数学 - Python - JavaScript - 面積、体積、長さ - 積分法の応用 – 面積 - 面積の公式(直線と放物線で囲まれる図形の面積、直線の傾き、最小値)

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, Microsoft Edge, 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20章(面積、体積、長さ - 積分法の応用)、20.1(面積)、面積の公式、問11.を取り組んでみる。

11. 
    

    点(1, 2)を通る傾きmの直線の方程式。

    $y=m\left(x-1\right)+2$

    直線と放物線で囲まれる面積S。

    $\begin{array}{l}m\left(x-1\right)+2=x^2\\ x^2-mx+m-2=0\\ \\ D=m^2-4\left(m-2\right)\\ =m^2-4m+8\\ \\ S=\frac{D^{\frac{3}{2}}}{6}\end{array}$

    Dが最小の時、面積Sは最小になる。

    Dが最小となるmを求める。

    $\begin{array}{l}D={\left(m-2\right)}^2+4\\ m=2\end{array}$

    Sの最小値を求める。

    $\begin{array}{l}S=\frac{4^{\frac{3}{2}}}{6}\\ =\frac{{\left(2^2\right)}^{\frac{3}{2}}}{6}\\ =\frac{2^3}{6}\\ =\frac{4}{3}\end{array}$

コード(Emacs)

Python 3

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

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

print('11.')
x, m = symbols('x m')
f = x ** 2
g = m * (x - 1) + 2

x1, x2 = solve(f - g, x)
S = Integral(g - f, (x, x1, x2))

for t in [S, S.doit()]:
    pprint(t.factor())

p = plot(S.doit(), (m, 1, 3), show=False, legend=True)
p.save('sample11.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample11.py && open sample11.svg 
11.
       ______________                        
      ╱  2                                   
m   ╲╱  m  - 4⋅m + 8                         
─ + ─────────────────                        
2           2                                
          ⌠                                  
          ⎮           ⎛             2    ⎞   
          ⎮           ⎝m⋅(x - 1) - x  + 2⎠ dx
          ⌡                                  
       ______________                        
      ╱  2                                   
m   ╲╱  m  - 4⋅m + 8                         
─ - ─────────────────                        
2           2                                
              3/2
⎛ 2          ⎞   
⎝m  - 4⋅m + 8⎠   
─────────────────
        6        
$

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="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<br>
<label for="m0">m = </label>
<input id="m0" type="number" value="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="sample11.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_m0 = document.querySelector('#m0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_m0],
    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) => x ** 2;

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),
        m0 = parseFloat(input_m0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [[2, y1, 2, y2, 'red']],
        g = (x) => m0 * (x - 1) + 2,
        h = (x) => Math.sqrt((x ** 2 - 4 * x + 8) ** 3) / 6,
        fns = [[f, 'green'],
               [h, 'orange']],
        fns1 = [[g, 'blue']],
        fns2 = [];

    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);

    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
    p(h(m0));
};

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

r = dx =
x1 = x2 =
y1 = y2 =
m =