大阪大学 前期理系 2011年度 問3

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解答作成者: 森 宏征

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入試情報

大学名 大阪大学
学科・方式 前期理系
年度 2011年度
問No 問3
学部 理学部 ・ 医学部 ・ 歯学部 ・ 薬学部 ・ 工学部 ・ 基礎工学部
カテゴリ
状態 解答 解説なし ウォッチリスト

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\documentclass[a4paper,12pt,fleqn]{jreport} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ascmac} \usepackage{vector3} \setlength{\topmargin}{-25mm} \setlength{\oddsidemargin}{2.5mm} \setlength{\textwidth}{420pt} \setlength{\textheight}{700pt} \usepackage{color} \ExecuteOptions{usename} \usepackage{graphicx} \usepackage{pifont} \usepackage{fancybox} \usepackage{custom_mori} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} \begin{FRAME}  実数の組$(p,\ q)$に対し, $f(x) = (x - p)^2 + q$ とおく. \begin{enumerate} \item[(1)]  放物線 $y = f(x)$ が点$(0,\ 1)$を通り, しかも直線 $y = x$ の $x > 0$ の部分と接するような実数の組$(p,\ q)$と 接点の座標を求めよ. %\medskip \item[(2)]  実数の組$(p_1,\ q_1),\enskip(p_2,\ q_2)$に対して, $f_1(x) = (x - p_1)^2 + q_1$ および % $f_2(x) = (x - p_2)^2 + q_2$ とおく. 実数 $\alpha,\ \beta\enskip(ただし\>\alpha < \beta)$ に対して \[ f_1(\alpha) < f_2(\alpha),\quad f_1(\beta) < f_2(\beta) \] であるならば, 区間 $\alpha \leqq x \leqq \beta$ において不等式 $f_1(x) < f_2(x)$ が つねに成り立つことを示せ. %\medskip \item[(3)]  長方形 $R : 0 \leqq x \leqq 1,\enskip 0 \leqq y \leqq 2$ を考える. また,4点$\P_0(0,\ 1),\\ \P_1(0,\ 0),\enskip \P_2(1,\ 1),\enskip \P_3(1,\ 0)$を この順に線分で結んで得られる折れ線を $L$ とする. 実数の組$(p,\ q)$を, 放物線 $y = f(x)$ と折れ線 $L$ に共有点がないようなすべての組にわたって 動かすとき, $R$ の点のうちで放物線 $y = f(x)$ が通過する点全体の集合を $T$ とする. $R$ から $T$ を除いた領域 $S$ を座標平面上に図示し, その面積を求めよ. %\hfill(配点率20%) \end{enumerate} \end{FRAME} \vskip 2mm \noindent{\color[named]{BurntOrange}\bfseries \Ovalbox{解答}} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  $y = f(x)$ が点$(0,\ 1)$を通るから, \begin{gather*} f(0) = 1 \qquad \therefore \enskip p^2 + q = 1 \tag*{$\cdott\MARU{1}$} \end{gather*} $y = f(x)$ と $y = x$ が $x > 0$ において接するから, \begin{gather*} (x - p)^2 + q = x \quad すなわち \quad x^2 - (2p + 1)x + p^2 + q = 0 \end{gather*} は正の重解をもつ. よって \begin{gather*} (判別式) = (2p + 1)^2 - 4(p^2 + q) = 0 \\ \therefore \enskip 4p - 4q + 1 = 0 \tag*{$\cdott\MARU{2}$} \end{gather*} このとき重解は $x = p + \dfrac{1}{2}$ だから, \begin{gather*} p + \frac{1}{2} > 0 \qquad \therefore \enskip p > -\frac{1}{2} \tag*{$\cdott\MARU{3}$} \end{gather*} \MARU{2}より $q = p + \dfrac{1}{4}$. これを\MARU{1}に代入して, \begin{gather*} p^2 + p + \frac{1}{4} = 1 \\ \qquad 4p^2 + 4p - 3 = 0 \qquad (2p - 1)(2p + 3) = 0 \\[1mm] \therefore \enskip p = -\frac{3}{2},\ \frac{1}{2} \end{gather*} \MARU{3}をみたすのは $p = \dfrac{1}{2}$ のみ. このとき重解は $p + \dfrac{1}{2} = 1$ だから, \begin{align*} (p,\ q) = \textcolor{red}{ \boldsymbol{\left(\frac{1}{2},\ \frac{3}{4} \right)}},\quad 接点のx座標 : \textcolor{red}{\boldsymbol{1}} \tag*{$\Ans$} \end{align*} \item  $g(x) = f_2(x) - f_1(x)$ とおく. $\alpha \leqq x \leqq \beta$ において つねに $g(x) > 0$ が成り立つことをいえばよい. \begin{align*} g(x) &= (x - p_2)^2 + q_2 - (x - p_1)^2 - q_1 \\ &= 2(p_1 - p_2)x + {p_2}^2 + q_2 - {p_1}^2 - q_1 \end{align*} $p_1 \geqq p_2$ ならば $g(x)$ は非減少関数だから, \begin{align*} g(x) \geqq g(\alpha) = f_2(\alpha) - f_1(\alpha) > 0 \end{align*} $p_1 \leqq p_2$ ならば $g(x)$ は非増加関数だから, \begin{align*} g(x) \geqq g(\beta) = f_2(\beta) - f_1(\alpha) > 0 \end{align*} ゆえに $g(x) > 0$ である. \hfill ■ \item  $f_+(x) = \left(x - \dfrac{1}{2} \right)^{\!\! 2} + \dfrac{3}{4}$ とおく. \smallskip また,2点$\P_1,\enskip\P_4$を通り, $x^2$ の係数が1の放物線を $y = f_-(x)$ とする. $f_-(x) = x(x - 1)$ である.  $y = f(x)$ と $L$ が共有点をもたないためには \begin{align*} \mbox{(i)}\ 「f(0) > 1 \enskip かつ \enskip f(1) > 1」\quad または \quad \mbox{(ii)}\ 「f(0) < 0 \enskip かつ \enskip f(1) < 0」 \end{align*} が必要.逆に, \begin{enumerate} \item[(i)] の場合  $f(0) > 1 = f_+(0),\enskip f(1) > 1 = f_+(1)$ だから (2)より $0 \leqq x \leqq 1$ においてつねに $f(x) > f_+(x)$. このとき $y = f(x)$ はつねに $y = f_+(x)$ の上側にあり $L$ と 共有点をもつことはない.\smallskip  特に $f(x) = \left(x - \dfrac{1}{2} \right)^{\!\! 2} + q,\enskip \dfrac{3}{4} < q$ とおけば,\smallskip $y = f(x)$ は $R$ の $y > f_+(x)$ をみたす部分をくまなく通過する. \item[(ii)] の場合  $f(0) < 0 = f_-(0),\enskip f(1) < 0 = f_-(1)$ だから (2)より $0 \leqq x \leqq 1$ においてつねに $f(x) < f_-(x)$. このとき $y = f(x)$ はつねに $y = f_-(x)$ の下側にあり $R$ と 共有点をもつことはない. もちろん $L$ とも共有点をもたない. \end{enumerate} (i),\enskip(ii)より $T$ は下左図のようになるから, $S$ は下右図のようになる. \begin{center} %\input{osaka2011s3f_zu_2} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 32.2000, 19.7000)( 8.4000,-23.1600) % STR 2 0 3 0 % 3 1110 382 1110 396 4 2400 % $y$ \put(11.1000,-3.9600){\makebox(0,0)[rt]{$y$}}% % STR 2 0 3 0 % 3 1944 1876 1944 1890 4 2400 % $x$ \put(19.4400,-18.9000){\makebox(0,0)[rt]{$x$}}% % VECTOR 2 0 3 0 % 2 1142 2156 1142 396 % \special{pn 8}% \special{pa 1142 2156}% \special{pa 1142 396}% \special{fp}% \special{sh 1}% \special{pa 1142 396}% \special{pa 1122 464}% \special{pa 1142 450}% \special{pa 1162 464}% \special{pa 1142 396}% \special{fp}% % VECTOR 2 0 3 0 % 2 982 1836 1942 1836 % \special{pn 8}% \special{pa 982 1836}% \special{pa 1942 1836}% \special{fp}% \special{sh 1}% \special{pa 1942 1836}% \special{pa 1876 1816}% \special{pa 1890 1836}% \special{pa 1876 1856}% \special{pa 1942 1836}% \special{fp}% % FUNC 2 2 3 0 % 9 982 396 1942 1996 1142 1836 1782 1836 1142 1196 982 396 1942 1996 0 3 0 0 % x^2-x+1 \special{pn 8}% \special{pa 980 994}% \special{pa 986 1000}% \special{pa 990 1008}% \special{pa 996 1016}% \special{pa 1000 1022}% \special{pa 1006 1030}% \special{pa 1010 1038}% \special{pa 1016 1044}% \special{pa 1020 1052}% \special{pa 1026 1058}% \special{pa 1030 1064}% \special{pa 1036 1072}% \special{pa 1040 1078}% \special{pa 1046 1084}% \special{pa 1050 1092}% \special{pa 1056 1098}% \special{pa 1060 1104}% \special{pa 1066 1110}% \special{pa 1070 1116}% \special{pa 1076 1122}% \special{pa 1080 1128}% \special{pa 1086 1134}% \special{pa 1090 1140}% \special{pa 1096 1146}% \special{pa 1100 1152}% \special{pa 1106 1158}% \special{pa 1110 1162}% \special{pa 1116 1168}% \special{pa 1120 1174}% \special{pa 1126 1180}% \special{pa 1130 1184}% \special{pa 1136 1190}% \special{pa 1140 1194}% \special{pa 1146 1200}% \special{pa 1150 1204}% \special{pa 1156 1210}% \special{pa 1160 1214}% \special{pa 1166 1218}% \special{pa 1170 1224}% \special{pa 1176 1228}% \special{pa 1180 1232}% \special{pa 1186 1236}% \special{pa 1190 1240}% \special{pa 1196 1246}% \special{pa 1200 1250}% \special{pa 1206 1254}% \special{pa 1210 1258}% \special{pa 1216 1262}% \special{pa 1220 1264}% \special{pa 1226 1268}% \special{pa 1230 1272}% \special{pa 1236 1276}% \special{pa 1240 1280}% \special{pa 1246 1282}% \special{pa 1250 1286}% \special{pa 1256 1290}% \special{pa 1260 1292}% \special{pa 1266 1296}% \special{pa 1270 1298}% \special{pa 1276 1302}% \special{pa 1280 1304}% \special{pa 1286 1308}% \special{pa 1290 1310}% \special{pa 1296 1312}% \special{pa 1300 1316}% \special{pa 1306 1318}% \special{pa 1310 1320}% \special{pa 1316 1322}% \special{pa 1320 1324}% \special{pa 1326 1328}% \special{pa 1330 1330}% \special{pa 1336 1332}% \special{pa 1340 1334}% \special{pa 1346 1336}% \special{pa 1350 1336}% \special{pa 1356 1338}% \special{pa 1360 1340}% \special{pa 1366 1342}% \special{pa 1370 1344}% \special{pa 1376 1344}% \special{pa 1380 1346}% \special{pa 1386 1348}% \special{pa 1390 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\special{pa 1600 1326}% \special{pa 1606 1324}% \special{pa 1610 1322}% \special{pa 1616 1320}% \special{pa 1620 1318}% \special{pa 1626 1314}% \special{pa 1630 1312}% \special{pa 1636 1310}% \special{pa 1640 1306}% \special{pa 1646 1304}% \special{pa 1650 1302}% \special{pa 1656 1298}% \special{pa 1660 1296}% \special{pa 1666 1292}% \special{pa 1670 1288}% \special{pa 1676 1286}% \special{pa 1680 1282}% \special{pa 1686 1278}% \special{pa 1690 1276}% \special{pa 1696 1272}% \special{pa 1700 1268}% \special{pa 1706 1264}% \special{pa 1710 1260}% \special{pa 1716 1256}% \special{pa 1720 1252}% \special{pa 1726 1248}% \special{pa 1730 1244}% \special{pa 1736 1240}% \special{pa 1740 1236}% \special{pa 1746 1232}% \special{pa 1750 1226}% \special{pa 1756 1222}% \special{pa 1760 1218}% \special{pa 1766 1214}% \special{pa 1770 1208}% \special{pa 1776 1204}% \special{pa 1780 1198}% \special{pa 1786 1194}% \special{pa 1790 1188}% \special{pa 1796 1184}% \special{pa 1800 1178}% \special{pa 1806 1172}% \special{pa 1810 1168}% \special{pa 1816 1162}% \special{pa 1820 1156}% \special{pa 1826 1150}% \special{pa 1830 1144}% \special{pa 1836 1140}% \special{pa 1840 1134}% \special{pa 1846 1128}% \special{pa 1850 1122}% \special{pa 1856 1116}% \special{pa 1860 1108}% \special{pa 1866 1102}% \special{pa 1870 1096}% \special{pa 1876 1090}% \special{pa 1880 1084}% \special{pa 1886 1076}% \special{pa 1890 1070}% \special{pa 1896 1064}% \special{pa 1900 1056}% \special{pa 1906 1050}% \special{pa 1910 1042}% \special{pa 1916 1036}% \special{pa 1920 1028}% \special{pa 1926 1022}% \special{pa 1930 1014}% \special{pa 1936 1006}% \special{pa 1940 1000}% \special{sp -0.045}% % LINE 2 0 3 0 % 4 1142 556 1782 556 1782 556 1782 1836 % \special{pn 8}% \special{pa 1142 556}% \special{pa 1782 556}% \special{fp}% \special{pa 1782 556}% \special{pa 1782 1836}% \special{fp}% % LINE 3 0 3 0 % 52 1782 924 1366 1340 1782 972 1406 1348 1782 1020 1446 1356 1782 1068 1494 1356 1782 1116 1558 1340 1782 1164 1646 1300 1782 876 1334 1324 1782 828 1294 1316 1782 780 1270 1292 1782 732 1238 1276 1782 684 1214 1252 1782 636 1182 1236 1782 588 1158 1212 1766 556 1142 1180 1718 556 1142 1132 1670 556 1142 1084 1622 556 1142 1036 1574 556 1142 988 1526 556 1142 940 1478 556 1142 892 1430 556 1142 844 1382 556 1142 796 1334 556 1142 748 1286 556 1142 700 1238 556 1142 652 1190 556 1142 604 % \special{pn 4}% \special{pa 1782 924}% \special{pa 1366 1340}% \special{fp}% \special{pa 1782 972}% \special{pa 1406 1348}% \special{fp}% \special{pa 1782 1020}% \special{pa 1446 1356}% \special{fp}% \special{pa 1782 1068}% \special{pa 1494 1356}% \special{fp}% \special{pa 1782 1116}% \special{pa 1558 1340}% \special{fp}% \special{pa 1782 1164}% \special{pa 1646 1300}% \special{fp}% \special{pa 1782 876}% \special{pa 1334 1324}% \special{fp}% \special{pa 1782 828}% \special{pa 1294 1316}% \special{fp}% \special{pa 1782 780}% \special{pa 1270 1292}% \special{fp}% \special{pa 1782 732}% \special{pa 1238 1276}% \special{fp}% \special{pa 1782 684}% \special{pa 1214 1252}% \special{fp}% \special{pa 1782 636}% \special{pa 1182 1236}% \special{fp}% \special{pa 1782 588}% \special{pa 1158 1212}% \special{fp}% \special{pa 1766 556}% \special{pa 1142 1180}% \special{fp}% \special{pa 1718 556}% \special{pa 1142 1132}% \special{fp}% \special{pa 1670 556}% \special{pa 1142 1084}% \special{fp}% \special{pa 1622 556}% \special{pa 1142 1036}% \special{fp}% \special{pa 1574 556}% \special{pa 1142 988}% \special{fp}% \special{pa 1526 556}% \special{pa 1142 940}% \special{fp}% \special{pa 1478 556}% \special{pa 1142 892}% \special{fp}% \special{pa 1430 556}% \special{pa 1142 844}% \special{fp}% \special{pa 1382 556}% \special{pa 1142 796}% \special{fp}% \special{pa 1334 556}% \special{pa 1142 748}% \special{fp}% \special{pa 1286 556}% \special{pa 1142 700}% \special{fp}% \special{pa 1238 556}% \special{pa 1142 652}% \special{fp}% \special{pa 1190 556}% \special{pa 1142 604}% \special{fp}% % STR 2 0 3 0 % 3 3226 382 3226 396 4 2400 % $y$ \put(32.2600,-3.9600){\makebox(0,0)[rt]{$y$}}% % STR 2 0 3 0 % 3 4060 1876 4060 1890 4 2400 % $x$ \put(40.6000,-18.9000){\makebox(0,0)[rt]{$x$}}% % VECTOR 2 0 3 0 % 2 3258 2156 3258 396 % \special{pn 8}% \special{pa 3258 2156}% \special{pa 3258 396}% \special{fp}% \special{sh 1}% \special{pa 3258 396}% \special{pa 3238 464}% \special{pa 3258 450}% \special{pa 3278 464}% \special{pa 3258 396}% \special{fp}% % VECTOR 2 0 3 0 % 2 3098 1836 4058 1836 % \special{pn 8}% \special{pa 3098 1836}% \special{pa 4058 1836}% \special{fp}% \special{sh 1}% \special{pa 4058 1836}% \special{pa 3992 1816}% \special{pa 4006 1836}% \special{pa 3992 1856}% \special{pa 4058 1836}% \special{fp}% % LINE 3 0 3 0 % 52 3722 1836 3258 1372 3770 1836 3258 1324 3818 1836 3258 1276 3866 1836 3258 1228 3898 1820 3370 1292 3898 1772 3466 1340 3898 1724 3530 1356 3898 1676 3578 1356 3898 1628 3626 1356 3898 1580 3666 1348 3898 1532 3698 1332 3898 1484 3738 1324 3898 1436 3762 1300 3898 1388 3794 1284 3898 1340 3826 1268 3898 1292 3850 1244 3898 1244 3874 1220 3674 1836 3258 1420 3626 1836 3258 1468 3578 1836 3258 1516 3530 1836 3258 1564 3482 1836 3258 1612 3434 1836 3258 1660 3386 1836 3258 1708 3338 1836 3258 1756 3290 1836 3258 1804 % \special{pn 4}% \special{pa 3722 1836}% \special{pa 3258 1372}% \special{fp}% \special{pa 3770 1836}% \special{pa 3258 1324}% \special{fp}% \special{pa 3818 1836}% \special{pa 3258 1276}% \special{fp}% \special{pa 3866 1836}% \special{pa 3258 1228}% \special{fp}% \special{pa 3898 1820}% \special{pa 3370 1292}% \special{fp}% \special{pa 3898 1772}% \special{pa 3466 1340}% \special{fp}% \special{pa 3898 1724}% \special{pa 3530 1356}% \special{fp}% \special{pa 3898 1676}% \special{pa 3578 1356}% \special{fp}% \special{pa 3898 1628}% \special{pa 3626 1356}% \special{fp}% \special{pa 3898 1580}% \special{pa 3666 1348}% \special{fp}% \special{pa 3898 1532}% \special{pa 3698 1332}% \special{fp}% \special{pa 3898 1484}% \special{pa 3738 1324}% \special{fp}% \special{pa 3898 1436}% \special{pa 3762 1300}% \special{fp}% \special{pa 3898 1388}% \special{pa 3794 1284}% \special{fp}% \special{pa 3898 1340}% \special{pa 3826 1268}% \special{fp}% \special{pa 3898 1292}% \special{pa 3850 1244}% \special{fp}% \special{pa 3898 1244}% \special{pa 3874 1220}% \special{fp}% \special{pa 3674 1836}% \special{pa 3258 1420}% \special{fp}% \special{pa 3626 1836}% \special{pa 3258 1468}% \special{fp}% \special{pa 3578 1836}% \special{pa 3258 1516}% \special{fp}% \special{pa 3530 1836}% \special{pa 3258 1564}% \special{fp}% \special{pa 3482 1836}% \special{pa 3258 1612}% \special{fp}% \special{pa 3434 1836}% \special{pa 3258 1660}% \special{fp}% \special{pa 3386 1836}% \special{pa 3258 1708}% \special{fp}% \special{pa 3338 1836}% \special{pa 3258 1756}% \special{fp}% \special{pa 3290 1836}% \special{pa 3258 1804}% \special{fp}% % FUNC 2 2 3 0 % 9 3098 396 4058 1996 3258 1836 3898 1836 3258 1196 3098 396 4058 1996 0 3 0 0 % x^2-x+1 \special{pn 8}% \special{pa 3096 992}% \special{pa 3100 1000}% \special{pa 3106 1006}% \special{pa 3110 1014}% \special{pa 3116 1022}% \special{pa 3120 1028}% \special{pa 3126 1036}% \special{pa 3130 1042}% \special{pa 3136 1050}% \special{pa 3140 1056}% \special{pa 3146 1064}% \special{pa 3150 1070}% \special{pa 3156 1076}% \special{pa 3160 1084}% \special{pa 3166 1090}% \special{pa 3170 1096}% \special{pa 3176 1102}% \special{pa 3180 1108}% \special{pa 3186 1116}% \special{pa 3190 1122}% \special{pa 3196 1128}% \special{pa 3200 1134}% \special{pa 3206 1140}% \special{pa 3210 1144}% \special{pa 3216 1150}% \special{pa 3220 1156}% \special{pa 3226 1162}% \special{pa 3230 1168}% \special{pa 3236 1172}% \special{pa 3240 1178}% \special{pa 3246 1184}% \special{pa 3250 1188}% \special{pa 3256 1194}% \special{pa 3260 1198}% \special{pa 3266 1204}% \special{pa 3270 1208}% \special{pa 3276 1214}% \special{pa 3280 1218}% \special{pa 3286 1222}% \special{pa 3290 1226}% \special{pa 3296 1232}% \special{pa 3300 1236}% \special{pa 3306 1240}% \special{pa 3310 1244}% \special{pa 3316 1248}% \special{pa 3320 1252}% \special{pa 3326 1256}% \special{pa 3330 1260}% \special{pa 3336 1264}% \special{pa 3340 1268}% \special{pa 3346 1272}% \special{pa 3350 1276}% \special{pa 3356 1278}% \special{pa 3360 1282}% \special{pa 3366 1286}% \special{pa 3370 1288}% \special{pa 3376 1292}% \special{pa 3380 1296}% \special{pa 3386 1298}% \special{pa 3390 1302}% \special{pa 3396 1304}% \special{pa 3400 1306}% \special{pa 3406 1310}% \special{pa 3410 1312}% \special{pa 3416 1314}% \special{pa 3420 1318}% \special{pa 3426 1320}% \special{pa 3430 1322}% \special{pa 3436 1324}% \special{pa 3440 1326}% \special{pa 3446 1328}% \special{pa 3450 1330}% \special{pa 3456 1332}% \special{pa 3460 1334}% \special{pa 3466 1336}% \special{pa 3470 1338}% \special{pa 3476 1340}% \special{pa 3480 1342}% \special{pa 3486 1342}% \special{pa 3490 1344}% \special{pa 3496 1346}% \special{pa 3500 1346}% \special{pa 3506 1348}% \special{pa 3510 1350}% \special{pa 3516 1350}% \special{pa 3520 1352}% \special{pa 3526 1352}% \special{pa 3530 1352}% \special{pa 3536 1354}% \special{pa 3540 1354}% \special{pa 3546 1354}% \special{pa 3550 1356}% \special{pa 3556 1356}% \special{pa 3560 1356}% \special{pa 3566 1356}% \special{pa 3570 1356}% \special{pa 3576 1356}% \special{pa 3580 1356}% \special{pa 3586 1356}% \special{pa 3590 1356}% \special{pa 3596 1356}% \special{pa 3600 1356}% \special{pa 3606 1356}% \special{pa 3610 1354}% \special{pa 3616 1354}% \special{pa 3620 1354}% \special{pa 3626 1354}% \special{pa 3630 1352}% \special{pa 3636 1352}% \special{pa 3640 1350}% \special{pa 3646 1350}% \special{pa 3650 1348}% \special{pa 3656 1348}% \special{pa 3660 1346}% \special{pa 3666 1344}% \special{pa 3670 1344}% \special{pa 3676 1342}% \special{pa 3680 1340}% \special{pa 3686 1338}% \special{pa 3690 1336}% \special{pa 3696 1336}% \special{pa 3700 1334}% \special{pa 3706 1332}% \special{pa 3710 1330}% \special{pa 3716 1328}% \special{pa 3720 1324}% \special{pa 3726 1322}% \special{pa 3730 1320}% \special{pa 3736 1318}% \special{pa 3740 1316}% \special{pa 3746 1312}% \special{pa 3750 1310}% \special{pa 3756 1308}% \special{pa 3760 1304}% \special{pa 3766 1302}% \special{pa 3770 1298}% \special{pa 3776 1296}% \special{pa 3780 1292}% \special{pa 3786 1290}% \special{pa 3790 1286}% \special{pa 3796 1282}% \special{pa 3800 1280}% \special{pa 3806 1276}% \special{pa 3810 1272}% \special{pa 3816 1268}% \special{pa 3820 1264}% \special{pa 3826 1262}% \special{pa 3830 1258}% \special{pa 3836 1254}% \special{pa 3840 1250}% \special{pa 3846 1246}% \special{pa 3850 1240}% \special{pa 3856 1236}% \special{pa 3860 1232}% \special{pa 3866 1228}% \special{pa 3870 1224}% \special{pa 3876 1218}% \special{pa 3880 1214}% \special{pa 3886 1210}% \special{pa 3890 1204}% \special{pa 3896 1200}% \special{pa 3900 1194}% \special{pa 3906 1190}% \special{pa 3910 1184}% \special{pa 3916 1180}% \special{pa 3920 1174}% \special{pa 3926 1168}% \special{pa 3930 1162}% \special{pa 3936 1158}% \special{pa 3940 1152}% \special{pa 3946 1146}% \special{pa 3950 1140}% \special{pa 3956 1134}% \special{pa 3960 1128}% \special{pa 3966 1122}% \special{pa 3970 1116}% \special{pa 3976 1110}% \special{pa 3980 1104}% \special{pa 3986 1098}% \special{pa 3990 1092}% \special{pa 3996 1084}% \special{pa 4000 1078}% \special{pa 4006 1072}% \special{pa 4010 1064}% \special{pa 4016 1058}% \special{pa 4020 1052}% \special{pa 4026 1044}% \special{pa 4030 1038}% \special{pa 4036 1030}% \special{pa 4040 1022}% \special{pa 4046 1016}% \special{pa 4050 1008}% \special{pa 4056 1000}% \special{sp -0.045}% % LINE 2 0 3 0 % 4 3258 556 3898 556 3898 556 3898 1836 % \special{pn 8}% \special{pa 3258 556}% \special{pa 3898 556}% \special{fp}% \special{pa 3898 556}% \special{pa 3898 1836}% \special{fp}% % FUNC 2 2 3 0 % 9 982 396 1942 2316 1142 1836 1782 1836 1142 1196 982 396 1942 2316 0 3 0 0 % x^2-x \special{pn 8}% \special{pa 980 1634}% \special{pa 986 1640}% \special{pa 990 1648}% \special{pa 996 1656}% \special{pa 1000 1662}% \special{pa 1006 1670}% \special{pa 1010 1678}% \special{pa 1016 1684}% \special{pa 1020 1692}% \special{pa 1026 1698}% \special{pa 1030 1704}% \special{pa 1036 1712}% \special{pa 1040 1718}% \special{pa 1046 1724}% \special{pa 1050 1732}% \special{pa 1056 1738}% \special{pa 1060 1744}% \special{pa 1066 1750}% \special{pa 1070 1756}% \special{pa 1076 1762}% \special{pa 1080 1768}% \special{pa 1086 1774}% \special{pa 1090 1780}% \special{pa 1096 1786}% \special{pa 1100 1792}% \special{pa 1106 1798}% \special{pa 1110 1802}% \special{pa 1116 1808}% \special{pa 1120 1814}% \special{pa 1126 1820}% \special{pa 1130 1824}% \special{pa 1136 1830}% \special{pa 1140 1834}% \special{pa 1146 1840}% \special{pa 1150 1844}% \special{pa 1156 1850}% \special{pa 1160 1854}% \special{pa 1166 1858}% \special{pa 1170 1864}% \special{pa 1176 1868}% \special{pa 1180 1872}% \special{pa 1186 1876}% \special{pa 1190 1880}% \special{pa 1196 1886}% \special{pa 1200 1890}% \special{pa 1206 1894}% \special{pa 1210 1898}% \special{pa 1216 1902}% \special{pa 1220 1904}% \special{pa 1226 1908}% \special{pa 1230 1912}% \special{pa 1236 1916}% \special{pa 1240 1920}% \special{pa 1246 1922}% \special{pa 1250 1926}% \special{pa 1256 1930}% \special{pa 1260 1932}% \special{pa 1266 1936}% \special{pa 1270 1938}% \special{pa 1276 1942}% \special{pa 1280 1944}% \special{pa 1286 1948}% \special{pa 1290 1950}% \special{pa 1296 1952}% \special{pa 1300 1956}% \special{pa 1306 1958}% \special{pa 1310 1960}% \special{pa 1316 1962}% \special{pa 1320 1964}% \special{pa 1326 1968}% \special{pa 1330 1970}% \special{pa 1336 1972}% \special{pa 1340 1974}% \special{pa 1346 1976}% \special{pa 1350 1976}% \special{pa 1356 1978}% \special{pa 1360 1980}% \special{pa 1366 1982}% \special{pa 1370 1984}% \special{pa 1376 1984}% \special{pa 1380 1986}% \special{pa 1386 1988}% \special{pa 1390 1988}% \special{pa 1396 1990}% \special{pa 1400 1990}% \special{pa 1406 1992}% \special{pa 1410 1992}% \special{pa 1416 1994}% \special{pa 1420 1994}% \special{pa 1426 1994}% \special{pa 1430 1994}% \special{pa 1436 1996}% \special{pa 1440 1996}% \special{pa 1446 1996}% \special{pa 1450 1996}% \special{pa 1456 1996}% \special{pa 1460 1996}% \special{pa 1466 1996}% \special{pa 1470 1996}% \special{pa 1476 1996}% \special{pa 1480 1996}% \special{pa 1486 1996}% \special{pa 1490 1996}% \special{pa 1496 1994}% \special{pa 1500 1994}% \special{pa 1506 1994}% \special{pa 1510 1992}% \special{pa 1516 1992}% \special{pa 1520 1992}% \special{pa 1526 1990}% \special{pa 1530 1990}% \special{pa 1536 1988}% \special{pa 1540 1986}% \special{pa 1546 1986}% \special{pa 1550 1984}% \special{pa 1556 1982}% \special{pa 1560 1982}% \special{pa 1566 1980}% \special{pa 1570 1978}% \special{pa 1576 1976}% \special{pa 1580 1974}% \special{pa 1586 1972}% \special{pa 1590 1970}% \special{pa 1596 1968}% \special{pa 1600 1966}% \special{pa 1606 1964}% \special{pa 1610 1962}% \special{pa 1616 1960}% \special{pa 1620 1958}% \special{pa 1626 1954}% \special{pa 1630 1952}% \special{pa 1636 1950}% \special{pa 1640 1946}% \special{pa 1646 1944}% \special{pa 1650 1942}% \special{pa 1656 1938}% \special{pa 1660 1936}% \special{pa 1666 1932}% \special{pa 1670 1928}% \special{pa 1676 1926}% \special{pa 1680 1922}% \special{pa 1686 1918}% \special{pa 1690 1916}% \special{pa 1696 1912}% \special{pa 1700 1908}% \special{pa 1706 1904}% \special{pa 1710 1900}% \special{pa 1716 1896}% \special{pa 1720 1892}% \special{pa 1726 1888}% \special{pa 1730 1884}% \special{pa 1736 1880}% \special{pa 1740 1876}% \special{pa 1746 1872}% \special{pa 1750 1866}% \special{pa 1756 1862}% \special{pa 1760 1858}% \special{pa 1766 1854}% \special{pa 1770 1848}% \special{pa 1776 1844}% \special{pa 1780 1838}% \special{pa 1786 1834}% \special{pa 1790 1828}% \special{pa 1796 1824}% \special{pa 1800 1818}% \special{pa 1806 1812}% \special{pa 1810 1808}% \special{pa 1816 1802}% \special{pa 1820 1796}% \special{pa 1826 1790}% \special{pa 1830 1784}% \special{pa 1836 1780}% \special{pa 1840 1774}% \special{pa 1846 1768}% \special{pa 1850 1762}% \special{pa 1856 1756}% \special{pa 1860 1748}% \special{pa 1866 1742}% \special{pa 1870 1736}% \special{pa 1876 1730}% \special{pa 1880 1724}% \special{pa 1886 1716}% \special{pa 1890 1710}% \special{pa 1896 1704}% \special{pa 1900 1696}% \special{pa 1906 1690}% \special{pa 1910 1682}% \special{pa 1916 1676}% \special{pa 1920 1668}% \special{pa 1926 1662}% \special{pa 1930 1654}% \special{pa 1936 1646}% \special{pa 1940 1640}% \special{sp -0.045}% % STR 2 0 3 0 % 3 1870 868 1870 948 2 0 % {\scriptsize $y=f_+(x)$} \put(18.7000,-9.4800){\makebox(0,0)[lb]{{\scriptsize $y=f_+(x)$}}}% % STR 2 0 3 0 % 3 1870 1508 1870 1588 2 0 % {\scriptsize $y=f_-(x)$} \put(18.7000,-15.8800){\makebox(0,0)[lb]{{\scriptsize $y=f_-(x)$}}}% % STR 2 0 3 0 % 3 3986 868 3986 948 2 0 % {\scriptsize $y=f_+(x)$} \put(39.8600,-9.4800){\makebox(0,0)[lb]{{\scriptsize $y=f_+(x)$}}}% % STR 2 0 3 0 % 3 3282 1092 3282 1172 2 0 % {\footnotesize 1} \put(32.8200,-11.7200){\makebox(0,0)[lb]{{\footnotesize 1}}}% % STR 2 0 3 0 % 3 1044 1210 1044 1290 2 0 % {\footnotesize 1} \put(10.4400,-12.9000){\makebox(0,0)[lb]{{\footnotesize 1}}}% % STR 2 0 3 0 % 3 1014 1890 1014 1970 2 0 % {\small O} \put(10.1400,-19.7000){\makebox(0,0)[lb]{{\small O}}}% % STR 2 0 3 0 % 3 3130 1890 3130 1970 2 0 % {\small O} \put(31.3000,-19.7000){\makebox(0,0)[lb]{{\small O}}}% % STR 2 0 3 0 % 3 3898 436 3898 516 2 0 % \resizebox{!}{2mm}{$(1,\,2)$} \put(38.9800,-5.1600){\makebox(0,0)[lb]{\resizebox{!}{2mm}{$(1,\,2)$}}}% % STR 2 0 3 0 % 3 1784 440 1784 520 2 0 % \resizebox{!}{2mm}{$(1,\,2)$} \put(17.8400,-5.2000){\makebox(0,0)[lb]{\resizebox{!}{2mm}{$(1,\,2)$}}}% % FUNC 0 0 3 0 % 9 3100 400 4060 2000 3260 1840 3900 1840 3260 1200 3260 400 3900 2000 0 5 0 0 % x^2-x+1 \special{pn 20}% \special{pa 3100 1000}% \special{pa 3106 1008}% \special{pa 3110 1016}% \special{pa 3116 1022}% \special{pa 3120 1030}% \special{pa 3126 1038}% \special{pa 3130 1044}% \special{pa 3136 1052}% \special{pa 3140 1058}% \special{pa 3146 1064}% \special{pa 3150 1072}% \special{pa 3156 1078}% \special{pa 3160 1084}% \special{pa 3166 1092}% \special{pa 3170 1098}% \special{pa 3176 1104}% \special{pa 3180 1110}% \special{pa 3186 1116}% \special{pa 3190 1122}% \special{pa 3196 1128}% \special{pa 3200 1134}% \special{pa 3206 1140}% \special{pa 3210 1146}% \special{pa 3216 1152}% \special{pa 3220 1158}% \special{pa 3226 1164}% \special{pa 3230 1170}% \special{pa 3236 1174}% \special{pa 3240 1180}% \special{pa 3246 1186}% \special{pa 3250 1190}% \special{pa 3256 1196}% \special{ip}% \special{pa 3260 1200}% \special{pa 3266 1206}% \special{pa 3270 1210}% \special{pa 3276 1216}% \special{pa 3280 1220}% \special{pa 3286 1224}% \special{pa 3290 1230}% \special{pa 3296 1234}% \special{pa 3300 1238}% \special{pa 3306 1242}% \special{pa 3310 1246}% \special{pa 3316 1250}% \special{pa 3320 1254}% \special{pa 3326 1258}% \special{pa 3330 1262}% \special{pa 3336 1266}% \special{pa 3340 1270}% \special{pa 3346 1274}% \special{pa 3350 1278}% \special{pa 3356 1282}% \special{pa 3360 1284}% \special{pa 3366 1288}% \special{pa 3370 1292}% \special{pa 3376 1294}% \special{pa 3380 1298}% \special{pa 3386 1302}% \special{pa 3390 1304}% \special{pa 3396 1308}% \special{pa 3400 1310}% \special{pa 3406 1312}% \special{pa 3410 1316}% \special{pa 3416 1318}% \special{pa 3420 1320}% \special{pa 3426 1322}% \special{pa 3430 1326}% \special{pa 3436 1328}% \special{pa 3440 1330}% \special{pa 3446 1332}% \special{pa 3450 1334}% \special{pa 3456 1336}% \special{pa 3460 1338}% \special{pa 3466 1340}% \special{pa 3470 1342}% \special{pa 3476 1344}% \special{pa 3480 1344}% \special{pa 3486 1346}% \special{pa 3490 1348}% \special{pa 3496 1350}% \special{pa 3500 1350}% \special{pa 3506 1352}% \special{pa 3510 1352}% \special{pa 3516 1354}% \special{pa 3520 1354}% \special{pa 3526 1356}% \special{pa 3530 1356}% \special{pa 3536 1358}% \special{pa 3540 1358}% \special{pa 3546 1358}% \special{pa 3550 1360}% \special{pa 3556 1360}% \special{pa 3560 1360}% \special{pa 3566 1360}% \special{pa 3570 1360}% \special{pa 3576 1360}% \special{pa 3580 1360}% \special{pa 3586 1360}% \special{pa 3590 1360}% \special{pa 3596 1360}% \special{pa 3600 1360}% \special{pa 3606 1360}% \special{pa 3610 1360}% \special{pa 3616 1358}% \special{pa 3620 1358}% \special{pa 3626 1358}% \special{pa 3630 1356}% \special{pa 3636 1356}% \special{pa 3640 1354}% \special{pa 3646 1354}% \special{pa 3650 1352}% \special{pa 3656 1352}% \special{pa 3660 1350}% \special{pa 3666 1350}% \special{pa 3670 1348}% \special{pa 3676 1346}% \special{pa 3680 1344}% \special{pa 3686 1344}% \special{pa 3690 1342}% \special{pa 3696 1340}% \special{pa 3700 1338}% \special{pa 3706 1336}% \special{pa 3710 1334}% \special{pa 3716 1332}% \special{pa 3720 1330}% \special{pa 3726 1328}% \special{pa 3730 1326}% \special{pa 3736 1322}% \special{pa 3740 1320}% \special{pa 3746 1318}% \special{pa 3750 1316}% \special{pa 3756 1312}% \special{pa 3760 1310}% \special{pa 3766 1308}% \special{pa 3770 1304}% \special{pa 3776 1302}% \special{pa 3780 1298}% \special{pa 3786 1294}% \special{pa 3790 1292}% \special{pa 3796 1288}% \special{pa 3800 1284}% \special{pa 3806 1282}% \special{pa 3810 1278}% \special{pa 3816 1274}% \special{pa 3820 1270}% \special{pa 3826 1266}% \special{pa 3830 1262}% \special{pa 3836 1258}% \special{pa 3840 1254}% \special{pa 3846 1250}% \special{pa 3850 1246}% \special{pa 3856 1242}% \special{pa 3860 1238}% \special{pa 3866 1234}% \special{pa 3870 1230}% \special{pa 3876 1224}% \special{pa 3880 1220}% \special{pa 3886 1216}% \special{pa 3890 1210}% \special{pa 3896 1206}% \special{pa 3900 1200}% \special{sp}% \special{pa 3900 1200}% \special{pa 3906 1196}% \special{pa 3910 1190}% \special{pa 3916 1186}% \special{pa 3920 1180}% \special{pa 3926 1174}% \special{pa 3930 1170}% \special{pa 3936 1164}% \special{pa 3940 1158}% \special{pa 3946 1152}% \special{pa 3950 1146}% \special{pa 3956 1140}% \special{pa 3960 1134}% \special{pa 3966 1128}% \special{pa 3970 1122}% \special{pa 3976 1116}% \special{pa 3980 1110}% \special{pa 3986 1104}% \special{pa 3990 1098}% \special{pa 3996 1092}% \special{pa 4000 1084}% \special{pa 4006 1078}% \special{pa 4010 1072}% \special{pa 4016 1064}% \special{pa 4020 1058}% \special{pa 4026 1052}% \special{pa 4030 1044}% \special{pa 4036 1038}% \special{pa 4040 1030}% \special{pa 4046 1022}% \special{pa 4050 1016}% \special{pa 4056 1008}% \special{pa 4060 1000}% \special{ip}% % LINE 0 0 3 0 % 2 3260 1840 3260 1200 % \special{pn 20}% \special{pa 3260 1840}% \special{pa 3260 1200}% \special{fp}% % LINE 0 0 3 0 % 2 3900 1200 3900 1840 % \special{pn 20}% \special{pa 3900 1200}% \special{pa 3900 1840}% \special{fp}% % LINE 0 0 3 0 % 2 3900 1840 3260 1840 % \special{pn 20}% \special{pa 3900 1840}% \special{pa 3260 1840}% \special{fp}% % STR 2 0 3 0 % 3 3950 1180 3950 1260 2 0 % \resizebox{!}{2mm}{$(1,\,1)$} \put(39.5000,-12.6000){\makebox(0,0)[lb]{\resizebox{!}{2mm}{$(1,\,1)$}}}% \end{picture}% \end{center} \vspace{-1zw}  ゆえに $S$ の面積は \begin{align*} 1 - \int_0^1 f_+(x)\,dx = 1 - \frac{1}{6}(1 - 0)^3 = \textcolor{red}{\boldsymbol{\frac{5}{6}}} \tag*{$\Ans$} \end{align*} \end{enumerate} \end{document}