数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 平均値の定理 - 増加・減少関数(廊下、直角、水平な棒、曲がることが可能な長さ)

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)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax
• 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5章(平均値の定理)、3(増加・減少関数)、補充問題21.を取り組んでみる。

21. 
    

    点(2, 4)を通り、負の傾きをもつ直線とx軸との交点を(a, 0)、y軸との交点を(b, 0)とする。

    $\begin{array}{l}y=-\frac{b}{a}\left(x-2\right)+4\\ \\ 0=-\frac{b}{a}\left(a-2\right)+4\\ 0=-ab+2b+4a\\ b=\frac{4a}{a-2}\\ \\ f\left(a\right)=\sqrt{a^2+b^2}\\ =\sqrt{a^2+\frac{4^2{a}^2}{{\left(a-2\right)}^2}}\\ =a\sqrt{1+\frac{4^2}{{\left(a-2\right)}^2}}\\ g\left(a\right)=a^2+\frac{4^2{a}^2}{{\left(a-2\right)}^2}\\ g\text{'}\left(a\right)=2a+\frac{4^2·2a{\left(a-2\right)}^2-4^2{a}^{2}2\left(a-2\right)}{{\left(a-2\right)}^4}\\ =2a\left(1+\frac{4^2\left(\left(a-2\right)-a\right)}{{\left(a-2\right)}^3}\right)\\ =2a\frac{{\left(a-2\right)}^3-2·4^2}{{\left(a-2\right)}^3}\\ {\left(a-2\right)}^3-2^5=0\\ a=2^{\frac{5}{3}}+2\\ f\left(2^{\frac{5}{3}}+2\right)==\left(2^{\frac{5}{3}}+2\right){\left(1+\frac{4^2}{{\left(2^{\frac{5}{3}}+2-2\right)}^2}\right)}^{\frac{1}{2}}\\ =\left(2^{\frac{5}{3}}+2\right){\left(1+\frac{2^4}{2^{\frac{10}{3}}}\right)}^{\frac{1}{2}}\\ =\left(2^{\frac{5}{3}}+2\right){\left(1+2^{\frac{2}{3}}\right)}^{\frac{1}{2}}\\ =\left(2^{\frac{5}{3}}+2\right){\left(1+2^{\frac{2}{3}}\right)}^{\frac{1}{2}}\\ =2\left(2^{\frac{2}{3}}+1\right){\left(1+2^{\frac{2}{3}}\right)}^{\frac{1}{2}}\\ =2{\left(1+2^{\frac{2}{3}}\right)}^{\frac{3}{2}}\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, Pow, Derivative, solve, sqrt

a = symbols('a', positive=True)

b = 4 * a / (a - 2)
f = sqrt(Pow(a, 2) + Pow(b, 2))
d = Derivative(f, a, 1)
f1 = d.doit()
s = solve(f1, a)

for a0 in s:
    pprint(a0)
    result = f.subs({a: a0})
    pprint(result)
    pprint(result.factor())
    pprint(result.expand())

入出力結果(Terminal, IPython)

$ ./sample21.py
       2/3
2 + 2⋅2   
    ____________________________________
   ╱             2                    2 
  ╱  ⎛       2/3⎞     2/3 ⎛       2/3⎞  
╲╱   ⎝2 + 2⋅2   ⎠  + 2   ⋅⎝2 + 2⋅2   ⎠  
            3/2
  ⎛     2/3⎞   
2⋅⎝1 + 2   ⎠   
   _________________________
  ╱     2/3           3 ___ 
╲╱  12⋅2    + 20 + 24⋅╲╱ 2
$

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="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="25">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="25">
<br>

<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0"  value="0.1">

<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="sample21.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'),
    input_dx0 = document.querySelector('#dx0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_dx0],
    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 f = (x) => Math.sqrt(x ** 2 + 4 ** 2 * x ** 2 / (x - 2) ** 2),
    f1 = (x) =>
    1 / 2 * (x ** 2 + 4 ** 2 * x ** 2 / (x - 2) ** 2) ** (-1 / 2) *
    (2 * x + (4 ** 2 * 2 * x * (x - 2) ** 2 - 4 ** 2 * x ** 2 * 2 * (x - 2)) / (x - 2) ** 4),
    g = (x0) => (x) => f1(x0) * (x - x0) + f(x0);
     
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),
        dx0 = parseFloat(input_dx0.value);

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

    let points = [],
        x3 = 2 ** (5 / 3) + 2,
        lines = [[x3, y1, x3, y2, 'red']],    
        fns = [[f, 'green']],
        fns1 = [],
        fns2 = [[g, 'blue']];

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

                if (Math.abs(y) < Infinity) {
                    points.push([x, y, 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 =
dx0 =