数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 指数関数と対数関数 - 大きさの程度(二階微分、導関数の導関数、極限、微分係数の存在)

解析入門 原書第3版
原著

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第2部(微分と基本的な関数)、第8章(指数関数と対数関数)、4(大きさの程度)、練習問題13.を取り組んでみる。

13. 
    
    $\begin{array}{}x\ne \; <!--\; not\; equal\; to\; -->0\\ f\text{'}\left(x\right)=e^{-\frac{1}{x^2}}·\; <!--\; middle\; dot\; -->\frac{2x}{x^4}=\frac{2e^{-\frac{1}{x^2}}}{x^3}\end{array}$

    よって、

    $f\text{'}\left(x\right)=\{\begin{array}{}0\left(x=0\right)\\ \frac{2e^{-\frac{1}{x^2}}}{x^3}\left(x\ne \; <!--\; not\; equal\; to\; -->0\right)\end{array}$

    微分係数について、 x が0の場合どうなるかを考える。

    $\begin{array}{}\underset{h\to \; <!--\; rightwards\; arrow\; -->0}{\lim }\frac{f\text{'}\left(h+0\right)-f\text{'}\left(0\right)}{h}\\ =\underset{h\to \; <!--\; rightwards\; arrow\; -->0}{\lim }\frac{f\text{'}\left(h\right)-0}{h}\\ =\underset{h\to \; <!--\; rightwards\; arrow\; -->0}{\lim }\frac{f\text{'}\left(h\right)}{h}\\ =\underset{h\to \; <!--\; rightwards\; arrow\; -->0}{\lim }\frac{\frac{2e^{-\frac{1}{h^2}}}{h^3}}{h}\\ =\underset{h\to \; <!--\; rightwards\; arrow\; -->0}{\lim }\frac{2}{h^4{e}^{\frac{1}{h^2}}}\\ =0\end{array}$

    x が0 ではない場合。

    $\begin{array}{}f\text{'}\text{'}\left(x\right)=2·\; <!--\; middle\; dot\; -->\frac{e^{-\frac{1}{x^2}}·\; <!--\; middle\; dot\; -->\frac{2x}{x^4}·\; <!--\; middle\; dot\; -->x^3-e^{-\frac{1}{x^2}}3x^2}{x^6}\\ =2·\; <!--\; middle\; dot\; -->\frac{e^{-\frac{1}{x^2}}·\; <!--\; middle\; dot\; -->2-e^{-\frac{1}{x^2}}3x^2}{x^6}\\ =\frac{2e^{-\frac{1}{x^2}}\left(2-3x^2\right)}{x^6}\end{array}$

    よって、導関数をもち、

    $f\text{'}\text{'}\left(x\right)=\{\begin{array}{}0\left(x=0\right)\\ \frac{2e^{-\frac{1}{x^2}}\left(2-3x^2\right)}{x^6}\left(x\ne \; <!--\; not\; equal\; to\; -->0\right)\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, exp, Limit, Derivative, plot

x = symbols('x')
f = exp(-1 / x ** 2)

for n in range(1, 3):
    D = Derivative(f, x, n)
    for t in [D, D.doit()]:
        pprint(t)
        print()

f1 = Derivative(f, x, 1).doit()
for dir in ['+', '-']:
    l = Limit(f1, x, 0, dir=dir)
    for t in [l, l.doit()]:
        pprint(t)
        print()
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample13.py
  ⎛ -1 ⎞
  ⎜ ───⎟
  ⎜   2⎟
d ⎜  x ⎟
──⎝ℯ   ⎠
dx      

   -1 
   ───
     2
    x 
2⋅ℯ   
──────
   3  
  x   

   ⎛ -1 ⎞
   ⎜ ───⎟
  2⎜   2⎟
 d ⎜  x ⎟
───⎝ℯ   ⎠
  2      
dx       

             -1 
             ───
               2
  ⎛     2 ⎞   x 
2⋅⎜-3 + ──⎟⋅ℯ   
  ⎜      2⎟     
  ⎝     x ⎠     
────────────────
        4       
       x        

     ⎛   -1 ⎞
     ⎜   ───⎟
     ⎜     2⎟
     ⎜    x ⎟
     ⎜2⋅ℯ   ⎟
 lim ⎜──────⎟
x─→0⁺⎜   3  ⎟
     ⎝  x   ⎠

0


     ⎛   -1 ⎞
     ⎜   ───⎟
     ⎜     2⎟
     ⎜    x ⎟
     ⎜2⋅ℯ   ⎟
 lim ⎜──────⎟
x─→0⁻⎜   3  ⎟
     ⎝  x   ⎠

0


$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample13.js"></script>

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let f1 = (x) => 2 * Math.exp(-1 / x ** 2) / x ** 3,
    f2 = (x) => 2 * Math.exp(-1 / x ** 2) * (2 - 3 * x ** 2) / x ** 6;

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [],
        fns = [[f1, 'red'],
               [f2, 'green']],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, color]);
            }
        });

    fns1
        .forEach((o) => {
            let [f, color] = o;
            
            lines.push([x1, f(x1), x2, f(x2), color]);
        });
    
    fns2
        .forEach((o) => {
           let [f, color] = o;

            for (let x = x1; x <= x2; x += dx0) {
                let g = f(x);
                lines.push([x1, g(x1), x2, g(x2), color]);
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');

    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =