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(距離の最小値、諸島幾何学的解法と微分法による解法、普遍性、優越)

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23.を取り組んでみる。

23. 
    $\begin{array}{l}0\le x\le a\\ f\left(x\right)=PR+QR\\ =\sqrt{x^2+p^2}+\sqrt{{\left(a-x\right)}^2+q^2}\\ =\sqrt{x^2+p^2}+\sqrt{x^2-2ax+a^2+q^2}\\ \\ f\text{'}\left(x\right)=\frac{1}{2}{\left(x^2+p^2\right)}^{-\frac{1}{2}}2x+\frac{1}{2}{\left(x^2-2ax+a^2+q^2\right)}^{-\frac{1}{2}}\left(2x-2a\right)\\ =x{\left(x^2+p^2\right)}^{-\frac{1}{2}}+{\left(x^2-2ax+a^2+q^2\right)}^{-\frac{1}{2}}\left(x-a\right)\\ x{\left(x^2+p^2\right)}^{-\frac{1}{2}}+{\left(x^2-2ax+a^2+q^2\right)}^{-\frac{1}{2}}\left(x-a\right)=0\\ x{\left(x^2+p^2\right)}^{-\frac{1}{2}}=-{\left(x^2-2ax+a^2+q^2\right)}^{-\frac{1}{2}}\left(x-a\right)\\ \frac{x^2}{x^2+p^2}=\frac{{\left(x-a\right)}^2}{x^2-2ax+a^2+q^2}\\ x^4-2a{x}^3+a^2{x}^2+q^2{x}^2=\left(x^2+p^2\right)\left(x^2-2ax+a^2\right)\\ x^4-2a{x}^3+a^2{x}^2+q^2{x}^2=x^4-2a{x}^3+\left(p^2+a^2\right)x^2-2a{p}^{2}x+p^2{a}^2\\ \left(p^2-q^2\right)x^2-2a{p}^{2}x+p^2{a}^2=0\\ x=\frac{a{p}^2\pm \sqrt{a^2{p}^4-\left(p^2-q^2\right)p^2{a}^2}}{p^2-q^2}\\ =\frac{a{p}^2\pm ap\sqrt{p^2-p^2+q^2}}{p^2-q^2}\\ =\frac{a{p}^2\pm apq}{p^2-q^2}\\ =\frac{ap\left(p\pm q\right)}{p^2-q^2}\\ x=\frac{ap}{p\pm q}\\ \\ x\le a\\ x=\frac{ap}{p+q}\\ f\text{'}\left(\frac{ap}{p+q}\right)=0\\ f\left(0\right)<0\\ f\left(a\right)>0\\ f\left(\frac{ap}{p+q}\right)=\sqrt{\frac{a^2{p}^2}{{\left(p+q\right)}^2}+p^2}+\sqrt{{\left(a-\frac{ap}{p+q}\right)}^2+q^2}\\ =\frac{p}{p+q}\sqrt{a^2+{\left(p+q\right)}^2}+\sqrt{a^2{\left(\frac{p+q-p}{p+q}\right)}^2+q^2}\\ =\frac{p}{p+q}\sqrt{a^2+{\left(p+q\right)}^2}+\frac{q}{p+q}\sqrt{a^2+{\left(p+q\right)}^2}\\ =\sqrt{a^2+{\left(p+q\right)}^2}\end{array}$

コード(Emacs)

Python 3

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

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

p, q, a, x = symbols('p q a x', real=True)

f = sqrt(x ** 2 + p ** 2) + sqrt((a - x) ** 2 + q ** 2)
d = Derivative(f, x, 1)
f1 = d.doit()

pprint(d)
pprint(f1)
x
xs = solve(f1, x)
pprint(xs)

for x0 in xs + [0, a]:
    pprint(x0)
    for func in [f, f1]:
        pprint(func.subs({x: x0}))
        print()
    print()

入出力結果(Terminal, IPython)

$ ./sample23.py
  ⎛   _________      _______________⎞
∂ ⎜  ╱  2    2      ╱  2          2 ⎟
──⎝╲╱  p  + x   + ╲╱  q  + (a - x)  ⎠
∂x                                   
     x               -a + x      
──────────── + ──────────────────
   _________      _______________
  ╱  2    2      ╱  2          2 
╲╱  p  + x     ╲╱  q  + (a - x)  
⎡ a⋅p    a⋅p ⎤
⎢─────, ─────⎥
⎣p - q  p + q⎦
 a⋅p 
─────
p - q
     _____________________         _______________
    ╱                   2         ╱   2  2        
   ╱   2   ⎛   a⋅p     ⎞         ╱   a ⋅p       2 
  ╱   q  + ⎜- ───── + a⎟   +    ╱   ──────── + p  
╲╱         ⎝  p - q    ⎠       ╱           2      
                             ╲╱     (p - q)       

                                         a⋅p              
                                        ───── - a         
             a⋅p                        p - q             
───────────────────────────── + ──────────────────────────
              _______________        _____________________
             ╱   2  2               ╱                   2 
            ╱   a ⋅p       2       ╱   2   ⎛   a⋅p     ⎞  
(p - q)⋅   ╱   ──────── + p       ╱   q  + ⎜- ───── + a⎟  
          ╱           2         ╲╱         ⎝  p - q    ⎠  
        ╲╱     (p - q)                                    


 a⋅p 
─────
p + q
     _____________________         _______________
    ╱                   2         ╱   2  2        
   ╱   2   ⎛   a⋅p     ⎞         ╱   a ⋅p       2 
  ╱   q  + ⎜- ───── + a⎟   +    ╱   ──────── + p  
╲╱         ⎝  p + q    ⎠       ╱           2      
                             ╲╱     (p + q)       

                                         a⋅p              
                                        ───── - a         
             a⋅p                        p + q             
───────────────────────────── + ──────────────────────────
              _______________        _____________________
             ╱   2  2               ╱                   2 
            ╱   a ⋅p       2       ╱   2   ⎛   a⋅p     ⎞  
(p + q)⋅   ╱   ──────── + p       ╱   q  + ⎜- ───── + a⎟  
          ╱           2         ╲╱         ⎝  p + q    ⎠  
        ╲╱     (p + q)                                    


0
   _________      
  ╱  2    2       
╲╱  a  + q   + │p│

    -a      
────────────
   _________
  ╱  2    2 
╲╱  a  + q  


a
   _________      
  ╱  2    2       
╲╱  a  + p   + │q│

     a      
────────────
   _________
  ╱  2    2 
╲╱  a  + p  


$

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="20">
<br>
<label for="p0">p = </label>
<input id="p0" type="number" min="0" value="5">
<label for="q0">q = </label>
<input id="q0" type="number" min="0" value="10">
<label for="a0">a0 = </label>
<input id="a0" type="number" min="0" value="10">
<label for="x0">x0 = </label>
<input id="x0" type="number" min="0" step="0.1" value="2">

<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="sample23.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_p0 = document.querySelector('#p0'),
    input_q0 = document.querySelector('#q0'),
    input_a0 = document.querySelector('#a0'),
    input_x0 = document.querySelector('#x0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_p0, input_q0, input_a0, input_x0],
    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.exp(x),
    g = (x) => x,
    h = (x) => Math.log(x);

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),
        p0 = parseFloat(input_p0.value),
        q0 = parseFloat(input_q0.value),
        a0 = parseFloat(input_a0.value),
        x0 = parseFloat(input_x0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        f = (x) =>
        Math.sqrt(x ** 2 + p0 ** 2) + Math.sqrt((a0 - x) ** 2 + q0 ** 2),
        f1 = (x) =>
        x * (x ** 2 + p0 ** 2) ** (- 1 / 2) +
        (x ** 2 - 2 * a0 * x + a0 ** 2 + q0 ** 2) ** (-1 / 2) * (x - a0),
        g = (x) => f1(x0) * (x - x0) + f(x0),
        lines = [[0, 0, 0, p0, 'green'], [a0, 0, a0, q0, 'green'],
                 [0, p0, x0, 0, 'green'], [x0, 0, a0, q0, 'green'],
                 [x0, 0, x0, y2, 'brown']],
        fns = [[f, 'red']],
        fns1 = [[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]);
    });
    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'));
    p(`f(${a0 * p0 / (p0 + q0)}) = ${f(a0 * p0 / (p0 + q0))}`);
    p(f(x0));    
};

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

r = dx =
x1 = x2 =
y1 = y2 =
p = q = a0 = x0 =