数学 - JavaScript - 解析学 - 各種の初等関数 - 三角関数(続き)、逆三角関数(大小、台形、面積、最大値)

解析入門〈1〉

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〈1〉(松坂 和夫(著)、岩波書店)の第5章(各種の初等関数)、5.4(三角関数(続き)、逆三角関数)、問題11、12.を取り組んでみる。

11. 
    $\begin{array}{l}0<x<\frac{\pi }{2}\\ \frac{d}{dx}\sin x=\cos x>0\\ \sin 0=0\\ \sin \frac{\pi }{2}=1\\ \frac{2}{\pi }0=0\\ \frac{2}{\pi }·\frac{\pi }{2}=1\\ \\ 0<x<\frac{\pi }{2}\\ \sin x<\frac{2}{\pi }\end{array}$
12. 
    $\begin{array}{l}0<\theta <\frac{\pi }{2}\\ f\left(\theta \right)=\frac{2a+2a\cos \theta }{2}·a\sin \theta \\ =a^2\left(1+\cos \theta \right)\sin \theta \\ \\ f\text{'}\left(\theta \right)=a^2\left(-{\sin }^2\theta +{\cos }^2\theta \right)\\ =a^2\left(2{\cos }^2\theta -1\right)\\ =a^2\left({\cos }^2\theta +1\right)\left(2\cos \theta -1\right)\\ \cos \theta =\frac{1}{2}\\ \theta =\frac{\pi }{3}\\ \\ f\left(\frac{\pi }{3}\right)=a^2\left(1+\frac{1}{2}\right)·\frac{\sqrt{3}}{2}=\frac{3\sqrt{3}{a}^2}{4}\end{array}$

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="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="0.01">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-1.5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1.5">
<br>
<label for="a0">a = </label>
<input id="a0" type="number" min="0" value="5">
<label for="theta0">θ = </label>
<input id="theta0" type="number" min="0" max="1.6" 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_a = document.querySelector('#a0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    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 factorial = (n) => range(1, n + 1).reduce((x, y) => x * y, 1);

let f = (n, x) =>
    range(1, n + 1)
    .map((i) => (-1) ** (i - 1) * x ** (2 * i - 1) / factorial(2 * i - 1))
    .reduce((x, y) => x + y, 0),
    g = (n, x) =>
    range(1, n + 1)
    .map((i) => (-1) ** (i - 1) * x ** (2 * i - 2) / factorial(2 * i - 2))
    .reduce((x, y) => x + y, 0);
    

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

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = 0,
        x2 = Math.PI / 2,
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value),
        a = parseFloat(input_a.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        f = (theta) => a ** 2 * (1 + Math.cos(theta)) * Math.sin(theta);
    
    for (let x = dx; x <= Math.PI / 2; x += dx) {
        let y = Math.sin(x);
        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t1 = points.length;

    for (let x = x1; x <= Math.PI / 2; x += dx) {
        let y = 2 / Math.PI * x;
        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t2 = points.length;

    for (let x = x1; x <= Math.PI / 2; x += dx) {
        let y = f(x);
        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    
    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 < t1 ? 'red':
              i < t2 ? 'green' : 'blue');

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]])
        .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', 'black');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

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

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

r = dx =
x1 = x2 =
y1 = y2 =
a = θ =