大阪大学 前期理系 1989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 大阪大学
学科・方式 前期理系
年度 1989年度
問No 問4
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 微分法の応用 ・ 積分法の応用
状態 解答 解説なし ウォッチリスト

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\documentclass[a4paper,12pt,fleqn]{jreport} \setlength{\topmargin}{-25mm} \setlength{\oddsidemargin}{2.5mm} \setlength{\textwidth}{420pt} \setlength{\textheight}{700pt} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ascmac} \usepackage{graphicx} \usepackage{delarray} \usepackage{multicol} \usepackage{amscd} \usepackage{pifont} \usepackage{color} \ExecuteOptions{usename} \usepackage{vector3} \usepackage{custom_mori} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} \begin{FRAME}  $a$ は正の定数とする. $t > 1$ に対し, 曲線 $y = x^a\log x$ 上の \\ 点 $\P = (t,\,\,t^a\log t)$ における接線が, $x$軸と交わる点をQとし, 点$(t,\,\,0)$をRとする.  三角形PQRの面積を $S_1(t)$, 曲線 $y = x^a\log x$ の $x \geqq 1$ の部分と, 2つの直線 $y = 0,\,\,\,x = t$ とで囲まれる部分の面積を $S_2(t)$ とする. \smallskip  $\lim\limits_{t \to +\infty} \dfrac{S_2(t)}{S_1(t)}$ の値を求めよ. \end{FRAME} \noindent{\color[named]{BurntOrange}\bfseries \fbox{解答}} \vskip 2mm $(x^a\log x)^\prime = ax^{a-1}\log x + x^a \cdot \dfrac{1}{x} = x^{a-1}(a + a\log x)$ より,\smallskip% Pにおける $y = x^a\log x$ の接線は,\\ \begin{minipage}{240pt} \begin{gather*} y = t^{a-1}(1 + a\log t)(x - t) + t^a\log t \end{gather*} よってQの$x$座標は, \begin{gather*} t^{a-1}(1 + a\log t)(x - t) + t^a\log t = 0 \\ \therefore \,\,\, x = t - \frac{t\log t}{1 + a\log t}\,\,(< t) \quad (\,\because\,\,\,t > 1) \end{gather*} したがって, \end{minipage} \begin{minipage}{120pt} %\input{osaka89s4f_zu_2} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 20.4700, 13.5800)( 4.7000,-17.5800) % STR 2 0 3 0 % 3 843 390 843 400 4 2800 % $y$ \put(8.4300,-4.0000){\makebox(0,0)[rt]{$y$}}% % STR 2 0 3 0 % 3 2520 1408 2520 1420 4 2600 % $x$ \put(25.2000,-14.2000){\makebox(0,0)[rt]{$x$}}% % VECTOR 2 0 3 0 % 2 902 1758 902 400 % \special{pn 8}% \special{pa 902 1758}% \special{pa 902 400}% \special{fp}% \special{sh 1}% \special{pa 902 400}% \special{pa 882 468}% \special{pa 902 454}% \special{pa 922 468}% \special{pa 902 400}% \special{fp}% % FUNC 2 0 3 0 % 9 470 400 2517 1758 909 1387 1786 1387 909 1264 470 400 2517 1758 0 3 0 0 % 10x^2ln(x) \special{pn 8}% \special{pa 910 1388}% \special{pa 916 1388}% \special{pa 920 1388}% \special{pa 926 1390}% \special{pa 930 1390}% \special{pa 936 1392}% \special{pa 940 1392}% \special{pa 946 1394}% \special{pa 950 1396}% \special{pa 956 1398}% \special{pa 960 1400}% \special{pa 966 1402}% \special{pa 970 1404}% \special{pa 976 1406}% \special{pa 980 1408}% \special{pa 986 1410}% \special{pa 990 1412}% \special{pa 996 1414}% \special{pa 1000 1418}% \special{pa 1006 1420}% \special{pa 1010 1422}% \special{pa 1016 1426}% \special{pa 1020 1428}% \special{pa 1026 1432}% \special{pa 1030 1434}% \special{pa 1036 1436}% \special{pa 1040 1440}% \special{pa 1046 1442}% \special{pa 1050 1446}% \special{pa 1056 1448}% \special{pa 1060 1452}% \special{pa 1066 1454}% \special{pa 1070 1458}% \special{pa 1076 1460}% \special{pa 1080 1464}% \special{pa 1086 1468}% \special{pa 1090 1470}% \special{pa 1096 1474}% \special{pa 1100 1476}% \special{pa 1106 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\special{pa 1730 1458}% \special{pa 1736 1452}% \special{pa 1740 1446}% \special{pa 1746 1442}% \special{pa 1750 1434}% \special{pa 1756 1428}% \special{pa 1760 1422}% \special{pa 1766 1416}% \special{pa 1770 1410}% \special{pa 1776 1402}% \special{pa 1780 1396}% \special{pa 1786 1388}% \special{pa 1790 1382}% \special{pa 1796 1374}% \special{pa 1800 1368}% \special{pa 1806 1360}% \special{pa 1810 1352}% \special{pa 1816 1344}% \special{pa 1820 1338}% \special{pa 1826 1330}% \special{pa 1830 1322}% \special{pa 1836 1312}% \special{pa 1840 1304}% \special{pa 1846 1296}% \special{pa 1850 1288}% \special{pa 1856 1280}% \special{pa 1860 1270}% \special{pa 1866 1262}% \special{pa 1870 1252}% \special{pa 1876 1244}% \special{pa 1880 1234}% \special{pa 1886 1224}% \special{pa 1890 1216}% \special{pa 1896 1206}% \special{pa 1900 1196}% \special{pa 1906 1186}% \special{pa 1910 1176}% \special{pa 1916 1166}% \special{pa 1920 1156}% \special{pa 1926 1144}% \special{pa 1930 1134}% \special{pa 1936 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532}% \special{pa 2156 516}% \special{pa 2160 498}% \special{pa 2166 482}% \special{pa 2170 464}% \special{pa 2176 446}% \special{pa 2180 428}% \special{pa 2186 412}% \special{pa 2188 400}% \special{sp}% % VECTOR 2 0 3 0 % 2 738 1387 2509 1387 % \special{pn 8}% \special{pa 738 1388}% \special{pa 2510 1388}% \special{fp}% \special{sh 1}% \special{pa 2510 1388}% \special{pa 2442 1368}% \special{pa 2456 1388}% \special{pa 2442 1408}% \special{pa 2510 1388}% \special{fp}% % LINE 2 0 3 0 % 2 1753 1758 2200 400 % \special{pn 8}% \special{pa 1754 1758}% \special{pa 2200 400}% \special{fp}% % DOT 0 0 3 0 % 1 2103 683 % \special{pn 20}% \special{sh 1}% \special{ar 2104 684 10 10 0 6.28318530717959E+0000}% % LINE 2 0 3 0 % 2 2103 1387 2103 683 % \special{pn 8}% \special{pa 2104 1388}% \special{pa 2104 684}% \special{fp}% % STR 2 0 3 0 % 3 1715 1266 1715 1357 2 0 % {\footnotesize 1} \put(17.1500,-13.5700){\makebox(0,0)[lb]{{\footnotesize 1}}}% % STR 2 0 3 0 % 3 2150 670 2150 760 2 0 % {\footnotesize P} \put(21.5000,-7.6000){\makebox(0,0)[lb]{{\footnotesize P}}}% % STR 2 0 3 0 % 3 2067 1419 2067 1510 2 0 % {\footnotesize R} \put(20.6700,-15.1000){\makebox(0,0)[lb]{{\footnotesize R}}}% % STR 2 0 3 0 % 3 1880 1440 1880 1530 2 0 % {\footnotesize Q} \put(18.8000,-15.3000){\makebox(0,0)[lb]{{\footnotesize Q}}}% % STR 2 0 3 0 % 3 1011 842 1011 932 2 0 % {\scriptsize$y=x^a\log x$} \put(10.1100,-9.3200){\makebox(0,0)[lb]{{\scriptsize$y=x^a\log x$}}}% % CIRCLE 3 0 3 0 % 4 1824 1212 1995 1862 659 788 587 2006 % \special{pn 4}% \special{ar 1824 1212 672 672 2.5709501 3.4906393}% % STR 2 0 3 0 % 3 780 1425 780 1520 2 0 % {\footnotesize O} \put(7.8000,-15.2000){\makebox(0,0)[lb]{{\footnotesize O}}}% \end{picture}% \end{minipage} \begin{align*} S_1(t) &= \frac{1}{2} \left\{t - \left(t - \frac{t\log t}{1 + a\log t} \right) \right\} \cdot t^a\log t \\[1mm] &= \frac{t^{a+1}(\log t)^2}{2(1 + a\log t)} \tag*{$\cdott\MARU{1}$} \end{align*} また, \begin{align*} S_2(t) &= \int_1^t x^a\log x\,dx = \int_1^t \left(\frac{x^{a+1}}{a+1} \right)^{\!\! \prime} \log x\,dx \\[1mm] &= \lll \frac{x^{a+1}}{a+1}\log x \rrr_1^t - \frac{1}{a+1}\int_1^t x^a\,dx \\[1mm] &= \frac{t^{a+1}\log t}{a + 1} - \frac{t^{a+1} - 1}{(a + 1)^2} \tag*{$\cdott\MARU{2}$} \end{align*} \MARU{1},\,\,\MARU{2}より, \begin{align*} \frac{S_2(t)}{S_1(t)} &= \frac{t^{a+1}\log t}{a + 1} \cdot \frac{2(1 + a\log t)}{t^{a+1}(\log t)^2} - \frac{t^{a+1} - 1}{(a + 1)^2} \cdot \frac{2(1 + a\log t)}{t^{a+1}(\log t)^2} \\[1mm] &= \frac{2}{a + 1}\! \left(\frac{1}{\log t} + a \right) - \frac{2}{(a + 1)^2}\! \left(1 - \frac{1}{t^{a+1}} \right)\! \left\{\frac{1}{(\log t)^2} + \frac{a}{\log t} \right\} \end{align*} $\lim\limits_{t \to \infty} \dfrac{1}{\log t} = 0$ より, \begin{align*} \lim_{t \to \infty} \frac{S_2(t)}{S_1(t)} = \textcolor{red}{\boldsymbol{\frac{2a}{a + 1}}} \tag*{$\Ans$} \end{align*} \end{document}