## 2017年12月19日火曜日

### 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(無限大、極限、n乗根)

1. $\begin{array}{}y={\left(\mathrm{log}n\right)}^{\frac{1}{n}}\\ \mathrm{log}y=\frac{1}{n}\mathrm{log}n\\ n\to \infty ⇒\mathrm{log}y=0\\ \underset{n\to \infty }{\mathrm{lim}}{\left(\mathrm{log}n\right)}^{\frac{1}{n}}=1\end{array}$

2. $\begin{array}{}y={\left(\frac{\mathrm{log}n}{n}\right)}^{\frac{1}{n}}\\ \mathrm{log}y=\frac{\mathrm{log}n}{{n}^{2}}\\ n\to \infty ⇒\mathrm{log}y=0\\ \underset{n\to \infty }{\mathrm{lim}}{\left(\frac{\mathrm{log}n}{n}\right)}^{\frac{1}{n}}=1\end{array}$

3. $\begin{array}{}y={\left(\frac{n}{{e}^{n}}\right)}^{\frac{1}{n}}\\ \mathrm{log}y=\frac{1}{n}\mathrm{log}\frac{n}{{e}^{n}}\\ \mathrm{log}y=\frac{1}{n}\left(\mathrm{log}n-\mathrm{log}{e}^{n}\right)\\ \mathrm{log}y=\frac{\mathrm{log}n}{n}-\mathrm{log}e\\ n\to \infty ⇒\mathrm{log}y=-\mathrm{log}e=\mathrm{log}{e}^{-1}\\ \underset{n\to \infty }{\mathrm{lim}}{\left(\frac{n}{{e}^{n}}\right)}^{\frac{1}{n}}={e}^{-1}\end{array}$

4. $\begin{array}{}y={\left(n\mathrm{log}n\right)}^{\frac{1}{n}}\\ \mathrm{log}y=\frac{1}{n}\mathrm{log}\left(n\mathrm{log}n\right)\\ \mathrm{log}y=\frac{1}{n}\left(\mathrm{log}n+\mathrm{log}\left(\mathrm{log}n\right)\right)\\ \mathrm{log}y=\frac{\mathrm{log}n}{n}+\frac{\mathrm{log}\left(\mathrm{log}n\right)}{n}\\ n\to \infty ⇒\mathrm{log}y=0\\ \underset{n\to \infty }{\mathrm{lim}}{\left(n\mathrm{log}n\right)}^{\frac{1}{n}}=1\end{array}$

コード(Emacs)

Python 3

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

n = symbols('n', real=True)
fs = [log(n) ** (1 / n),
(log(n) / n) ** (1 / n),
(n / exp(n)) ** (1 / n),
(n * log(n)) ** (1 / n)]

for i, f in enumerate(fs):
print(f'({chr(ord("a") + i)})')
l = Limit(f, n, oo)
for t in [l, l.doit()]:
pprint(t)
print()
print()

p = plot(1, 1 / E, *fs, (n, 1, 100), show=False, legend=True)
for i, color in enumerate(['red', 'red', 'green', 'blue', 'orange', 'brown']):
p[i].line_color = color

p.save('sample22.svg')


$./sample22.py (a) n ________ lim ╲╱ log(n) n─→∞ 1 (b) ________ ╱ log(n) lim n ╱ ────── n─→∞╲╱ n 1 (c) _______ n ╱ -n lim ╲╱ n⋅ℯ n─→∞ -1 ℯ (d) n __________ lim ╲╱ n⋅log(n) n─→∞ 1$


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="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<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="sample22.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 f1 = (n) => Math.log(n) ** (1 / n),
f2 = (n) => (Math.log(n) / n) ** (1 / n),
f3 = (n) => (n / Math.exp(n)) ** (1 / n),
f4 = (n) => (n * Math.log(n)) ** (1 / n);

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 = [[x1, 1, x2, 1, 'red'],
[x1, 1 / Math.E, x2, 1 / Math.E, 'red']],
fns = [[f1, 'green'],
[f2, 'blue'],
[f3, 'orange'],
[f4, 'brown']],
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])
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();