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(壁と塀、梯子の長さ)

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35.を取り組んでみる。

35. 
    

    壁をy軸とし、x = 1に4mの塀があり、梯子のと地面の接点を(a, 0)、壁との接点を(0, b)とする。

    (0, b)、(1, 4)、(a, 0)を通る直線の方程式を求める。

    $\begin{array}{l}y=-\frac{b}{a}\left(x-1\right)+4\\ 0=-\frac{b}{a}\left(a-1\right)+4\\ 0=-b\left(a-1\right)+4a\\ b=\frac{4a}{a-1}\end{array}$

    梯子の長さについて考える。

    $\begin{array}{l}f\left(a\right)=\sqrt{a^2+b^2}\\ =\sqrt{a^2+{\left(\frac{4a}{a-1}\right)}^2}\\ \\ g\left(a\right)=a^2+{\left(\frac{4a}{a-1}\right)}^2\\ g\text{'}\left(a\right)=2a+2·\frac{4a}{a-1}·\frac{4\left(a-1\right)-4a}{{\left(a-1\right)}^2}\\ =2a\left(1+\frac{-16}{{\left(a-1\right)}^3}\right)\\ 1+\frac{-16}{{\left(a-1\right)}^3}=0\\ {\left(a-1\right)}^3=16\\ a-1={16}^{\frac{1}{3}}\\ a={16}^{\frac{1}{3}}+1\\ \\ f\left({16}^{\frac{1}{3}}+1\right)={\left({\left({16}^{\frac{1}{3}}+1\right)}^2+{\left(\frac{4\left({16}^{\frac{1}{3}}+1\right)}{{16}^{\frac{1}{3}}}\right)}^2\right)}^{\frac{1}{2}}\\ ={\left({\left({16}^{\frac{1}{3}}+1\right)}^2\left(1+{16}^{\frac{1}{3}}\right)\right)}^{\frac{1}{2}}\\ ={\left({16}^{\frac{1}{3}}+1\right)}^{\frac{3}{2}}\end{array}$

コード(Emacs)

Python 3

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

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

print('35.')
x = symbols('x', positive=True)

f = sqrt(x ** 2 + (4 * x / (x - 1)) ** 2)
d = Derivative(f, x, 1)
pprint(d)
f1 = d.doit()
pprint(f1)
xs = solve(f1, x)
pprint(xs)

for x0 in xs:
    if x0 > 0:
        y = f.subs({x: x0})
        pprint(y)
        pprint(y.expand())
        pprint(y.factor())

入出力結果(Terminal, IPython)

$ ./sample35.py
35.
  ⎛      _______________⎞
  ⎜     ╱           2   ⎟
d ⎜    ╱   2    16⋅x    ⎟
──⎜   ╱   x  + ──────── ⎟
dx⎜  ╱                2 ⎟
  ⎝╲╱          (x - 1)  ⎠
       2                 
   16⋅x            16⋅x  
- ──────── + x + ────────
         3              2
  (x - 1)        (x - 1) 
─────────────────────────
        _______________  
       ╱           2     
      ╱   2    16⋅x      
     ╱   x  + ────────   
    ╱                2   
  ╲╱          (x - 1)    
⎡      3 ___⎤
⎣1 + 2⋅╲╱ 2 ⎦
    _________________________________________
   ╱              2                        2 
  ╱  ⎛      3 ___⎞      3 ___ ⎛      3 ___⎞  
╲╱   ⎝1 + 2⋅╲╱ 2 ⎠  + 2⋅╲╱ 2 ⋅⎝1 + 2⋅╲╱ 2 ⎠  
   ________________________
  ╱   3 ___            2/3 
╲╱  6⋅╲╱ 2  + 17 + 12⋅2    
             3/2
⎛      3 ___⎞   
⎝1 + 2⋅╲╱ 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="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" step="0.01" value="0.1">
<label for="a0">a = </label>
<input id="a0" type="number" min="0" step="0.1" 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="sample35.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'),
    input_a0 = document.querySelector('#a0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_dx0, input_a0],
    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 * x / (x - 1)) ** 2),
    f1 = (x) => 1 / 2 * (x ** 2 + (4 * x / (x - 1)) ** 2) ** (-1 / 2) *
    (2 * x + 2 * 4 * x / (x - 1) * (4 * (x - 1) - 4 * x) / (x - 1) ** 2),
    g = (x0) => (x) => f1(x0) * (x - x0) + f(x0),
    b = (a) => 4 * a / (a - 1);

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),
        a0 = parseFloat(input_a0.value);

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

    let points = [],
        x3 = 16 ** (1 / 3) + 1,
        lines = [[1, 0, 1, 4, 'brown'],
                 [0, b(a0), a0, 0, 'blue'],
                 [x3, y1, x3, y2, 'red'],
                 [a0, y1, a0, y2, 'purple']],
        fns = [[f, 'green']],
        fns1 = [[]],
        fns2 = [[g, 'orange']];

    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 = a =