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(長方形、正の数の和、一定、平方、ワイヤーの切断、正方形と円の面積の最大値、最小値)

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5、6、7.を取り組んでみる。

5. 
   $\begin{array}{l}ab=64\\ b=\frac{64}{a}\\ f\left(a\right)=b^2+{\left(\frac{a}{2}\right)}^2\\ =\frac{{64}^2}{a^2}+\frac{a^2}{4}\\ \\ f\text{'}\left(a\right)=-2·{64}^2{a}^{-3}+\frac{a}{2}\\ -2·{64}^2{a}^{-3}+\frac{a}{2}=0\\ -4·{64}^2+a^4=0\\ a^4=4^7\\ a^4=2^{14}\\ a=2^{\frac{7}{2}}\\ a=8\sqrt{2}\\ b=\frac{64}{8\sqrt{2}}=\frac{8}{\sqrt{2}}=4\sqrt{2}\end{array}$
6. 
   $\begin{array}{l}a+b=4\\ b=4-a\\ f\left(a\right)=a^2+b^2\\ =a^2+16-8a+a^2\\ =2\left(a^2-4a+8\right)\\ f\text{'}\left(a\right)=2\left(2a-4\right)\\ =4\left(a-2\right)\\ a=2\\ b=4-2=2\end{array}$
7. 
   $\begin{array}{l}x,24-x\\ 0\le x\le 24\\ 2\pi r=x\\ r=\frac{x}{2\pi }\\ \\ 4a=24-x\\ a=6-\frac{x}{4}\\ f\left(x\right)=r^2\pi +a^2\\ =\frac{x^2}{4}+36-\frac{x^2}{16}-3x\\ =\frac{3}{16}{x}^2-3x+36\\ f\text{'}\left(x\right)=\frac{3}{8}x-3\\ \frac{3}{8}x-3=0\\ x=8\\ f\left(0\right)=36\\ f\left(8\right)=12-24+36=24\\ f\left(24\right)=72\end{array}$

   最小値: 8 cm で円、16 cm で正方形を作る。

   最大値: 24 cm で円を作る。

コード(Emacs)

Python 3

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

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

x = symbols('x', negative=False)

fs = [65 ** 2 / x ** 2 + x ** 2 / 4,
      2 * (x ** 2 - 4 * x + 8)]

for i, f in enumerate(fs, 5):
    print('{0}.'.format(i))
    d = Derivative(f, x, 1)
    f1 = d.doit()
    pprint(d)
    pprint(f1)
    pprint(solve(f1, x))
    p = plot(f, (x, 0, 5), show=False)
    p.save('sample{0}.svg'.format(i))

入出力結果(Terminal, IPython)

$ ./sample5.py
5.
  ⎛ 2       ⎞
d ⎜x    4225⎟
──⎜── + ────⎟
dx⎜4      2 ⎟
  ⎝      x  ⎠
x   8450
─ - ────
2     3 
     x  
[√130, -√130⋅ⅈ, √130⋅ⅈ]
6.
d ⎛   2           ⎞
──⎝2⋅x  - 8⋅x + 16⎠
dx                 
4⋅x - 8
[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="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-25">
<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="50">
<br>
<label for="a0">a = </label>
<input id="a0" type="number" min="0" max="24" value="10">
b = <span id="b0"></span>

<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="sample5.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_a0 = document.querySelector('#a0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0],
    span_b0 = document.querySelector('#b0'),
    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 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),
        a = parseFloat(input_a0.value),
        b = 24 - a,
        r0 = a / (2 * Math.PI);

    span_b0.innerText = b;
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;n
    }    
    
    let points = [[x2 / 2, y2 / 2, 'green']],
        lines = [[0, 0, 0, b, 'green'],
                 [0, b, -b, b, 'green'],
                 [-b, b, -b, 0, 'green'],
                 [-b, 0, 0, 0, 'green']],
        fns = [];

    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]);
                }
            }
        });                 
    
    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');

    r = xscale(r0) - xscale(0);
    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);

    p(fns.join('\n'));
};

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

r = dx =
x1 = x2 =
y1 = y2 =
a = b =