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 - 大きさの程度(底、対数関数の微分、導関数、最大値)

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)、4(大きさの程度)、練習問題16.を取り組んでみる。

16. 
    
    $\begin{array}{}xy={\log }_{a}x\\ x=a^{xy}\\ \log x=\log a^{xy}\\ \log x=xy\log a\\ y=\frac{\log x}{x\log a}\end{array}$
    $\begin{array}{}y\text{'}=\frac{\frac{1}{x}·\; <!--\; middle\; dot\; -->x\log a-\left(\log x\right)\left(\log a\right)}{x^2{\left(\log a\right)}^2}\\ =\frac{\log a\left(1-\log x\right)}{x^2{\left(\log a\right)}^2}\\ =\frac{1-\log x}{x^2\log a}\\ \log x=1\\ x=e\\ x<e\\ y\text{'}>0\\ x>e\\ y\text{'}<0\end{array}$

    x が e のとき最大値をもつことがわかる。

    $\begin{array}{}y\text{'}\text{'}=\frac{-\frac{1}{x}·\; <!--\; middle\; dot\; -->x^2\log a-\left(1-\log x\right)2x\log a}{x^4{\left(\log a\right)}^2}\\ =\frac{-1+2\left(\log x-1\right)}{x^3\log a}\\ =\frac{2\log x-3}{x^3\log a}\\ 2\log x-3=0\\ \log x=\frac{3}{2}\\ x=e^{\frac{3}{2}}\\ x<e^{\frac{3}{2}}\\ y\text{'}\text{'}<0\\ x>e^{\frac{3}{2}}\\ y\text{'}\text{'}>0\end{array}$

コード(Emacs)

Python 3

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

x = symbols('x', real=True)
a = symbols('a', real=True)

f = log(x, a) / x
for n in range(1, 3):
    Dn = Derivative(f, x, n)
    fn = Dn.doit()
    for t in [Dn, fn, fn.factor()]:
        pprint(t)
        print()
    try:
        for s in solve(fn):
            pprint(s)
            print()
    except Exception as err:
        print(type(err), err)
    print()

# l1 = Limit(f, x, 0, dir='+')
# l2 = Limit(f, x, oo)

# for l in [l1, l2]:
#     for t in [l, l.doit()]:
#         pprint(t)
#         print()
#     print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample15.py
d ⎛ x⎞
──⎝x ⎠
dx    

 x             
x ⋅(log(x) + 1)

 -1
ℯ  

 LambertW(zoo)
ℯ             


  2    
 d ⎛ x⎞
───⎝x ⎠
  2    
dx     

 x ⎛            2   1⎞
x ⋅⎜(log(x) + 1)  + ─⎟
   ⎝                x⎠

<class 'NotImplementedError'> multiple generators [x, log(x)]
No algorithms are implemented to solve equation (log(x) + 1)**2 + 1/x

      x
 lim x 
x─→0⁺  

1


     x
lim x 
x─→∞  

∞


$

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="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">
<br>
<label for="a0">a0 = </label>
<input id="a0" type="number" value="2">
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" value="0.05">

<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="sample16.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_dx0 = document.querySelector('#dx0'),
    input_a0 = document.querySelector('#a0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_dx0, input_a0],
    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) => x ** x,
    f1 = (x) => Math.exp(x * Math.log(x)) * (Math.log(x) + 1),
    f2 = (x) => Math.exp(x * Math.log(x)) * ((Math.log(x) + 1) ** 2 + 1 / x),
    g = (x0) => (x) => f1(x0) * (x - x0) + f(x0);

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),
        dx0 = parseFloat(input_dx0.value),
        a0 = parseFloat(input_a0.value);
        

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        f = (x) => Math.log(x, a0) / x,
        lines = [[Math.E, y1, Math.E, y2, 'red'],
                 [x1, Math.log(Math.E, a0) / Math.E,
                  x2, Math.log(Math.E, a0) / Math.E, 'blue']],
        fns = [[f, 'green']],
        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 =
a0 = dx0 =