数学 - JavaScript - 解析学 - 各種の初等関数 - 累乗関数、大きさの比較(累乗と指数関数の積、極限)

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

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)、5.2(累乗関数、大きさの比較)、問題5.2-1.を取り組んでみる。

1. 
   $\underset{y\to \infty }{\lim }\frac{y^{\alpha }}{{\left(\log a\right)}^{\alpha }}\frac{1}{e^y}=0$

コード(Emacs)

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
r = <input id="r0" type="number" min="0.01" max="0.99" step="0.01" value="0.5">
α = <input id="alpha0" type="number" value="5">
n = <input id="n0" type="number" min="0" step="1" value="20">
<br>
<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="sample1.js"></script>    

JavaScript

let div0 = document.querySelector('#graph0'),
    input_r = document.querySelector('#r0'),
    input_a = document.querySelector('#alpha0'),
    input_n = document.querySelector('#n0'),
    inputs = [input_r, input_a, input_n],
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    p = (x) => pre0.textContent += x + '\n',
    f = (n, a, r) => Math.pow(n, a) * Math.pow(r, n);
    
let draw = () => {
    pre0.textContent = '';
    
    let points = [],
        r = parseFloat(input_r.value),
        a = parseFloat(input_a.value),
        n = parseInt(input_n.value, 10);

    for (let i = 0; i <= n; i += 1) {
        points.push([i, f(i, a, r)]);
    }
    
    let xscale = d3.scaleLinear()
        .domain([0, n + 1])
        .range([padding, width - padding]);
    let ys = points.map((a) => a[1]);
    let yscale = d3.scaleLinear()
        .domain([Math.min(...ys), Math.max(...ys)])
        .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', 2)
        .attr('fill', 'green');

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

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

    points.forEach((x) => {
        p(`${x[0]}, ${x[1]}`);
    });
}

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

r = α = n =