©ºé¢˜ï¼ˆæ¯å°é¢˜3åˆ†ï¼Œå…±18åˆ†ï¼‰}

\textbf{è¯·å°†ä½&nbsp;çš„ç­”æ¡ˆå¯¹åº”å¡«åœ¨æ¨ªçº¿ä¸Šï¼š}

\textbf{1.} \fillblank{$e$}, 
\textbf{2.} \fillblank{$2x\cos(x^2+1)\diff x$}, 
\textbf{3.} \fillblank{$\dfrac12$}, 

\textbf{4.} \fillblank{$y=x-1+2\ln 2$},
\textbf{5.} \fillblank{$1$},
\textbf{6.} \fillblank{$0$}.

\tigan{äºŒã€é€‰æ‹©é¢˜ï¼ˆæ¯å°é¢˜3åˆ†ï¼Œå…±18åˆ†ï¼‰}

\textbf{è¯·å°†ä½&nbsp;æ‰€é€‰æ‹©çš„å­—æ¯ A, B, C, D ä¹‹ä¸€å¯¹åº”å¡«åœ¨ä¸‹åˆ—è¡¨æ&nbsp;¼é‡Œï¼š}

\xuanzeti{\textbf{é¢˜å·}}{\textbf{ç­”æ¡ˆ}}%
\xuanzeti{1}{A}%
\xuanzeti{2}{D}%
\xuanzeti{3}{B}%
\xuanzeti{4}{A}%
\xuanzeti{5}{C}%
\xuanzeti{6}{D}

\tigan{ä¸‰ã€è§£ç­”é¢˜ï¼ˆæ¯å°é¢˜8åˆ†ï¼Œå…±64åˆ†ï¼‰}
% å¾—åˆ†ç‚¹å‘½ä»¤ \score1, å¾—åˆ†ç‚¹é•¿åº¦æ˜¯è‡ªåŠ¨çš„
% \Score{(2åˆ†, ç¼ºå°‘å¸¸æ•°å¾—1åˆ†)} ç”¨äºŽè‡ªå®šä¹‰å¾—åˆ†è¯´æ˜Ž

\textbf{1. ï¼ˆ8åˆ†ï¼‰ã€è§£ã€‘}
\begin{align*}
\lim_{x\to-1}\frac{x^2-1}{x^2+3x+2}&amp;=\lim_{x\to-1}\frac{(x-1)(x+1)}{(x+2)(x+1)} \score3\\
&amp;=\lim_{x\to-1}\frac{x-1}{x+2} \score3\\
&amp;=\frac{-2}1=-2. \score2
\end{align*}

\textbf{2. ï¼ˆ8åˆ†ï¼‰ã€è§£ã€‘}
\begin{align*}
\lim_{x\to0}\frac{e^x-1-x}{\arcsin x^2}&amp;=\lim_{x\to0}\frac{e^x-1-x}{x^2} \score3\\
&amp;\xlongequal{\text{æ´›å¿…è¾¾}}\lim_{x\to0}\frac{e^x-1}{2x} \score3\\
&amp;=\lim_{x\to0}\frac{x}{2x}=\frac12. \score2
\end{align*}

\newpage

\textbf{3. ï¼ˆ8åˆ†ï¼‰ã€è§£ã€‘}
\begin{align*}
\frac{\diff y}{\diff x}&amp;=\frac{\diff y/\diff t}{\diff x/\diff t} \score2\\
&amp;=\frac{3t^2+1}{2t+1}, \score2\\
\frac{\diff^2 y}{\diff x^2}&amp;=\frac{\diff y'/\diff t}{\diff x/\diff t} \score2\\
&amp;=\frac{6t(2t+1)-(3t^2+1)2}{(2t+1)^3}=\frac{6t^2+6t-2}{(2t+1)^3}. \score2
\end{align*}

\textbf{4. ï¼ˆ8åˆ†ï¼‰ã€è§£ã€‘}

\indent ç”±äºŽ $f(x)$ åœ¨ $x=0$ å¤„è¿žç»­, å›&nbsp;æ­¤
\begin{align*}
f(0)&amp;=f(0^+) \score1\\
		&amp;=b=\lim_{x\to0^-}x\arctan\frac1x=0\times\left(-\frac\pi2\right)=0. \score1
\end{align*}

\indent ç”±äºŽ $f(x)$ åœ¨ $x=0$ å¤„å¯å¯¼, å›&nbsp;æ­¤
\begin{align*}
f'_-(0)&amp;=f'_+(0), \score1\\
f'_-(0)&amp;=\lim_{x\to0^-}\frac{x\arctan\frac1x}x=\lim_{x\to0^-}\arctan\frac1x=-\frac\pi2 \score1\\
f'_+(0)&amp;=(2x+a)|_{x=0}=a, \score1
\end{align*}
å›&nbsp;æ­¤ $a=-\dfrac\pi2$. \score1

\indent ç”±äºŽ
\begin{align*}
\lim_{x\to+\infty}\frac yx&amp;=\lim_{x\to+\infty}\left(x-\frac\pi2\right)=+\infty, \score1\\
\lim_{x\to-\infty}\frac yx&amp;=\lim_{x\to-\infty}\arctan\frac1x=0,\\
\lim_{x\to-\infty}y&amp;=\lim_{x\to-\infty}x\arctan\frac1x=\lim_{t\to0^-}\frac{\arctan t}t=1,
\end{align*}
å›&nbsp;æ­¤æ›²çº¿ $y=f(x)$ çš„æ¸è¿‘çº¿åªæœ‰ $y=1$. \score1

\newpage

\textbf{5. ï¼ˆ8åˆ†ï¼‰ã€è§£ã€‘}

\indent ç”±
\[f'(x)=3x^2-2x-1=(3x+1)(x-1)=0 \score2\]
å¯å¾—é©»ç‚¹ $x=-\dfrac13,1$. \score2

\indent ç”±äºŽ
\[f(-2)=-10,\quad f(2)=2,\quad f\left(-\frac13\right)=\frac5{27},\quad f(1)=-1, \score2\]
å›&nbsp;æ­¤æœ€å¤§å€¼ä¸º $2$, æœ€å°å€¼ä¸º $-10$. \score2

\textbf{6. ï¼ˆ8åˆ†ï¼‰ã€è¯æ˜Žã€‘}

\textbf{è¯æ³•ä¸€}: è®¾ $f(x)=\tan x-x$, åˆ™ \score2
\begin{align*}
f'(x)=\frac1{\cos^2x}-1=\tan^2x\ge0. \score2
\end{align*}
å›&nbsp;æ­¤ $f(x)$ åœ¨ $\left(-\dfrac\pi2,\dfrac\pi2\right)$ ä¸Šå•è°ƒé€’å¢ž, ä»Žè€Œ \score2
\begin{align*}
f(x_2)\ge f(x_1),\quad\tan x_2-\tan x_1\ge x_2-x_1. \score2
\end{align*}

\textbf{è¯æ³•äºŒ}: è®¾ $f(x)=\tan x$, åˆ™ $f(x)$ åœ¨ $[x_1,x_2]$ ä¸Šè¿žç»­, $(x_1,x_2)$ å†…å¯å¯¼. \score2

\indent ç”±æ‹‰æ&nbsp;¼æœ—æ—¥ä¸­å€¼å®šç†, å­˜åœ¨ $\xi\in(x_1,x_2)$ ä½¿å¾—
\begin{align*}
\frac{f(x_2)-f(x_1)}{x_2-x_1}=f'(\xi), \score2
\end{align*}
å³
\begin{align*}
\frac{\tan x_2-\tan x_1}{x_2-x_1}=\frac1{\cos^2\xi}\ge1. \score2
\end{align*}
æ‰€ä»¥ $\tan x_2-\tan x_1\ge x_2-x_1$. \score2
\newpage

\textbf{7. ï¼ˆ8åˆ†ï¼‰ã€è¯æ˜Žã€‘}

\indent è®¾ $F(x)=x^{2022}f(x)$, \score2\\
åˆ™ $F(x)$ åœ¨ $[0,1]$ ä¸Šè¿žç»­, $(0,1)$ å†…å¯å¯¼, \score1\\
ä¸” $F(0)=0,F(1)=f(1)=0$. \score1

\indent ç”±ç½—å°”ä¸­å€¼å®šç†, å­˜åœ¨ $\xi\in(0,1)$ ä½¿å¾— $F'(\xi)=0$. \score2\\
ç”±äºŽ $F'(x)=x^{2022}f'(x)+2022x^{2021}f(x)$ ä¸” $\xi\neq0$, \score1\\
æ‰€ä»¥ $\xi f'(\xi)+2022f(\xi)=1$. \score1


\textbf{8. ï¼ˆ8åˆ†ï¼‰ã€è§£ã€‘}

(1) 
\[f'(x)=\frac1x-\frac4{x^3}=\frac{x^2-4}{x^3}=\frac{(x+2)(x-2)}{x^3}. \score1\]
å½“ $0&lt;x&lt;2$ æ—¶, $f'(x)&lt;0$. å½“ $x&gt;2$ æ—¶, $f'(x)&gt;0$. \score1\\
å›&nbsp;æ­¤ $(0,2]$ æ˜¯ $f(x)$ çš„å•å‡åŒºé—´, $[2,+\infty)$ æ˜¯ $f(x)$ çš„å•å¢žåŒºé—´. \Score{(1åˆ†, å†™æˆå¼€åŒºé—´ä¸æ‰£åˆ†)}\\
æ‰€ä»¥ $f(x)$ åªæœ‰å”¯ä¸€çš„æžå°å€¼ $f(2)=\ln2+\dfrac12$. \score1

(2) 
\[f''(x)=-\frac1{x^2}+\frac{12}{x^4}=-\frac{x^2-12}{x^4}=-\frac{(x-2\sqrt3)(x+2\sqrt3)}{x^4}. \score1\]
å½“ $0&lt;x&lt;2\sqrt3$ æ—¶, $f''(x)&gt;0$. å½“ $x&gt;2\sqrt3$ æ—¶, $f''(x)&lt;0$. \score1\\
å›&nbsp;æ­¤ $(0,2\sqrt3]$ æ˜¯æ›²çº¿ $y=f(x)$ çš„å‡¹åŒºé—´, \\
$[2\sqrt3,+\infty)$ æ˜¯æ›²çº¿ $y=f(x)$ çš„å‡¸åŒºé—´, \Score{(1åˆ†, å†™æˆå¼€åŒºé—´ä¸æ‰£åˆ†)}\\
æ‹ç‚¹ä¸º $\left(2\sqrt3,\ln(2\sqrt3)+\dfrac16\right)$. \score1

\end{document}

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