{"cells":[{"cell_type":"markdown","metadata":{"toc":"true"},"source":["<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n","<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#問1:-$2^x$のplot\" data-toc-modified-id=\"問1:-$2^x$のplot-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>問1: $2^x$のplot</a></span></li><li><span><a href=\"#問2:-平均変化率\" data-toc-modified-id=\"問2:-平均変化率-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>問2: 平均変化率</a></span></li><li><span><a href=\"#問3:-接線\" data-toc-modified-id=\"問3:-接線-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>問3: 接線</a></span></li><li><span><a href=\"#問4:-関数とその一次導関数の同時プロット\" data-toc-modified-id=\"問4:-関数とその一次導関数の同時プロット-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>問4: 関数とその一次導関数の同時プロット</a></span></li><li><span><a href=\"#問5:-一次導関数が恒等的にほぼ一致する$b$\" data-toc-modified-id=\"問5:-一次導関数が恒等的にほぼ一致する$b$-5\"><span class=\"toc-item-num\">5&nbsp;&nbsp;</span>問5: 一次導関数が恒等的にほぼ一致する$b$</a></span></li><li><span><a href=\"#問6:-ネイピア数のオイラーによる定義\" data-toc-modified-id=\"問6:-ネイピア数のオイラーによる定義-6\"><span class=\"toc-item-num\">6&nbsp;&nbsp;</span>問6: ネイピア数のオイラーによる定義</a></span><ul class=\"toc-item\"><li><span><a href=\"#解答例\" data-toc-modified-id=\"解答例-6.1\"><span class=\"toc-item-num\">6.1&nbsp;&nbsp;</span>解答例</a></span></li></ul></li><li><span><a href=\"#問7:-おまけ(時間が余ったらやってください)\" data-toc-modified-id=\"問7:-おまけ(時間が余ったらやってください)-7\"><span class=\"toc-item-num\">7&nbsp;&nbsp;</span>問7: おまけ(時間が余ったらやってください)</a></span></li></ul></div>"]},{"cell_type":"markdown","metadata":{},"source":["<br />\n","\n","<div style=\"text-align: center;\">\n","<font size=\"5\">数式処理group work-2(関数と微分)</font>\n","</div>\n","<br />\n","<div style=\"text-align: right;\">\n","<font size=\"4\">file:/~/python/doing_math_with_python/symbolic_math/gw2_diff_ans.ipynb</font>\n","<br />\n","<font size=\"4\">cc by Shigeto R. Nishitani 2009-2021  </font>\n","</div> \n","\n",": ruby bin/pick_works_from_ans.rb gitignores/gw2_diff_ans.ipynb -1 '4 17 23' '13'\n"]},{"cell_type":"markdown","metadata":{},"source":["対数関数の狙いが先週の課題で明らかになったでしょうか．\n","主な応用分野は制御系の計算で，その昔は大砲の着弾距離を計算するのが主な仕事でした．\n","では，対数・指数でexpってのがありますが，\n","なんであんな中途半端な数を使わにゃいかんニャロって思ったことないですか．\n","それを導いてもらおうというのが今日の狙いです．\n","\n","# 問1: $2^x$のplot\n","指数関数\n","$$\n","f(x; b=2) = b^x\n","$$\n","について考える．\n","$f(x; b=2)$の表記は，$b$が助変数で，そのデフォルト値が2であることを意図している．\n","1. 次の表を埋めよ．また，\n","2. 関数$f(x; 2)$をプロットして確認せよ．\n","\n","| x   | -1 | 0 | 1 | 2 | 3 |\n","| ---|---|---|---|---|---|\n","| f(x; 2)|"]},{"cell_type":"markdown","metadata":{},"source":["# 問2: 平均変化率\n","問1.の指数関数に対して，平均変化率\n","$$\n","m0(x; h) = \\frac{f(x+h)-f(x)}{x+h-x}\n","$$\n","を求める式を考える．\n","\n","1. x=0..2(x=0からx=2)の傾きを求めよ．\n","2. それらの点の間を結ぶ直線の方程式を求めよ．\n","3. また，元の関数と同時にプロットせよ．\n","4. x=0..1(x=0からx=1)についても同様に求め，同時にプロットせよ．\n"]},{"cell_type":"markdown","metadata":{},"source":["# 問3: 接線\n","1. 問1.の関数に対して，x=0における接線の傾きを微分により求めよ．また問2.でx=0..0.1に対して求めた平均変化率と小数点以下5桁で比べよ．\n","   1. なお浮動小数点数での表示は，`print(\"%10.5f\" % log(2))`が便利．\n","2. 問1.の関数に対して，x=0における接線の方程式を求め，同時にプロットせよ．\n","3. 問2.で求めた接線をx=0..0.1の平均変化率に対して求め, ~~1.~~ 2.で作成した関数および問1.の関数と同時プロットして比較せよ．"]},{"cell_type":"markdown","metadata":{},"source":["\n","問3，4，5の解答例では問1,2の解答例と違って，テキストの表記に従っているので注意せよ．"]},{"cell_type":"markdown","metadata":{},"source":["# 問4: 関数とその一次導関数の同時プロット\n","問1.の関数とその一次導関数を同時にプロットすると次のようになる．一次導関数の意味を言葉で述べよ．"]},{"cell_type":"code","execution_count":6,"metadata":{},"outputs":[{"name":"stdout","output_type":"stream","text":["2**x\n"]},{"data":{"image/png":"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","text/plain":["<Figure size 432x288 with 1 Axes>"]},"metadata":{"needs_background":"light"},"output_type":"display_data"}],"source":["%matplotlib inline\n","from sympy.plotting import plot\n","from sympy import *\n","\n","x = symbols('x')\n","\n","f0 = 2**x\n","print(f0)\n","f1 = diff(f0,x)\n","p = plot(f0,f1,\n","         (x, -1,3), ylim=[0,8],\n","         show=False)\n","p[0].line_color = 'b'\n","p[1].line_color = 'r'\n","p.show()"]},{"cell_type":"markdown","metadata":{},"source":["# 問5: 一次導関数が恒等的にほぼ一致する$b$\n","関数\n","$$\n","f(x) = b^x, \\, \\textrm{where} \\,2 < b < 3\n","$$\n","と，その一次導関数が恒等的にほぼ一致する$b$を，\n","上のグラフの指数関数の底’2’を色々変えて小数点以下1桁で求めよ．"]},{"cell_type":"markdown","metadata":{},"source":["# 問6: ネイピア数のオイラーによる定義\n","問5で求めた数は自然対数の底e(ネイピア数)のオイラーによる定義である．\n","1. ネイピア数の定義を，この導出にしたがって，言葉で述べよ．\n","2. $f(x; b)=b^x$を問2.の平均変化率の定義に代入し変形することによって，\n","$$\n","m2(h; b) = \\frac{b^h-1}{h}\n","$$\n","としたときに，$b$がネイピア数であるならば$m2(h; b)$が$h \\rightarrow 0$で取るべき値はなんであるか．\n","3. また，\n","\n","> plot(m2(h, 2),m2(h, exp(1)),m2(h, 3),(h, -1, 3))\n","\n","で結果を確かめよ． "]},{"cell_type":"markdown","metadata":{},"source":["この$m1(x; h, b)$には少し注意が必要．\n","$m1(x, h; b)$として扱った方がいいかも．\n","なぜならば，平均変化率を取るときには，\n","$h$を助変数としていたが，最後のplotでは\n","独立変数として導いている．"]},{"cell_type":"markdown","metadata":{},"source":["# 問7: おまけ(時間が余ったらやってください)\n","関数\n","$$\n","f_1(x) = \\frac{x}{x^2-2x+4}\n","$$\n","を考える．\n","1. $f_1(x)$を積分した関数$f_0(x)$を求めよ．\n","2. 関数$f_1(x)$の１次導関数$f_2(x)$を求め，３つの関数を同時にプロットせよ．\n","3. この$f_0$関数の極小値，変曲点の$x$座標をグラフから求めよ．\n","4. この$f_0$関数の増減表を作れ．"]}],"metadata":{"kernelspec":{"display_name":"Python 3.8.5 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