数学 - Python - JavaScript - 曲線の性質、最大・最小 - 微分法の応用 – 関数の増減の判定およびその応用 - 最大・最小問題(正接の加法定理、扇形、半径、中心角、光源、照度、光の投射角、距離、比例、反比例、円錐、側面積、体積)

数学読本〈5〉

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

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第18章(曲線の性質、最大・最小 - 微分法の応用)、18.2(関数の増減の判定およびその応用)、最大・最小問題、問25、26、27、28.を取り組んでみる。

25. 
    $\begin{array}{l}\angle OPA=\alpha \\ \tan \alpha =\frac{9}{x}\\ \tan \left(\alpha +\theta \right)=\frac{16}{x}\\ \tan \left(\alpha +\theta \right)=\frac{\tan \alpha +\tan \theta }{1-\tan \alpha \tan \theta }\\ \frac{16}{x}=\frac{\frac{9}{x}+\tan \theta }{1-\frac{9}{x}\tan \theta }\\ \frac{16}{x}-\frac{144}{x^2}\tan \theta =\frac{9}{x}+\tan \theta \\ \frac{7}{x}=\left(\frac{144}{x^2}+1\right)\tan \theta \\ \tan \theta =\frac{7}{x\left(\frac{144}{x^2}+1\right)}\\ =\frac{7x}{144+x^2}\\ \frac{d}{dx}\tan \theta =7\frac{144+x^2-x·2x}{{\left(144+x^2\right)}^2}\\ =-7\frac{x^2-144}{{\left(144+x^2\right)}^2}\\ x=12\end{array}$
26. 
    

    半径の長さをr、中心角の大きさをθ、面積をSとする。

    $\begin{array}{l}2r+2\pi r\frac{\theta }{2\pi }=12\\ 2r+r\theta =12\\ \theta =\frac{12-2r}{r}\\ S=\pi r^2\frac{\theta }{2\pi }\\ =\frac{\theta r^2}{2}\\ =\left(6-r\right)r\\ =-r^2+6r\\ \frac{d}{dr}S=-2r+6\\ =-2\left(r-3\right)\\ \\ r=3\left(m\right)\\ \theta =2\left(rad\right)\\ S=9\left(m^2\right)\end{array}$
27. 
    

    光源の高さをhとする。

    $\begin{array}{l}f\left(h\right)=\frac{\frac{h}{\sqrt{h^2+a^2}}}{\left(h^2+a^2\right)}\\ =\frac{h}{{\left(h^2+a^2\right)}^{\frac{3}{2}}}\\ f\text{'}\left(h\right)=\frac{{\left(h^2+a^2\right)}^{\frac{3}{2}}-h\frac{3}{2}{\left(h^2+a^2\right)}^{\frac{1}{2}}2h}{{\left(h^2+a^2\right)}^3}\\ =\frac{{\left(h^2+a^2\right)}^{\frac{1}{2}}\left(h^2+a^2-3h^2\right)}{{\left(h^2+a^2\right)}^3}\\ -2h^2+a^2=0\\ h=\frac{a}{\sqrt{2}}\end{array}$
28. 
    $\begin{array}{l}S=\pi \left(h^2+r^2\right)\frac{2\pi r}{2\pi \sqrt{h^2+r^2}}\\ =\pi r{\left(h^2+r^2\right)}^{\frac{1}{2}}\\ S^2={\pi }^2{r}^2\left(h^2+r^2\right)\\ h^2=\frac{S^2-{\pi }^2{r}^4}{\pi r^2}\\ V=\frac{1}{3}\pi r^{2}h\\ V^2=\frac{1}{9}{\pi }^2{r}^4{h}^2\\ =\frac{1}{9}\pi r^4\frac{S^2-{\pi }^2{r}^4}{\pi r^2}\\ =\frac{1}{9}{r}^2\left(S^2-{\pi }^2{r}^4\right)\\ =\frac{1}{9}\left(r^2{S}^2-{\pi }^2{r}^6\right)\\ \frac{d}{dr}\left(r^2{S}^2-{\pi }^2{r}^6\right)=2S^{2}r-6{\pi }^2{r}^5\\ =2r\left(S^2-3{\pi }^2{r}^4\right)\\ S^2=3{\pi }^2{r}^4\\ h^2=\frac{S^2-{\pi }^2{r}^4}{\pi r^2}\\ =\frac{3{\pi }^2{r}^4-{\pi }^2{r}^4}{\pi r^2}\\ =2r^2\\ r:h=1:\sqrt{2}\end{array}$

コード(Emacs)

Python 3

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

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

x = symbols('x')

fs = [(25, 7 * x / (144 + x ** 2), (0, 20)),
      (27, x / (x ** 2 + 10 ** 2) ** (3 / 2), (0, 10))]

for i, f, (x1, x2) in fs:
    d = Derivative(f, x, 1)
    pprint(d)
    f1 = d.doit()
    pprint(f1)
    pprint(solve(f1, x))
    p = plot(f, (x, x1, x2), show=False, legend=True)
    p.save('sample{0}.svg'.format(i))

入出力結果(Terminal, IPython)

$ ./sample25.py
d ⎛  7⋅x   ⎞
──⎜────────⎟
dx⎜ 2      ⎟
  ⎝x  + 144⎠
         2              
     14⋅x          7    
- ─────────── + ────────
            2    2      
  ⎛ 2      ⎞    x  + 144
  ⎝x  + 144⎠            
[-12, 12]
  ⎛            -1.5⎞
d ⎜  ⎛ 2      ⎞    ⎟
──⎝x⋅⎝x  + 100⎠    ⎠
dx                  
                   -2.5             -1.5
       2 ⎛ 2      ⎞       ⎛ 2      ⎞    
- 3.0⋅x ⋅⎝x  + 100⎠     + ⎝x  + 100⎠    
[-7.07106781186548, 7.07106781186548]
$

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="20">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="0.02">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" value="0.1">

<label for="a0">a = </label>
<input id="a0" type="number" min="0" 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="sample25.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 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 = [],        
        lines = [],
        f = (x) => x / (x ** 2 + a0 ** 2) ** (3 / 2),
        f1 = (x) =>
        (x ** 2 + a0 ** 2) ** (1 / 2) * (x ** 2 + a0 ** 2 - 3 * x ** 2) /
        (x ** 2 + a0 ** 2) ** 3,
        g = (x0) => (x) => f1(x0) * (x - x0) + f(x0),
        fns = [[f, 'green']],
        fns1 = [],
        fns2 = [[g, 'blue']];

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

            if (Math.abs(y) < Infinity) {
                points.push([x, y, color]);
            }
        }
    });
    fns1.forEach((o) => {
        let [fn, color] = o;

        lines.push([x1, fn(x1), x2, fn(x2), color]);
    });
    fns2.forEach((o) => {
        let [fn, color] = o;
        
        for (let x = x1; x <= x2; x += dx0) {
            let g = fn(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);
    
    p(fns.join('\n'));
    p(fns1.join('\n'));
};

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

(x) => x / (x ** 2 + a0 ** 2) ** (3 / 2),green

r = dx =
x1 = x2 =
y1 = y2 =
dx0 = a =