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(強増加、対数の対数、逆関数)

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

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19.を取り組んでみる。

$\begin{array}{l} f (x) = e^{x \log x} \\ f' (x) = e^{x \log x} (\log x + x \cdot \frac{1}{x}) \\ = e^{x \log x} (\log x + 1) \\ f'' (x) = e^{x \log x} (\log x + 1) (\log x + 1) + e^{x \log x} (\frac{1}{x} + 1) \\ = e^{x \log x} ({(\log x + 1)}^{2} + \frac{1}{x} + 1) \end{array}$

x が1より大きい場合。

$\begin{array}{l} e^{x \log x} > 0 \\ \log x + 1 > 0 + 1 = 1 \\ {(\log x + 1)}^{2} + \frac{1}{x} + 1 > 1 + 1 > 0 \\ f' (x) > 0 \\ f'' (x) > 0 \end{array}$

よって、関数 f は強増加である。
1. $\begin{array}{l} h (x) \\ = \frac{\log (\log x)}{\log (\log e^{x \log x})} \\ = \frac{\log (\log x)}{\log (x \log x)} \\ \lim_{x \to \infty} \frac{\log x}{x \log x} = 0 \\ \lim_{x \to \infty} h (x) = 0 \end{array}$
2. $\begin{array}{l} x = g {(y)}^{g (y)} \\ \log y = x \log x \\ \log (\log y) = \log (x \log x) \\ \log (\log y) = \log x + \log (\log x) \\ \log (\log y) - \log (\log x) = \log x \\ \log x = \log \frac{\log y}{\log x} \\ x = \frac{\log y}{\log x} \\ x = \frac{\log y}{\log (\log y)} \cdot \frac{\log (\log y)}{\log x} \end{array}$

コード(Emacs)

Python 3

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

x, y = symbols('x, y', real=True)
f = x ** x
l = Limit(log(log(x)) / log(log(f)), x, oo)
g = solve(y - f, x)[0]
for t in [f, l, l.doit(), g]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample18.py
(a)
d           
──(x⋅log(x))
dx          

log(x) + 1

⎡ -1⎤
⎣ℯ  ⎦


  2          
 d           
───(x⋅log(x))
  2          
dx           

1
─
x

[]


 lim (x⋅log(x))
x─→0⁺          

0


lim (x⋅log(x))
x─→∞          

∞



(b)
d ⎛ 2       ⎞
──⎝x ⋅log(x)⎠
dx           

2⋅x⋅log(x) + x

⎡ -1/2⎤
⎣ℯ    ⎦


  2           
 d ⎛ 2       ⎞
───⎝x ⋅log(x)⎠
  2           
dx            

2⋅log(x) + 3

⎡ -3/2⎤
⎣ℯ    ⎦


     ⎛ 2       ⎞
 lim ⎝x ⋅log(x)⎠
x─→0⁺           

0


    ⎛ 2       ⎞
lim ⎝x ⋅log(x)⎠
x─→∞           

∞



(c)
d ⎛     2   ⎞
──⎝x⋅log (x)⎠
dx           

   2              
log (x) + 2⋅log(x)

⎡    -2⎤
⎣1, ℯ  ⎦


  2           
 d ⎛     2   ⎞
───⎝x⋅log (x)⎠
  2           
dx            

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

⎡ -1⎤
⎣ℯ  ⎦


     ⎛     2   ⎞
 lim ⎝x⋅log (x)⎠
x─→0⁺           

0


    ⎛     2   ⎞
lim ⎝x⋅log (x)⎠
x─→∞           

∞



(d)
d ⎛  x   ⎞
──⎜──────⎟
dx⎝log(x)⎠

  1         1   
────── - ───────
log(x)      2   
         log (x)

[ℯ]


  2        
 d ⎛  x   ⎞
───⎜──────⎟
  2⎝log(x)⎠
dx         

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

⎡ 2⎤
⎣ℯ ⎦


     ⎛  x   ⎞
 lim ⎜──────⎟
x─→1⁺⎝log(x)⎠

∞


    ⎛  x   ⎞
lim ⎜──────⎟
x─→∞⎝log(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">

<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="sample19.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) => x ** x,
    h = (x) => Math.log(Math.log(x)) / Math.log(Math.log(f(x)));

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']],
        fns = [[f, 'green'],
               [h, 'blue']],
        fns1 = [[(x) => x, 'orange']],
        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 =

Kamimura's blog

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2017年12月16日土曜日

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(強増加、対数の対数、逆関数)

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