## 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)

1. $\begin{array}{}f\left(x\right)={e}^{x\mathrm{log}x}\\ f\text{'}\left(x\right)={e}^{x\mathrm{log}x}\left(\mathrm{log}x+x·\frac{1}{x}\right)\\ ={e}^{x\mathrm{log}x}\left(\mathrm{log}x+1\right)\\ f\text{'}\text{'}\left(x\right)={e}^{x\mathrm{log}x}\left(\mathrm{log}x+1\right)\left(\mathrm{log}x+1\right)+{e}^{x\mathrm{log}x}\left(\frac{1}{x}+1\right)\\ ={e}^{x\mathrm{log}x}\left({\left(\mathrm{log}x+1\right)}^{2}+\frac{1}{x}+1\right)\end{array}$

x が1より大きい場合。

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

よって、関数 f は強増加である。

1. $\begin{array}{}h\left(x\right)\\ =\frac{\mathrm{log}\left(\mathrm{log}x\right)}{\mathrm{log}\left(\mathrm{log}{e}^{x\mathrm{log}x}\right)}\\ =\frac{\mathrm{log}\left(\mathrm{log}x\right)}{\mathrm{log}\left(x\mathrm{log}x\right)}\\ \underset{x\to \infty }{\mathrm{lim}}\frac{\mathrm{log}x}{x\mathrm{log}x}=0\\ \underset{x\to \infty }{\mathrm{lim}}h\left(x\right)=0\end{array}$

2. $\begin{array}{}x=g{\left(y\right)}^{g\left(y\right)}\\ \mathrm{log}y=x\mathrm{log}x\\ \mathrm{log}\left(\mathrm{log}y\right)=\mathrm{log}\left(x\mathrm{log}x\right)\\ \mathrm{log}\left(\mathrm{log}y\right)=\mathrm{log}x+\mathrm{log}\left(\mathrm{log}x\right)\\ \mathrm{log}\left(\mathrm{log}y\right)-\mathrm{log}\left(\mathrm{log}x\right)=\mathrm{log}x\\ \mathrm{log}x=\mathrm{log}\frac{\mathrm{log}y}{\mathrm{log}x}\\ x=\frac{\mathrm{log}y}{\mathrm{log}x}\\ x=\frac{\mathrm{log}y}{\mathrm{log}\left(\mathrm{log}y\right)}·\frac{\mathrm{log}\left(\mathrm{log}y\right)}{\mathrm{log}x}\end{array}$

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()


$./sample18.py (a) d ──(x⋅log(x)) dx log(x) + 1 ⎡ -1⎤ ⎣ℯ ⎦ 2 d ───(x⋅log(x)) 2 dx 1 ─ x [] lim (x⋅log(x)) x─→0⁺ 0 lim (x⋅log(x)) x─→∞ ∞ (b) d ⎛ 2 ⎞ ──⎝x ⋅log(x)⎠ dx 2⋅x⋅log(x) + x ⎡ -1/2⎤ ⎣ℯ ⎦ 2 d ⎛ 2 ⎞ ───⎝x ⋅log(x)⎠ 2 dx 2⋅log(x) + 3 ⎡ -3/2⎤ ⎣ℯ ⎦ ⎛ 2 ⎞ lim ⎝x ⋅log(x)⎠ x─→0⁺ 0 ⎛ 2 ⎞ lim ⎝x ⋅log(x)⎠ x─→∞ ∞ (c) d ⎛ 2 ⎞ ──⎝x⋅log (x)⎠ dx 2 log (x) + 2⋅log(x) ⎡ -2⎤ ⎣1, ℯ ⎦ 2 d ⎛ 2 ⎞ ───⎝x⋅log (x)⎠ 2 dx 2⋅(log(x) + 1) ────────────── x ⎡ -1⎤ ⎣ℯ ⎦ ⎛ 2 ⎞ lim ⎝x⋅log (x)⎠ x─→0⁺ 0 ⎛ 2 ⎞ lim ⎝x⋅log (x)⎠ x─→∞ ∞ (d) d ⎛ x ⎞ ──⎜──────⎟ dx⎝log(x)⎠ 1 1 ────── - ─────── log(x) 2 log (x) [ℯ] 2 d ⎛ x ⎞ ───⎜──────⎟ 2⎝log(x)⎠ dx 2 -1 + ────── log(x) ─────────── 2 x⋅log (x) ⎡ 2⎤ ⎣ℯ ⎦ ⎛ x ⎞ lim ⎜──────⎟ x─→1⁺⎝log(x)⎠ ∞ ⎛ x ⎞ lim ⎜──────⎟ x─→∞⎝log(x)⎠ ∞$


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,
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])
let yscale = d3.scaleLinear()
.domain([y1, y2])

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();