数学 - JavaScript - 解析学 - 各種の初等関数 - 累乗関数、大きさの比較(指数関数、対数関数、微分、グラフの描画)

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5.2-3.を取り組んでみる。

3. 1. 
      $\begin{array}{l}f\text{'}\left(x\right)=2x{e}^{-x}-x^2{e}^{-x}=x{e}^{-x}\left(2-x\right)\\ f\text{'}\left(x\right)=0\\ x=0,x=2\\ \\ f\text{'}\text{'}\left(x\right)=2e^{-x}-2x{e}^{-x}-2x{e}^{-x}+x^2{e}^{-x}\\ =e^{-x}\left(x^2-4x+2\right)\\ f\text{'}\text{'}\left(x\right)=0\\ x=2\pm \sqrt{4-2}=2\pm \sqrt{2}\end{array}$

      x		0		2-√2		2		2+√2
      f'(x)	-	0	+	+	+	0	-	-	-
      f''(x)	+	+	+	0	-	-	-	0	+
      f(x)		0				4 / e^2

   2. 
      $\begin{array}{l}f\text{'}\left(x\right)=\frac{1-\log x}{x^2}\\ f\text{'}\left(x\right)=0\\ x=e\\ \\ f\text{'}\text{'}\left(x\right)=\frac{-\frac{1}{x}{x}^2-\left(1-\log x\right)2x}{x^4}\\ =\frac{-1-2+2\log x}{x^3}\\ =\frac{-3+2\log x}{x^3}\\ f\text{'}\text{'}\left(x\right)=0\\ -3+2\log x=0\\ \log x=\frac{2}{3}\\ x=e^{\frac{2}{3}}\end{array}$
   3. 
      $\begin{array}{l}f\text{'}\left(x\right)=e^{\frac{1}{x}}\frac{-1}{x^2}=-\frac{e^{\frac{1}{x}}}{x^2}\\ f\text{'}\text{'}\left(x\right)=-\frac{e^{\frac{1}{x}}\frac{-1}{x^2}{x}^2-e^{\frac{1}{x}}2x}{x^4}\\ =-\frac{-e^{\frac{1}{x}}-e^{\frac{1}{x}}2x}{x^4}\\ =\frac{e^{\frac{1}{x}}+e^{\frac{1}{x}}2x}{x^4}\\ =\frac{e^{\frac{1}{x}}\left(1+2x\right)}{x^4}\\ f\text{'}\text{'}\left(x\right)=0\\ x=-\frac{1}{2}\end{array}$

コード(Emacs)

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<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="sample3.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'),
    p = (x) => pre0.textContent += x + '\n';

let f = (x) => Math.pow(x, 2) * Math.exp(-x);

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

    for (let x = -0.5; x <= 4; x += 0.001) {
        points.push([x, f(x)]);
    }
    
    let xscale = d3.scaleLinear()
        .domain([-0.5, 4.5])
        .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);

    let t = points.length / 2;
    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.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

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

btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();