九州大学 文系 2009年度 問4

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解答作成者: 森 宏征

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入試情報

大学名 九州大学
学科・方式 文系
年度 2009年度
問No 問4
学部 文 ・ 教育 ・ 法 ・ 経済
カテゴリ 微分法の応用
状態 解答 解説なし ウォッチリスト

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\documentclass[a4paper,12pt,fleqn,dvipdfmx]{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} \def\cdotts{{\cdots\cdotssp}} \usepackage{graphicx} \usepackage{pifont} \usepackage{fancybox} \usepackage{custom_mori} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} \begin{FRAME}  曲線 $y = x^2$ の点$\P(a,\ a^2)$における接線と 点$\Q(b,\ b^2)$における接線が点Rで交わるとする. ただし,$a < 0 < b$ とする. このとき,次の問いに答えよ. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  点Rの座標および三角形PRQの面積を求めよ. \item  線分PRと線分QRを2辺とする平行四辺形をPRQSとする. 折れ線PSQと曲線 $y = x^2$ で囲まれた図形の面積を求めよ. \item  $\angle\P\R\Q = 60^\circ$ をみたしながらPとQが動くとき, (2)で求めた面積の最小値を求めよ. \end{enumerate} \end{FRAME} \vskip 2mm \noindent{\color[named]{BurntOrange}\bfseries \Ovalbox{解答}} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  $(x^2)' = 2x$ より$\P(a,\ a^2)$における $y = x^2$ の接線の式は \begin{align*} y = 2a(x - a) + a^2 \qquad \therefore \,\,\, y = 2ax - a^2 \end{align*} 同様に$\Q(b,\ b^2)$における $y = x^2$ の 接線の式は $y = 2bx - b^2$ だから, Rの$x$座標は \begin{align*} 2ax - a^2 = 2bx - b^2 \qquad \therefore \,\,\, x = \frac{a^2 - b^2}{2(a - b)} = \frac{a+b}{2} \quad(\,\because\,\,\,a \neq b) \end{align*} Rの$y$座標は $y = 2a \cdot \dfrac{a+b}{2} - a^2 = ab$ である. ゆえに \\ \begin{minipage}{270pt} \begin{align*} \textcolor{red}{\boldsymbol{ \R\!\left(\frac{a + b}{2},\,\,ab \right) }} \tag*{$\Ans$} \end{align*} PQの中点をMとすると% $\M\!\left(\dfrac{a+b}{2},\ \dfrac{a^2+b^2}{3} \right)$だから, \begin{align*} \triangle\P\Q\R &= \frac{1}{2}\M\R \times (\P と\Q のx座標の差) \\[1mm] &= \frac{1}{2}\!\left(\frac{a^2 + b^2}{2} - ab \right) \cdot (b - a) \\[1mm] &= \frac{1}{4}(a^2 - 2ab + b^2) \cdot (b - a) \\[1mm] &= \textcolor{red}{\boldsymbol{\frac{1}{4}(b - a)^3}} \tag*{$\Ans$} \end{align*} \end{minipage} 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1965 1936 1965 % \special{pn 8}% \special{pa 400 1966}% \special{pa 1936 1966}% \special{fp}% \special{sh 1}% \special{pa 1936 1966}% \special{pa 1870 1946}% \special{pa 1884 1966}% \special{pa 1870 1986}% \special{pa 1936 1966}% \special{fp}% % STR 2 0 3 0 % 3 1180 1909 1180 1920 2 0 % {\footnotesize O} \put(11.8000,-19.2000){\makebox(0,0)[lb]{{\footnotesize O}}}% % LINE 2 0 3 0 % 2 1936 1018 816 2733 % \special{pn 8}% \special{pa 1936 1018}% \special{pa 816 2734}% \special{fp}% % STR 2 0 3 0 % 3 950 2286 950 2350 2 0 % {\footnotesize R} \put(9.5000,-23.5000){\makebox(0,0)[lb]{{\footnotesize R}}}% % STR 2 0 3 0 % 3 600 1476 600 1540 2 0 % {\footnotesize P} \put(6.0000,-15.4000){\makebox(0,0)[lb]{{\footnotesize P}}}% % STR 2 0 3 0 % 3 1494 1766 1494 1830 2 0 % {\footnotesize Q} \put(14.9400,-18.3000){\makebox(0,0)[lb]{{\footnotesize Q}}}% % LINE 2 0 3 0 % 2 1475 1722 726 1459 % \special{pn 8}% \special{pa 1476 1722}% \special{pa 726 1460}% \special{fp}% % LINE 2 0 3 0 % 2 1096 2308 1096 1588 % \special{pn 8}% \special{pa 1096 2308}% \special{pa 1096 1588}% \special{fp}% % STR 2 0 3 0 % 3 950 1646 950 1710 2 0 % {\footnotesize M} \put(9.5000,-17.1000){\makebox(0,0)[lb]{{\footnotesize M}}}% % CIRCLE 3 0 3 0 % 4 1112 2176 1384 2696 1640 1512 1092 1380 % \special{pn 4}% \special{ar 1112 2176 588 588 4.6872686 5.3841844}% % CIRCLE 3 0 3 0 % 4 728 2048 1004 2568 1180 1488 728 1264 % \special{pn 4}% \special{ar 728 2048 590 590 4.7123890 5.3914702}% % LINE 0 0 3 0 % 2 1300 1580 1284 1630 % \special{pn 20}% \special{pa 1300 1580}% \special{pa 1284 1630}% \special{fp}% % LINE 0 0 3 0 % 2 1323 1587 1307 1637 % \special{pn 20}% \special{pa 1324 1588}% \special{pa 1308 1638}% \special{fp}% % LINE 0 0 3 0 % 2 908 1448 892 1498 % \special{pn 20}% \special{pa 908 1448}% \special{pa 892 1498}% \special{fp}% % LINE 0 0 3 0 % 2 931 1455 915 1505 % \special{pn 20}% \special{pa 932 1456}% \special{pa 916 1506}% \special{fp}% % STR 2 0 3 0 % 3 460 480 460 580 2 0 % {\scriptsize$y=x^2$} \put(4.6000,-5.8000){\makebox(0,0)[lb]{{\scriptsize$y=x^2$}}}% \end{picture}% %\input{q2009l4f_zu_2} \end{minipage} %\vskip 0.5zw \newpage \item  題意の面積を $S$ とする. $S$ は $y = x^2$ と直線ABで囲まれる図形の面積と, $\triangle\P\Q\R$の面積を合わせたものである. ABの式は \\ \begin{minipage}{270pt} \begin{gather*} y = \frac{a^2 - b^2}{a - b}(x - a) + a^2 \\ \therefore \,\,\, y = (a + b)x - ab \end{gather*} だから, \begin{align*} S &= \int_a^b \{(a + b)x - ab - x^2\}\,dx + \frac{1}{4}(b - a)^3 \\[1mm] &= -\int_a^b (x - a)(x - b)\,dx + \frac{1}{4}(b - a)^3 \\[1mm] &= \frac{1}{6}(b - a)^3 + \frac{1}{4}(b - a)^3 \\[1mm] &= \textcolor{red}{\boldsymbol{\frac{5}{12}(b - a)^3}} \tag*{$\Ans$} \end{align*} \end{minipage} \begin{minipage}{120pt} \hspace*{1zw} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 15.3600, 22.5300)( 4.0000,-27.3300) % STR 2 0 3 0 % 3 1940 2009 1940 2020 4 3000 % $x$ \put(19.4000,-20.2000){\makebox(0,0)[rt]{$x$}}% % STR 2 0 3 0 % 3 1130 469 1130 480 4 3800 % $y$ \put(11.3000,-4.8000){\makebox(0,0)[rt]{$y$}}% % VECTOR 2 0 3 0 % 2 400 1965 1936 1965 % \special{pn 8}% \special{pa 400 1966}% \special{pa 1936 1966}% \special{fp}% \special{sh 1}% \special{pa 1936 1966}% \special{pa 1870 1946}% \special{pa 1884 1966}% \special{pa 1870 1986}% \special{pa 1936 1966}% \special{fp}% % VECTOR 2 0 3 0 % 2 1168 2733 1168 480 % \special{pn 8}% \special{pa 1168 2734}% \special{pa 1168 480}% \special{fp}% \special{sh 1}% \special{pa 1168 480}% \special{pa 1148 548}% \special{pa 1168 534}% \special{pa 1188 548}% \special{pa 1168 480}% \special{fp}% % STR 2 0 3 0 % 3 1050 2079 1050 2090 2 0 % {\footnotesize O} \put(10.5000,-20.9000){\makebox(0,0)[lb]{{\footnotesize O}}}% % FUNC 2 0 3 0 % 9 400 480 1936 2605 1168 1965 1296 1965 1168 1837 400 480 1936 2605 0 3 0 0 % 1/3x^2 \special{pn 8}% \special{pa 414 480}% \special{pa 416 488}% \special{pa 420 508}% \special{pa 426 528}% \special{pa 430 548}% \special{pa 436 566}% \special{pa 440 586}% \special{pa 446 604}% \special{pa 450 622}% \special{pa 456 642}% \special{pa 460 660}% \special{pa 466 678}% \special{pa 470 696}% \special{pa 476 714}% \special{pa 480 732}% \special{pa 486 750}% \special{pa 490 768}% \special{pa 496 786}% \special{pa 500 804}% \special{pa 506 820}% 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\special{pa 1570 1544}% \special{pa 1576 1534}% \special{pa 1580 1524}% \special{pa 1586 1512}% \special{pa 1590 1502}% \special{pa 1596 1490}% \special{pa 1600 1480}% \special{pa 1606 1468}% \special{pa 1610 1456}% \special{pa 1616 1446}% \special{pa 1620 1434}% \special{pa 1626 1422}% \special{pa 1630 1410}% \special{pa 1636 1398}% \special{pa 1640 1386}% \special{pa 1646 1372}% \special{pa 1650 1360}% \special{pa 1656 1348}% \special{pa 1660 1336}% \special{pa 1666 1322}% \special{pa 1670 1310}% \special{pa 1676 1296}% \special{pa 1680 1282}% \special{pa 1686 1270}% \special{pa 1690 1256}% \special{pa 1696 1242}% \special{pa 1700 1228}% \special{pa 1706 1214}% \special{pa 1710 1200}% \special{pa 1716 1186}% \special{pa 1720 1172}% \special{pa 1726 1158}% \special{pa 1730 1142}% \special{pa 1736 1128}% \special{pa 1740 1114}% \special{pa 1746 1098}% \special{pa 1750 1084}% \special{pa 1756 1068}% \special{pa 1760 1052}% \special{pa 1766 1038}% \special{pa 1770 1022}% \special{pa 1776 1006}% \special{pa 1780 990}% \special{pa 1786 974}% \special{pa 1790 958}% \special{pa 1796 942}% \special{pa 1800 926}% \special{pa 1806 908}% \special{pa 1810 892}% \special{pa 1816 876}% \special{pa 1820 858}% \special{pa 1826 842}% \special{pa 1830 824}% \special{pa 1836 806}% \special{pa 1840 790}% \special{pa 1846 772}% \special{pa 1850 754}% \special{pa 1856 736}% \special{pa 1860 718}% \special{pa 1866 700}% \special{pa 1870 682}% \special{pa 1876 664}% \special{pa 1880 646}% \special{pa 1886 626}% \special{pa 1890 608}% \special{pa 1896 590}% \special{pa 1900 570}% \special{pa 1906 550}% \special{pa 1910 532}% \special{pa 1916 512}% \special{pa 1920 492}% \special{pa 1924 480}% \special{sp}% % LINE 2 0 3 0 % 2 406 736 1283 2733 % \special{pn 8}% \special{pa 406 736}% \special{pa 1284 2734}% \special{fp}% % LINE 2 0 3 0 % 2 1936 1018 816 2733 % \special{pn 8}% \special{pa 1936 1018}% \special{pa 816 2734}% \special{fp}% % POLYLINE 2 0 3 0 % 5 726 1459 1110 870 1110 870 1475 1722 1475 1722 % \special{pn 8}% \special{pa 726 1460}% \special{pa 1110 870}% \special{pa 1110 870}% \special{pa 1476 1722}% \special{pa 1476 1722}% \special{fp}% % STR 2 0 3 0 % 3 940 2286 940 2350 2 0 % {\footnotesize R} \put(9.4000,-23.5000){\makebox(0,0)[lb]{{\footnotesize R}}}% % STR 2 0 3 0 % 3 600 1476 600 1540 2 0 % {\footnotesize P} \put(6.0000,-15.4000){\makebox(0,0)[lb]{{\footnotesize P}}}% % STR 2 0 3 0 % 3 1494 1766 1494 1830 2 0 % {\footnotesize Q} \put(14.9400,-18.3000){\makebox(0,0)[lb]{{\footnotesize Q}}}% % STR 2 0 3 0 % 3 1030 766 1030 830 2 0 % {\footnotesize S} \put(10.3000,-8.3000){\makebox(0,0)[lb]{{\footnotesize S}}}% % LINE 2 0 3 0 % 2 1475 1722 726 1459 % \special{pn 8}% \special{pa 1476 1722}% \special{pa 726 1460}% \special{fp}% % LINE 3 0 3 0 % 52 1302 1664 1046 1920 1334 1670 1066 1939 1360 1683 1091 1952 1392 1690 1123 1958 1418 1702 1155 1965 1443 1715 1194 1965 1462 1734 1251 1946 1277 1651 1021 1907 1245 1645 1002 1888 1219 1632 982 1869 1187 1626 963 1850 1162 1613 944 1830 1136 1600 925 1811 1104 1594 912 1786 1078 1581 893 1766 1046 1574 880 1741 1021 1562 861 1722 989 1555 848 1696 963 1542 835 1670 931 1536 822 1645 906 1523 803 1626 880 1510 790 1600 848 1504 778 1574 822 1491 765 1549 790 1485 752 1523 765 1472 746 1491 % \special{pn 4}% \special{pa 1302 1664}% \special{pa 1046 1920}% \special{fp}% \special{pa 1334 1670}% \special{pa 1066 1940}% \special{fp}% \special{pa 1360 1684}% \special{pa 1092 1952}% \special{fp}% \special{pa 1392 1690}% \special{pa 1124 1958}% \special{fp}% \special{pa 1418 1702}% \special{pa 1156 1966}% \special{fp}% \special{pa 1444 1716}% \special{pa 1194 1966}% \special{fp}% \special{pa 1462 1734}% \special{pa 1252 1946}% \special{fp}% \special{pa 1278 1652}% \special{pa 1022 1908}% \special{fp}% \special{pa 1246 1646}% \special{pa 1002 1888}% \special{fp}% \special{pa 1220 1632}% \special{pa 982 1870}% \special{fp}% \special{pa 1188 1626}% \special{pa 964 1850}% \special{fp}% \special{pa 1162 1614}% \special{pa 944 1830}% \special{fp}% \special{pa 1136 1600}% \special{pa 926 1812}% \special{fp}% \special{pa 1104 1594}% \special{pa 912 1786}% \special{fp}% \special{pa 1078 1582}% \special{pa 894 1766}% \special{fp}% \special{pa 1046 1574}% \special{pa 880 1742}% \special{fp}% \special{pa 1022 1562}% \special{pa 862 1722}% \special{fp}% \special{pa 990 1556}% \special{pa 848 1696}% \special{fp}% \special{pa 964 1542}% \special{pa 836 1670}% \special{fp}% \special{pa 932 1536}% \special{pa 822 1646}% \special{fp}% \special{pa 906 1524}% \special{pa 804 1626}% \special{fp}% \special{pa 880 1510}% \special{pa 790 1600}% \special{fp}% \special{pa 848 1504}% \special{pa 778 1574}% \special{fp}% \special{pa 822 1492}% \special{pa 766 1550}% \special{fp}% \special{pa 790 1486}% \special{pa 752 1524}% \special{fp}% \special{pa 766 1472}% \special{pa 746 1492}% \special{fp}% % STR 2 0 3 0 % 3 490 560 490 660 2 0 % {\scriptsize $y=x^2$} \put(4.9000,-6.6000){\makebox(0,0)[lb]{{\scriptsize $y=x^2$}}}% % LINE 3 0 3 0 % 30 1430 1610 950 1130 1380 1500 970 1090 1340 1400 990 1050 1290 1290 1020 1020 1240 1180 1040 980 1200 1080 1060 940 1150 970 1090 910 1460 1700 920 1160 1390 1690 900 1200 1290 1650 880 1240 1200 1620 850 1270 1110 1590 830 1310 1020 1560 810 1350 930 1530 780 1380 830 1490 760 1420 % \special{pn 4}% \special{pa 1430 1610}% \special{pa 950 1130}% \special{fp}% \special{pa 1380 1500}% \special{pa 970 1090}% \special{fp}% \special{pa 1340 1400}% \special{pa 990 1050}% \special{fp}% \special{pa 1290 1290}% \special{pa 1020 1020}% \special{fp}% \special{pa 1240 1180}% \special{pa 1040 980}% \special{fp}% \special{pa 1200 1080}% \special{pa 1060 940}% \special{fp}% \special{pa 1150 970}% \special{pa 1090 910}% \special{fp}% \special{pa 1460 1700}% \special{pa 920 1160}% \special{fp}% \special{pa 1390 1690}% \special{pa 900 1200}% \special{fp}% \special{pa 1290 1650}% \special{pa 880 1240}% \special{fp}% \special{pa 1200 1620}% \special{pa 850 1270}% \special{fp}% \special{pa 1110 1590}% \special{pa 830 1310}% \special{fp}% \special{pa 1020 1560}% \special{pa 810 1350}% \special{fp}% \special{pa 930 1530}% \special{pa 780 1380}% \special{fp}% \special{pa 830 1490}% \special{pa 760 1420}% \special{fp}% \end{picture}% %\input{q2009l4f_zu_1} \end{minipage} \vskip 0.5zw \item  $\angle\P\R\Q = 90^\circ$ のとき $y = x^2$ のP,\enskip Qにおける 接線は直交するから, \begin{align*} 2a \cdot 2b = -1 \qquad \therefore \,\,\, a = -\frac{1}{4b} \end{align*} このとき $S = \dfrac{5}{12}\!\left(b + \dfrac{1}{4b} \right)^{\!\! 3}$ である. 相加平均と相乗平均の関係より \begin{align*} b + \frac{1}{4b} \geqq 2\sqrt{\vphantom{b} b \cdot \frac{1}{4b}} = 1 \end{align*} 等号は \smallskip$b = \dfrac{1}{4b}$ すなわち $b = \dfrac{1}{2}$ のとき 成り立つから $b + \dfrac{1}{4b}$ は最小値1をとる. このとき $S$ の値は最小になり最小値は, \begin{align*} \frac{5}{12} \cdot 1^3 = \textcolor{red}{\boldsymbol{\frac{5}{12}}} \tag*{$\Ans$} \end{align*} \end{enumerate} \end{document}