数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 指数関数と対数関数 - 指数関数(微分可能な関数、区間、導関数、二階導関数、増減の判定、漸近線、グラフの描画)

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)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 参考書籍
  • Pythonからはじめる数学入門
  • 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8章(指数関数と対数関数)、2(指数関数)、練習問題31.を取り組んでみる。

31. 
    
    $f\text{'}\left(x\right)>0$

    よって、 f は問題の区間で増加である。

    $\begin{array}{}f\text{'}\text{'}\left(x\right)=\frac{-2\left(1-x^2\right)·\; <!--\; middle\; dot\; -->\left(-2x\right)}{{\left(1-x^2\right)}^4}\\ =\frac{4x}{{\left(1-x^2\right)}^3}\end{array}$

    $\begin{array}{}-1<x<0\\ f\text{'}\text{'}\left(x\right)<0\\ 0<x<1\\ f\text{'}\text{'}\left(x\right)>0\end{array}$

    x が 1に近づくとき、 f (x)は限りなく大きくなる。

    また、 x が-1に近づくとき、 f(x）は限りなく小さくなる。

    $\begin{array}{}f\text{'}\left(x\right)=\frac{1}{{\left(1+x\right)}^2{\left(1-x\right)}^2}\\ \frac{a}{1+x}+\frac{b}{1-x}+\frac{c}{{\left(1+x\right)}^2}+\frac{d}{{\left(1-x\right)}^2}\\ =\frac{\left(1+x\right){\left(1-x\right)}^{2}a+b\left(1-x\right){\left(1+x\right)}^2+c{\left(1-x\right)}^2+d{\left(1+x\right)}^2}{{\left(1+x\right)}^2{\left(1-x\right)}^2}\\ =\frac{\left(a-b\right)x^3+\left(-a-b+c+d\right)x^2+\left(-a+b-2c+2d\right)x+a+b+c+d}{{\left(1+x\right)}^2{\left(1-x\right)}^2}\end{array}$

    $\begin{array}{}a-b=0\\ b=a\\ -a-a+c+d=0\\ -a+a-2c+2d=0\\ -c+d=0\\ d=c\\ -2a+2c=0\\ c=a\\ d=a\\ 4a=1\\ a=\frac{1}{4}\\ b=c=d=\frac{1}{4}\end{array}$

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

    よって、

    $f\left(x\right)=\frac{1}{4}\left(\log \frac{1+x}{1-x}+\frac{1}{1-x}-\frac{1}{1+x}\right)$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, log, Derivative, Rational, plot

print('31.')
x = symbols('x')
f = Rational(1, 4) * (log((1 + x) / (1 - x)) + 1 / (1 - x) - 1 / (1 + x))

for n in range(1, 3):
    D = Derivative(f, x, n)
    for t in [D, D.doit().factor()]:
        pprint(t)
        print()
    print()

p = plot(f, ylim=(-10, 10), show=False, legend=True)
p.save('sample31.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample31.py && open sample31.svg
31.
  ⎛   ⎛x + 1 ⎞                         ⎞
  ⎜log⎜──────⎟                         ⎟
d ⎜   ⎝-x + 1⎠       1           1     ⎟
──⎜─────────── - ───────── + ──────────⎟
dx⎝     4        4⋅(x + 1)   4⋅(-x + 1)⎠

        1        
─────────────────
       2        2
(x - 1) ⋅(x + 1) 


   ⎛   ⎛x + 1 ⎞                         ⎞
  2⎜log⎜──────⎟                         ⎟
 d ⎜   ⎝-x + 1⎠       1           1     ⎟
───⎜─────────── - ───────── + ──────────⎟
  2⎝     4        4⋅(x + 1)   4⋅(-x + 1)⎠
dx                                       

      -4⋅x       
─────────────────
       3        3
(x - 1) ⋅(x + 1) 


$

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="0.001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">

<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="sample31.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'),
    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 f = (x) => 1 / 4 * (Math.log((1 + x) / (1 - x)) + 1 / (1 - x) - 1 / (1 + x)),
    f1 = (x) => 1 / (1 - x ** 2) ** 2,
    f2 = (x) => 4 * x / (1 - x** 2) ** 3;

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

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        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 = [],
        lines = [[-1, y1, -1, y2, 'red'],
                 [1, y1, 1, y2, 'red']],
        fns = [[f, 'green'],
               [f1, 'blue'],
               [f2, 'orange']],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, color]);
            }
        });

    fns1
        .forEach((o) => {
            let [f, color] = o;
            
            lines.push([x1, f(x1), x2, f(x2), color]);
        });
        
    fns2
        .forEach((o) => {
           let [f, color] = o;

            for (let x = x1; x <= x2; x += dx0) {
                let g = f(x);
                lines.push([x1, g(x1), x2, g(x2), color]);
            }
        });
    
    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('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .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', (d) => d[4] || 'black');

    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) => d[2] || 'green');

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

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

    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

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

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