{"cells":[{"cell_type":"markdown","metadata":{},"source":["# 17のpre試験\n","\n","1234 西谷滋人\n","\n","# 1(a)\n","(Einstein 結晶のエネルギー) 次の関数 E(x) を求めて x=0..2 でプロットせよ.\n","\\begin{align*}\n","Z(x) = \\frac{\\exp(1/x)}{1-\\exp(-1/x)} \\\\\n","E(x) = x^2 \\frac{\\rm d}{{\\rm d}x} \\log \\left(Z(x)\\right)\n","\\end{align*}\n"]},{"cell_type":"markdown","metadata":{},"source":["# 1(b)\n","資料を参考にして，次の2重積分を求めよ．(15点)\n","\\begin{equation*}\n","\\int \\int_D \\sqrt{2x^2-y^2}dxdy,\\hspace{5mm} D:0\\leqq y \\leqq x \\leqq 1 \n","\\end{equation*}\n"]},{"cell_type":"markdown","metadata":{},"source":["# 2(a)\n","\n","行列$\\displaystyle A= \n","\\left( \\begin {array}{ccc} \n","1&1&3\\\\ \n","-1&0&1\\\\ \n","1&2&1\n","\\end {array} \\right) $\n","の対角化行列を求めて，対角化せよ．(15点)\n"]},{"cell_type":"markdown","metadata":{},"source":["# 2(b)\n","\n","資料を参考にして，行列$\\displaystyle  \n","\\left[ \\begin {array}{cc} \n","1/\\sqrt{2}&a\\\\ \n","b&-1/\\sqrt{2}\n","\\end {array} \\right] $が直交行列であるとき，$a,b$を求めよ．(15点)\n"]},{"cell_type":"markdown","metadata":{},"source":["# 3(a)\n","\n","$p$を実数とし，$f(x)=x^3-p\\,\\, x$とする．\n","\n","関数$f(x)$が極値をもつための$p$の条件を求めよう．$f(x)$の導関数は，\n","\\begin{equation*}\n","f'(x) = \\fbox{ ア }\\,\\,  x^{\\,\\,\\fbox{ イ }}-p\n","\\end{equation*}\n","である．したがって，$f(x)$が$x=a$で極値をとるならば，\n","\\begin{equation*}\n","\\fbox{ ア }\\,\\,  a^{\\,\\,\\fbox{ イ }}-p=\\fbox{ ウ }\n","\\end{equation*}\n","が成り立つ．さらに$x=a$の前後での$f'(x)$の符号の変化を考えることにより，\n","$p$が条件$\\fbox{ エ $(p>0)$}$を満たす場合は$f(x)$は必ず極値を持つことがわかる．"]},{"cell_type":"markdown","metadata":{},"source":["# 3(b)\n","関数$f(x)$が$\\displaystyle x=\\frac{p}{3}$で極値をとるとする．また，曲線$y=f(x)$を$C$とし，$C$上の点$\\displaystyle \\left(\\frac{p}{3}, f\\left(\\frac{p}{3}\\right) \\right)$をAとする．\n","\n","$f(x)$が$\\displaystyle x=\\frac{p}{3}$で極値をとることから，$p=\\fbox{ オ }$であり，$f(x)$は$x=\\fbox{ カキ }$で極大値をとり，$x=\\fbox{ ク }$で極小値をとる．\n","\n"]},{"cell_type":"markdown","metadata":{},"source":["\n","曲線$C$の接線で，点Aを通り傾きが0でないものを$l$とする．$l$の方程式を求めよう．$l$と$C$の接点の$x$座標を$b$とすると，$l$は点$(b, f(b))$における$C$の接線であるから，$l$の方程式は$b$を用いて\n","\\begin{equation*}\n","y= \\left(\\fbox{ ケ }\\,\\,  b^2 - \\fbox{ コ }\\right)(x-b)+f(b)\n","\\end{equation*}\n","と表すことができる．また，$l$は点Aを通るから，方程式\n","\\begin{equation*}\n","\\fbox{ サ }\\,\\, b^3-\\fbox{ シ }\\,\\, b^2+1=0\n","\\end{equation*}\n","を得る．この方程式を解くと，\n","\\begin{equation*}\n","b = \\fbox{ ス }\\,\\,, \\frac{\\fbox{ セソ }}{\\fbox{ タ }}\n","\\end{equation*}\n","であるが，$l$の傾きが0でないことから，$l$の方程式は\n","\\begin{equation*}\n","y = \\frac{\\fbox{ チツ }}{\\fbox{ テ }}\\,\\, x+\\frac{\\fbox{ ト }}{\\fbox{ ナ }}\n","\\end{equation*}\n","である．\n","\n"]},{"cell_type":"markdown","metadata":{},"source":["点Aを頂点とし，原点を通る放物線を$D$とする．$l$と$D$で囲まれた図形のうち，不等式$x \\geqq 0$の表す領域に含まれる部分の面積$S$を求めよう．$D$の方程式は，\n","\\begin{equation*}\n","y = \\fbox{ ニ }\\,\\, x^2 -\\fbox{ ヌ }\\,\\, x\n","\\end{equation*}\n","であるから，定積分を計算することにより，$\\displaystyle S=\\frac{\\fbox{ ネノ }}{24}$となる．(10点)\n","\n","(2014年度大学入試センター試験　本試験　数学II・B第2問)"]},{"cell_type":"markdown","metadata":{},"source":["# 4\n","\n","前問3(b)の$C$上の頂点Aの座標を$\\displaystyle \\left(\\frac{p}{4}, f\\left(\\frac{p}{4}\\right) \\right)$と変えて問題を解け．ただし数値を変えたので，それほど複雑な数字にはならないが，\\fbox{ オ }，\\fbox{ カキ }等には箱にこだわらず数字がはいる．最後は$\\displaystyle S=\\frac{11}{27}$ではなく，$\\displaystyle S=\\frac{352}{243}$になる．(30点)\n"]},{"cell_type":"markdown","metadata":{},"source":["# 4(b)\n","関数$f(x)$が$\\displaystyle x=\\frac{p}{4}$で極値をとるとする．また，曲線$y=f(x)$を$C$とし，$C$上の点$\\displaystyle \\left(\\frac{p}{4}, f\\left(\\frac{p}{4}\\right) \\right)$をAとする．\n","\n","$f(x)$が$\\displaystyle x=\\frac{p}{4}$で極値をとることから，$p=\\fbox{ オ }$であり，$f(x)$は$x=\\fbox{ カキ }$で極大値をとり，$x=\\fbox{ ク }$で極小値をとる．\n","\n"]},{"cell_type":"markdown","metadata":{},"source":["\n","曲線$C$の接線で，点Aを通り傾きが0でないものを$l$とする．$l$の方程式を求めよう．$l$と$C$の接点の$x$座標を$b$とすると，$l$は点$(b, f(b))$における$C$の接線であるから，$l$の方程式は$b$を用いて\n","\\begin{equation*}\n","y= \\left(\\fbox{ ケ }\\,\\,  b^2 - \\fbox{ コ }\\right)(x-b)+f(b)\n","\\end{equation*}\n","と表すことができる．また，$l$は点Aを通るから，方程式\n","\\begin{equation*}\n","\\fbox{ サ }\\,\\, b^3-\\fbox{ シ }\\,\\, b^2+1=0\n","\\end{equation*}\n","を得る．この方程式を解くと，\n","\\begin{equation*}\n","b = \\fbox{ ス }\\,\\,, \\frac{\\fbox{ セソ }}{\\fbox{ タ }}\n","\\end{equation*}\n","であるが，$l$の傾きが0でないことから，$l$の方程式は\n","\\begin{equation*}\n","y = \\frac{\\fbox{ チツ }}{\\fbox{ テ }}\\,\\, x+\\frac{\\fbox{ ト }}{\\fbox{ ナ }}\n","\\end{equation*}\n","である．\n","\n"]},{"cell_type":"markdown","metadata":{},"source":["点Aを頂点とし，原点を通る放物線を$D$とする．$l$と$D$で囲まれた図形のうち，不等式$x \\geqq 0$の表す領域に含まれる部分の面積$S$を求めよう．$D$の方程式は，\n","\\begin{equation*}\n","y = \\fbox{ ニ }\\,\\, x^2 -\\fbox{ ヌ }\\,\\, x\n","\\end{equation*}\n","であるから，定積分を計算することにより，$\\displaystyle S=\\frac{\\fbox{ ネノ }}{24}$となる．(10点)\n","\n","(2014年度大学入試センター試験　本試験　数学II・B第2問)"]}],"metadata":{"kernelspec":{"display_name":"Python 3","language":"python","name":"python3"},"language_info":{"codemirror_mode":{"name":"ipython","version":3},"file_extension":".py","mimetype":"text/x-python","name":"python","nbconvert_exporter":"python","pygments_lexer":"ipython3","version":"3.8.3"},"toc":{"base_numbering":1,"nav_menu":{},"number_sections":true,"sideBar":true,"skip_h1_title":false,"title_cell":"Table of Contents","title_sidebar":"Contents","toc_cell":false,"toc_position":{},"toc_section_display":true,"toc_window_display":true}},"nbformat":4,"nbformat_minor":4}