数学 - JavaScript - 解析学 - 微分と基本的な関数 - 微分係数、導関数(曲線、接線の方程式)

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
  • 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3章(微分係数、導関数)、補充問題 41-50.を取り組んでみる。

41. 
    $\begin{array}{l}\frac{dy}{dx}=\frac{1}{4}{x}^{-\frac{3}{4}}+\frac{3}{2}{x}^{-\frac{1}{4}}\\ y-\left(2+2·2^3\right)=\left(\frac{1}{4}·\frac{1}{8}+\frac{3}{2}·\frac{1}{2}\right)\left(x-16\right)\\ y-18=\left(\frac{1}{32}+\frac{3}{4}\right)\left(x-16\right)\\ y-18=\frac{25}{32}\left(x-16\right)\end{array}$
42. 
    $\begin{array}{l}\frac{dy}{dx}=6x^2\\ y-\left(\frac{2}{8}+3\right)=\frac{3}{2}\left(x-\frac{1}{2}\right)\\ y-\frac{26}{8}=\frac{3}{2}\left(x-\frac{1}{2}\right)\end{array}$
43. 
    $\begin{array}{l}\frac{dy}{dx}=\left(x-3\right)\left(x-4\right)+\left(x-1\right)\left(x-4\right)+\left(x-1\right)\left(x-3\right)\\ y+12=\left(12+4+3\right)x\\ y+12=19x\end{array}$
44. 
    $\begin{array}{l}\frac{dy}{dx}=4x+5\\ y-17=13\left(x-2\right)\end{array}$
45. 
    $\begin{array}{l}\frac{dy}{dx}=2x\left(2x+3\right)+\left(x^2+1\right)2\\ y-10=14\left(x-1\right)\end{array}$
46. 
    $\begin{array}{l}\frac{dy}{dx}=\frac{x+5-\left(x-1\right)}{{\left(x+5\right)}^2}\\ y-\frac{1}{7}=\frac{6}{49}\left(x-2\right)\end{array}$
47. 
    $\begin{array}{l}\frac{dy}{dx}=\frac{2x\left(x^3+1\right)-x^2·3x^2}{{\left(x^3+1\right)}^2}\\ y-\frac{4}{9}=\frac{36-4·12}{81}\left(x-2\right)\\ y-\frac{4}{9}=\frac{12-16}{27}\left(x-2\right)\\ y-\frac{4}{9}=-\frac{4}{27}\left(x-2\right)\end{array}$
48. 
    $\begin{array}{l}\frac{dy}{dx}=\frac{2x\left(x^3+1\right)-\left(x^2+1\right)·3x^2}{{\left(x^3+1\right)}^2}\\ y-\frac{5}{9}=\frac{-24}{81}\left(x-2\right)\\ y-\frac{5}{9}=\frac{-8}{37}\left(x-2\right)\end{array}$
49. 
    $\begin{array}{l}\frac{dy}{dx}=\frac{2x\left(x^2-1\right)-x^2·2x}{{\left(x^2-1\right)}^2}\\ y-\frac{4}{3}=\frac{12-16}{9}\left(x-2\right)\\ y-\frac{4}{3}=-\frac{4}{9}\left(x-2\right)\end{array}$
50. 
    $\begin{array}{l}\frac{dy}{dx}=\frac{x^2+1-\left(x-1\right)·2x}{{\left(x^2+1\right)}^2}\\ y=\frac{1}{2}\left(x-1\right)\end{array}$

コード(Emacs)

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.01" value="0.01">
<label for="n0">n = </label>
<input id="n0" type="number" min="41" max="50" step="1" value="41">
<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="20">

<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="sample41.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_n = document.querySelector('#n0'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dx, input_n, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let f10 = (x) => x ** (1 / 4) + 2 * x ** (3 / 4),
    f11 = (x) => 25 / 32 * (x - 16) + 18,
    f20 = (x) => 2 * x ** 3 + 3,
    f21 = (x) => 3 / 2 * (x - 1 / 2) + 26 / 8,
    f30 = (x) => (x - 1) * (x - 3) * (x - 4),
    f31 = (x) => 19 * x - 12,
    f40 = (x) => 2 * x ** 2 + 5 * x - 1,
    f41 = (x) => 13 * (x - 2) + 17,
    f50 = (x) => (x ** 2 + 1) * (2 * x + 3),
    f51 = (x) => 14 * (x - 1) + 10,
    f60 = (x) => (x - 1) / (x + 5),
    f61 = (x) => 6 / 49 * (x - 2) + 1 / 7,
    f70 = (x) => x ** 2 / (x ** 3 + 1),
    f71 = (x) => - 4 / 27 * (x - 2) + 4 / 9,
    f80 = (x) => (x ** 2 + 1) / (x ** 3 + 1),
    f81 = (x) => - 8 / 37 * (x - 2) + 5 / 9,
    f90 = (x) => x ** 2 / (x ** 2 - 1),
    f91 = (x) => -4 / 9 * (x - 2) + 4 / 3,
    f00 = (x) => (x - 1) / (x ** 2 + 1),
    f01 = (x) => 1 / 2 *(x - 1)
    fns = [[f10, f11],
           [f20, f21],
           [f30, f31],
           [f40, f41],
           [f50, f51],
           [f60, f61],
           [f70, f71],
           [f80, f81],
           [f90, f91],
           [f00, f01]];

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        n = parseInt(input_n.value, 10) - 41,
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        [f0, f1] = fns[n];

    for (let x = x1; x <= x2; x += dx) {
        points.push([x, f0(x)]);
    }
    let t = points.length;
    for (let x = x1; x <= x2; x += dx) {
        points.push([x, f1(x)]);
    }
    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('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d, i) => i < t ? 'green': 'blue');
    
    svg.append('g')
        .attr('transform', `translate(0, ${yscale(0)})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${xscale(0)}, 0)`)
        .call(yaxis);
}

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

r = dx = n =
x1 = x2 =
y1 = y2 =