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数学博物館『すうじあむ』にようこそ。 『すうじあむ』は数学の問題・解答・解説のデータベースをはじめ 数学に関するあらゆる情報が詰まったポータルサイトです。
京都府立医科大学 前期 1993年度 問4
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入試情報
| 大学名 | 京都府立医科大学 |
|---|---|
| 学科・方式 | 前期 |
| 年度 | 1993年度 |
| 問No | 問4 |
| 学部 | 医学部 |
| カテゴリ | 関数と極限 ・ 微分法 ・ 積分法 |
| 状態 |    |
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%
% すうじあむはアップするファイルで完結したTeXソースでないといけない．
% このひな形で表示確認をしよう．
%
\documentclass[b5paper,fleqn,papersize]{jsarticle}
%===========================================================
% ■余白の設定 B5
%-----------------------------------------------------------
\setlength{\oddsidemargin}{-5.4truemm} % 左マージン
\setlength{\topmargin}{-15.4truemm} % 上マージン
\setlength{\textwidth}{14.2cm} % B5 サイズ用
\setlength{\headheight}{2zw}
\setlength{\headsep}{2zw}
\setlength{\textheight}{217mm}
%===========================================================
\usepackage[dvipdfm]{graphicx,color}
%\usepackage{wallpaper}
\usepackage{wrapfig}
\usepackage{amsmath,amsthm,amssymb}
\usepackage{multicol}
\usepackage{longtable}
\usepackage{ascmac}
\usepackage{fancybox}
\usepackage{framed}
\usepackage{ifthen}
\usepackage{setspace} % setspaceパッケージのインクルード
\usepackage{amssymb}% ≒を表示するために必要
% \usepackage{comment}% コメント% mebio.styと競合する
\usepackage{booktabs}
\usepackage{ascmac}
\usepackage{itembbox}
%
% \usepackage{ custom_suseum}% すうじあむ
%=========================================================
%============================================================
% ■ 分数の横棒を長く，分数を立てに狭く（emath.styの古い記述）
%------------------------------------------------------------
% 分数記号(分数罫を少し長めに)
\newcommand{\bunsuu}[2]{\dfrac{\,#1\,}{\,#2\,}}%
% さらに罫線の上下を狭く．
\newcommand\Dfrac[2]{\bunsuu{\lower.6ex\hbox{$#1$}}{\lower-.3ex\hbox{$#2$}}}%
%============================================================
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% enumerate の自動ラベルの変更
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\labelenumi{(\theenumi)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 増減表の矢印
% http://www.biwako.shiga-u.ac.jp/sensei/kumazawa/tex/arrow.html
%
% $\cvinc$ increase
% $\ccinc$ concavity 凹状の
% $\cvdec$ convex 凸形の
% $\ccdec$ decrease
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\makeatletter
\newcommand{\ccinc}{\ifx\@currsize\small
\setlength{\unitlength}{1.1pt}
\begin{picture}(10,10)(1,2)
\put(1,2){\line(0,1){3}}
\put(5,5){\oval(8,8)[lt]}
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\setlength{\unitlength}{1.2pt}
\begin{picture}(10,10)(1,2)
\put(1,2){\line(0,1){3}}
\put(5,5){\oval(8,8)[lt]}
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\newcommand{\cvinc}{\ifx\@currsize\small
\setlength{\unitlength}{1.1pt}
\begin{picture}(10,10)(2,1)
\put(2,1){\line(1,0){3}}
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\setlength{\unitlength}{1.2pt}
\begin{picture}(10,10)(2,1)
\put(2,1){\line(1,0){3}}
\put(5,5){\oval(8,8)[rb]}
\put(9,5){\vector(0,1){4}}
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\begin{picture}(10,10)(1,0)
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\put(5,5){\oval(8,8)[lb]}
\put(5,1){\vector(1,0){4}}
\end{picture}\fi}
\makeatother
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%======================================================================
\newcommand{\解説印}{\fboxsep=1.3pt\noindent\ovalbox{\bf \,解\,説\,}\hspace{0.5em}}
% \newcommand{\解答印}{\fboxsep=1.3pt\noindent\fbox{\bf \,解\,答\,}\hspace{0.5em}}
\newcommand{\解答印}{\shadowsize=0.15\shadowsize{\fboxsep=1.3pt\noindent\shadowbox{\bf \,解\,答\,}\hspace{0.5em}}}
\newcommand{\別解印}{\fboxsep=1.3pt\noindent\fbox{\bf \,別\,解\,}\hspace{0.5em}}
\newcommand{\注意印}{\noindent{\bf [注意]}\hspace{0.5em}}
%======================================================================
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%■ベクトルの矢印の高さを揃える定義
% \overrightarrow の矢印の高さを揃える
% \vec, \Vec で定義する
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\vec#1{\overrightarrow{\mathstrut #1}}
\def\Vec#1{\overrightarrow{\mathstrut {\rm #1}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%===========================================================
% ■二重根号
%-----------------------------------------------------------
\def\tsqrt#1{\textstyle\sqrt{#1}}
%===========================================================
%%% 平行 // の記号%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\para{%
\setlength{\unitlength}{1pt}%
\thinlines %
\begin{picture}(10, 12)%
\put(1,0){/}%
\put(4,0){/}%
\end{picture}%
}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%■ 答えを太文字
\newcommand{\kotae}[1]{\mbox{\boldmath$#1$}}
% \def\kotae#1{\mbox{\boldmath $#1$}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%===========================================================
% ■枠囲み
%-----------------------------------------------------------
\newcommand{\枠囲}[1]{\,\fboxsep1pt\fbox{ #1 }\,}% 文章中でも数式中でも使える？
% \newcommand{\枠}[1]{\fboxsep1pt\fbox{ #1 }\,}
%===========================================================
%===========================================================
% ■≦と≧の定義
%-----------------------------------------------------------
\def\le{\leqq}
\def\ge{\geqq}
\newcommand{\LEQQ}{\leqq}
\newcommand{\GEQQ}{\geqq}
% \newcommand{\LEQQ}{\mathrel{\mathpalette\gl@align<}}
% \newcommand{\GEQQ}{\mathrel{\mathpalette\gl@align>}}
% \newcommand{\gl@align}[2]{\lower.6ex\vbox{\baselineskip\z@skip\lineskip\z@
% \ialign{$\m@th#1\hfil##\hfil$\crcr#2\crcr=\crcr}}}
% エラーになる。理由はよく解らん。
%===========================================================
%===========================================================
% ■行列 2×2
%-----------------------------------------------------------
\newcommand{\matrixTT}[4]{
\begin{pmatrix}
#1 & #2 \\
#3 & #4
\end{pmatrix}
}
%===========================================================
%===========================================================
% ■行列 3×3
%-----------------------------------------------------------
\newcommand{\matrixTTT}[9]{
\begin{pmatrix}
#1 & #2 & #3\\
#4 & #5 & #6\\
#7 & #8 & #9
\end{pmatrix}
}
%===========================================================
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ■ 3*1行列の定義
\newcommand{\matrixthreeone}[3]{
\left(\begin{array}{c}
#1\\
#2\\
#3
\end{array}\right)
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%===========================================================
% ■△ABCとか
%
% 「$\∠{ABC}=\Dfrac12\pi$」のような使い方をするので「$」は定義に含めない
%
%-----------------------------------------------------------
\def\△#1{\triangle{\rm #1}}
\def\∠#1{\angle{\rm #1}}
%===========================================================
%===========================================================
% ■角括弧 複数行 #1 は横幅％，#2は内容
%
%-----------------------------------------------------------
\newcommand{\角括弧}[2]{%
$%
\left[ \begin{tabular}{@{}p{.#1\linewidth}}#2\end{tabular} \right]%
$%
}
%===========================================================
%===========================================================
% ■丸括弧 複数行 #1 は横幅％，#2は内容
%-----------------------------------------------------------
\newcommand{\丸括弧}[2]{%
$%
\left( \begin{tabular}{@{}p{.#1\linewidth}}#2\end{tabular} \right)%
$%
}
%===========================================================
%===========================================================
% ■丸囲み数字 ①とか
%-----------------------------------------------------------
\newcommand{\tyoMaru}[1]{\mbox{{\normalsize \textcircled{\raisebox{-.25ex}{#1}}}}}
%===========================================================
%===========================================================
% ■ 微分 df/dt
%-----------------------------------------------------------
\newcommand{\dd}[2]{%
\dfrac{{\rm d}#1}{{\rm d}#2}%
}
%===========================================================
%===========================================================
% ■ ∫記号のdisplaystyleマクロ
%-----------------------------------------------------------
% \newcommand{\dint}{\displaystyle\int}% emathと干渉しないように\defで定義
\def\dint{\displaystyle\int}
%===========================================================
%===========================================================
% ■ Σ記号のdisplaystyleマクロ
%-----------------------------------------------------------
\newcommand{\dsum}{\displaystyle\sum}
%===========================================================
%===========================================================
% ■ lim記号のdisplaystyleマクロ
%-----------------------------------------------------------
\newcommand{\dlim}{\displaystyle\lim}
\newcommand{\Dsum}{\sum\limits}% ちっさいΣ
%===========================================================
%===========================================================
% ■ nCr マクロ
%-----------------------------------------------------------
\newcommand{\nCr}[2]{%
{}_{#1}\mathrm{C}_{#2}%
}
%===========================================================
%===========================================================
% ■ローマ数字 (i) (ii) (iii) (iv) (v) (vi) (vii) まで定義
%-----------------------------------------------------------
\providecommand{\RMi}{\mbox{(\hspace{.1em}i\hspace{.1em})}}
\providecommand{\RMii}{\mbox{(ii)}}
\providecommand{\RMiii}{\mbox{(i\hspace{-.1em}i\hspace{-.1em}i)}}
\providecommand{\RMiv}{\mbox{(i\hspace{-.1em}v)}}
\providecommand{\RMv}{\mbox{(\hspace{.1em}v\hspace{.1em})}}
\providecommand{\RMvi}{\mbox{(v\hspace{-.1em}i)}}
\providecommand{\RMvii}{\mbox{(\hspace{-.1em}v\hspace{-.1em}i\hspace{-.1em}i)}}
%===========================================================
%===========================================================
% ■点A(t, t^3)とか
% #1 は点の名前（「A」とか）
% #2 は「(t,\,t^3)」とか「(s,\,t,\,u)」と（ ）内を全部書く
% （）の大きさとか，の後のスペースとか微調整ができるように
%
%-----------------------------------------------------------
\providecommand{\点}[2]{${\rm#1} #2$}
%===========================================================
%===========================================================
% ■ 証明終了記号 ■
%-----------------------------------------------------------
\newcommand{\■}{{\tiny \text{■}}}
%===========================================================
\begin{document}
%%%%% ■ 本文開始 ■ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{FRAME}% この間に設問を書く% \usepackage{ custom_suseum} をゆうこうにすること
\begin{framed}
%
%----------------------------------------------------------------
%
\begin{enumerate}
\item $y=e^{-x}\sin{x}\ (0\LEQQ x \LEQQ \pi)$のグラフを，
下図の$xy$平面に書き入れよ．
\begin{center}
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% STR 2 0 3 0 Black White
% 4 650 384 650 400 4 2800 0 0
% $y$
\put(6.5000,-4.0000){\makebox(0,0)[rt]{$y$}}%
% STR 2 0 3 0 Black White
% 4 4310 1872 4310 1888 4 2800 0 0
% $x$
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% VECTOR 2 0 3 0 Black White
% 2 710 1840 710 400
%
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% FUNC 2 0 3 0 Black White
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% exp(-x)
\special{pn 8}%
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\special{pa 3640 1767}%
\special{pa 3646 1767}%
\special{pa 3656 1768}%
\special{pa 3660 1768}%
\special{pa 3670 1769}%
\special{pa 3676 1769}%
\special{pa 3686 1770}%
\special{pa 3690 1770}%
\special{pa 3700 1771}%
\special{pa 3706 1771}%
\special{pa 3716 1772}%
\special{pa 3720 1772}%
\special{pa 3730 1773}%
\special{pa 3736 1773}%
\special{pa 3746 1774}%
\special{pa 3750 1774}%
\special{pa 3760 1775}%
\special{pa 3766 1775}%
\special{pa 3780 1776}%
\special{pa 3786 1776}%
\special{pa 3796 1777}%
\special{pa 3800 1777}%
\special{pa 3810 1778}%
\special{pa 3816 1778}%
\special{pa 3830 1779}%
\special{pa 3836 1779}%
\special{pa 3846 1780}%
\special{pa 3850 1780}%
\special{pa 3866 1781}%
\special{pa 3870 1781}%
\special{pa 3880 1782}%
\special{pa 3886 1782}%
\special{pa 3900 1783}%
\special{pa 3906 1783}%
\special{pa 3916 1784}%
\special{pa 3920 1784}%
\special{pa 3936 1785}%
\special{pa 3940 1785}%
\special{pa 3956 1786}%
\special{pa 3960 1786}%
\special{pa 3976 1787}%
\special{pa 3980 1787}%
\special{pa 3996 1788}%
\special{pa 4000 1788}%
\special{pa 4010 1789}%
\special{fp}%
\special{pn 8}%
\special{pa 4019 1789}%
\special{pa 4020 1789}%
\special{pa 4036 1790}%
\special{pa 4040 1790}%
\special{pa 4056 1791}%
\special{pa 4060 1791}%
\special{pa 4060 1791}%
\special{ip}%
\special{pa 4069 1791}%
\special{pa 4080 1792}%
\special{pa 4086 1792}%
\special{pa 4100 1793}%
\special{pa 4106 1793}%
\special{pa 4110 1793}%
\special{ip}%
\special{pa 4119 1794}%
\special{pa 4120 1794}%
\special{pa 4126 1794}%
\special{pa 4146 1795}%
\special{pa 4150 1795}%
\special{pa 4160 1795}%
\special{ip}%
\special{pa 4169 1796}%
\special{pa 4170 1796}%
\special{pa 4176 1796}%
\special{pa 4190 1797}%
\special{pa 4196 1797}%
\special{pa 4210 1798}%
\special{ip}%
\special{pa 4219 1798}%
\special{pa 4220 1798}%
\special{pa 4240 1799}%
\special{pa 4246 1799}%
\special{pa 4260 1800}%
\special{ip}%
\special{pa 4269 1800}%
\special{pa 4270 1800}%
\special{pa 4296 1801}%
\special{pa 4300 1801}%
\special{pa 4310 1802}%
\special{ip}%
% FUNC 2 0 3 0 Black White
% 9 410 400 4310 2080 710 1840 4010 1840 710 640 410 640 4310 1840 0 4 1 1
% pi
\special{pn 8}%
\special{pn 8}%
\special{pa 4010 400}%
\special{pa 4010 409}%
\special{ip}%
\special{pa 4010 448}%
\special{pa 4010 457}%
\special{ip}%
\special{pa 4010 496}%
\special{pa 4010 505}%
\special{ip}%
\special{pa 4010 544}%
\special{pa 4010 553}%
\special{ip}%
\special{pa 4010 592}%
\special{pa 4010 601}%
\special{ip}%
\special{ip}%
\special{pa 4010 640}%
\special{pa 4010 1840}%
\special{fp}%
\special{pn 8}%
\special{pa 4010 1849}%
\special{pa 4010 1888}%
\special{ip}%
\special{pa 4010 1897}%
\special{pa 4010 1936}%
\special{ip}%
\special{pa 4010 1945}%
\special{pa 4010 1984}%
\special{ip}%
\special{pa 4010 1993}%
\special{pa 4010 2032}%
\special{ip}%
\special{pa 4010 2041}%
\special{pa 4010 2080}%
\special{ip}%
% FUNC 2 0 3 0 Black White
% 9 410 400 4310 2080 710 1840 4010 1840 710 640 410 640 4310 1840 0 4 1 1
% pi/2
\special{pn 8}%
\special{pn 8}%
\special{pa 2360 400}%
\special{pa 2360 409}%
\special{ip}%
\special{pa 2360 448}%
\special{pa 2360 457}%
\special{ip}%
\special{pa 2360 496}%
\special{pa 2360 505}%
\special{ip}%
\special{pa 2360 544}%
\special{pa 2360 553}%
\special{ip}%
\special{pa 2360 592}%
\special{pa 2360 601}%
\special{ip}%
\special{ip}%
\special{pa 2360 640}%
\special{pa 2360 1840}%
\special{fp}%
\special{pn 8}%
\special{pa 2360 1849}%
\special{pa 2360 1888}%
\special{ip}%
\special{pa 2360 1897}%
\special{pa 2360 1936}%
\special{ip}%
\special{pa 2360 1945}%
\special{pa 2360 1984}%
\special{ip}%
\special{pa 2360 1993}%
\special{pa 2360 2032}%
\special{ip}%
\special{pa 2360 2041}%
\special{pa 2360 2080}%
\special{ip}%
% FUNC 2 0 3 0 Black White
% 9 410 400 4310 2080 710 1840 4010 1840 710 640 410 640 4310 1840 0 4 1 1
% pi/4
\special{pn 8}%
\special{pn 8}%
\special{pa 1536 400}%
\special{pa 1536 409}%
\special{ip}%
\special{pa 1536 448}%
\special{pa 1536 457}%
\special{ip}%
\special{pa 1536 496}%
\special{pa 1536 505}%
\special{ip}%
\special{pa 1536 544}%
\special{pa 1536 553}%
\special{ip}%
\special{pa 1536 592}%
\special{pa 1536 601}%
\special{ip}%
\special{ip}%
\special{pa 1536 640}%
\special{pa 1536 1840}%
\special{fp}%
\special{pn 8}%
\special{pa 1536 1849}%
\special{pa 1536 1888}%
\special{ip}%
\special{pa 1536 1897}%
\special{pa 1536 1936}%
\special{ip}%
\special{pa 1536 1945}%
\special{pa 1536 1984}%
\special{ip}%
\special{pa 1536 1993}%
\special{pa 1536 2032}%
\special{ip}%
\special{pa 1536 2041}%
\special{pa 1536 2080}%
\special{ip}%
% FUNC 2 0 3 0 Black White
% 9 410 400 4310 2080 710 1840 4010 1840 710 640 410 640 4310 1840 0 4 1 1
% pi/4*3
\special{pn 8}%
\special{pn 8}%
\special{pa 3185 400}%
\special{pa 3185 409}%
\special{ip}%
\special{pa 3185 448}%
\special{pa 3185 457}%
\special{ip}%
\special{pa 3185 496}%
\special{pa 3185 505}%
\special{ip}%
\special{pa 3185 544}%
\special{pa 3185 553}%
\special{ip}%
\special{pa 3185 592}%
\special{pa 3185 601}%
\special{ip}%
\special{ip}%
\special{pa 3185 640}%
\special{pa 3185 1840}%
\special{fp}%
\special{pn 8}%
\special{pa 3185 1849}%
\special{pa 3185 1888}%
\special{ip}%
\special{pa 3185 1897}%
\special{pa 3185 1936}%
\special{ip}%
\special{pa 3185 1945}%
\special{pa 3185 1984}%
\special{ip}%
\special{pa 3185 1993}%
\special{pa 3185 2032}%
\special{ip}%
\special{pa 3185 2041}%
\special{pa 3185 2080}%
\special{ip}%
% FUNC 2 0 3 0 Black White
% 10 410 400 4310 2080 710 1840 4010 1840 710 640 710 400 4010 2080 0 4 1 0 0 0
% 1
\special{pn 8}%
\special{pn 8}%
\special{pa 410 640}%
\special{pa 419 640}%
\special{ip}%
\special{pa 460 640}%
\special{pa 469 640}%
\special{ip}%
\special{pa 510 640}%
\special{pa 519 640}%
\special{ip}%
\special{pa 560 640}%
\special{pa 569 640}%
\special{ip}%
\special{pa 610 640}%
\special{pa 619 640}%
\special{ip}%
\special{pa 660 640}%
\special{pa 669 640}%
\special{ip}%
\special{pa 710 640}%
\special{pa 710 640}%
\special{ip}%
\special{pa 710 640}%
\special{pa 4010 640}%
\special{fp}%
\special{pn 8}%
\special{pa 4019 640}%
\special{pa 4060 640}%
\special{ip}%
\special{pa 4069 640}%
\special{pa 4110 640}%
\special{ip}%
\special{pa 4119 640}%
\special{pa 4160 640}%
\special{ip}%
\special{pa 4169 640}%
\special{pa 4210 640}%
\special{ip}%
\special{pa 4219 640}%
\special{pa 4260 640}%
\special{ip}%
\special{pa 4269 640}%
\special{pa 4310 640}%
\special{ip}%
% STR 2 0 3 0 Black White
% 4 1450 2020 1450 2120 2 0 0 0
% $\Dfrac14\pi$
\put(14.5000,-21.2000){\makebox(0,0)[lb]{$\Dfrac14\pi$}}%
% STR 2 0 3 0 Black White
% 4 2320 2020 2320 2120 2 0 0 0
% $\Dfrac12\pi$
\put(23.2000,-21.2000){\makebox(0,0)[lb]{$\Dfrac12\pi$}}%
% STR 2 0 3 0 Black White
% 4 3100 2010 3100 2110 2 0 0 0
% $\Dfrac34\pi$
\put(31.0000,-21.1000){\makebox(0,0)[lb]{$\Dfrac34\pi$}}%
% STR 2 0 3 0 Black White
% 4 3940 1920 3940 2020 2 0 0 0
% $\pi$
\put(39.4000,-20.2000){\makebox(0,0)[lb]{$\pi$}}%
\end{picture}}%
\end{center}
ただし，図の曲線は$y=e^{-x}\ (0\LEQQ x \LEQQ \pi)$である．
\item 正の整数$n$について，
\[I_n=\dint_0^{n\pi}e^{-x}|\sin{x}|dx\]
とする．このとき，極限値$\dlim_{n\to\infty}I_n$を求めよ．
\end{enumerate}
%--------------------------------------------------------------
\end{framed}
%\end{FRAME}
%--- 解答 ------------------------------------------------------------------------
{\footnotesize
}% footnotesize
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\end{document}
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