数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 逆関数 - 逆正接関数(交差点、光線、速度、回転)

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, 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

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7章(逆関数)、4(逆正接関数)、練習問題26.を取り組んでみる。

26. 
    

    時間をt秒、交差点、警察の車、第2の車のなす角をθとする。

    $\begin{array}{l}\tan \theta =\frac{21t}{70-28t}\\ =\frac{3t}{10-4t}\\ =\frac{3}{2}·\frac{t}{5-2t}\\ \\ \theta =\arctan \left(\frac{3}{2}·\frac{t}{5-2t}\right)\\ \\ \frac{d\theta }{dt}=\frac{1}{1+{\left(\frac{3}{2}·\frac{t}{5-2t}\right)}^2}·\frac{3}{2}·\frac{5-2t-t\left(-2\right)}{{\left(5-2t\right)}^2}\\ =\frac{1}{1+{\left(\frac{3}{2}·\frac{t}{5-2t}\right)}^2}·\frac{3}{2}·\frac{5}{{\left(5-2t\right)}^2}\\ \\ t=2\\ \frac{1}{1+{\left(\frac{3}{2}·\frac{2}{5-2·2}\right)}^2}·\frac{3}{2}·\frac{5}{{\left(5-2·2\right)}^2}\\ =\frac{1}{1+{\left(\frac{3}{2}·2\right)}^2}·\frac{3}{2}·5\\ =\frac{1}{1+3^2}·\frac{3}{2}·5\\ =\frac{1}{10}·\frac{3}{2}·5\\ =\frac{1}{2}·\frac{3}{2}\\ =\frac{3}{4}\left(\theta /秒\right)\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3

from sympy import pprint, symbols, atan, Derivative, Rational

t = symbols('t')
θ = atan(3 * t / (2 * (5 - 2 * t)))
Dt = Derivative(θ, t, 1)
for t in [Dt, Dt.doit(), Dt.doit().subs({t: 2})]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample26.py
d ⎛    ⎛   3⋅t   ⎞⎞
──⎜atan⎜─────────⎟⎟
dt⎝    ⎝-4⋅t + 10⎠⎠

    12⋅t           3    
──────────── + ─────────
           2   -4⋅t + 10
(-4⋅t + 10)             
────────────────────────
           2            
        9⋅t             
    ──────────── + 1    
               2        
    (-4⋅t + 10)         

3/4

$

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="0.1">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-100">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="100">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-100">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="100">
<br>
<label for="x0">x = </label>
<input id="x0" type="number" step="2.8" value="-70">

<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="sample26.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_x0 = document.querySelector('#x0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              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 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),
        x0 = parseFloat(input_x0.value);

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

    let points = [],
        lines = [[x0, 0, 0, (70 + x0) / 28 * 21, 'green']],
        fns = [],
        fns1 = [],
        fns2 = [];

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

                points.push([x, y, color]);
            }
        });

    fns1
        .forEach((o) => {
            let [f, color] = o;
            
            lines.push([x1, f(x1), x2, f(x2), 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')));

    p(Math.atan(200 / (1000 - x0)));
};

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

r = dx =
x1 = x2 =
y1 = y2 =
x =