数学 - JavaScript - 解析学 - 各種の初等関数 - 累乗関数、大きさの比較(複利、利率、元金合計、ネイピア数(オイラー数、自然対数の底))

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5.2-4.を取り組んでみる。

4. 
   $\begin{array}{l}\underset{n\to \infty }{\lim }{\left(1+\frac{r}{n}\right)}^n\\ =\underset{n\to \infty }{\lim }{\left(1+\frac{r}{n}\right)}^{\frac{n}{r}·r}\\ =\underset{n\to \infty }{\lim }{\left({\left(1+\frac{r}{n}\right)}^{\frac{n}{r}}\right)}^r\\ =e^r\end{array}$

コード(Emacs)

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label><input id="r0" type="number" value="0.1">
<label for="n0">n = </label><input id="n0" type="number" min="1" value="100">

<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="sample4.js"></script>    

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    input_r = document.querySelector('#r0'),
    input_n = document.querySelector('#n0'),
    inputs = [input_r, input_n],
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    p = (x) => pre0.textContent += x + '\n';

let f = (r, n) => Math.pow(1 + r / n, n);

let draw = () => {
    pre0.textContent = '';    
    
    let r = parseFloat(input_r.value),
        n = parseInt(input_n.value, 10),
        y0 = Math.exp(r),
        points = [];

    for (let x = 1; x <= n; x += 0.01) {
        points.push([x, f(r, x)]);
    }
    
    let xscale = d3.scaleLinear()
        .domain([1, n])
        .range([padding, width - padding]);
    let ys = points.map((a) => a[1]).concat([y0]);
    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', 1)
        .attr('fill', 'green');

    svg.selectAll('line')
        .data([[[1, y0], [n, y0]]])
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0][0]))
        .attr('y1', (d) => yscale(d[0][1]))
        .attr('x2', (d) => xscale(d[1][0]))
        .attr('y2', (d) => yscale(d[1][1]))
        .attr('stroke', 'blue');

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

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

    let t = points[points.length - 1][1];
    p(`(1 + ${r} / ${n}) ^ ${n} = ${t}`);
    p(`e^${r} = ${y0}`);
    p(`difference: ${y0 - t}`);
}

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

r = n =