## 2017年12月15日金曜日

### 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(対数、累乗、積、曲線、極値点、変曲点、導関数、二回微分、極限)

1. $\begin{array}{}f\text{'}\left(x\right)=\mathrm{log}x+x·\frac{1}{x}=\mathrm{log}x+1\\ f\text{'}\text{'}\left(x\right)=\frac{1}{x}\\ f\text{'}\left(x\right)=0\\ \mathrm{log}x=-1\\ x=\frac{1}{e}\\ x<\frac{1}{e}\\ f\text{'}\left(x\right)<0\\ x>\frac{1}{e}\\ f\text{'}\left(x\right)>0\\ \underset{x\to +0}{\mathrm{lim}}f\left(x\right)=0\\ \underset{x\to \infty }{\mathrm{lim}}f\left(x\right)=\infty \end{array}$

曲線の描画。

2. $\begin{array}{}f\text{'}\left(x\right)=2x\mathrm{log}x+{x}^{2}\frac{1}{x}\\ =2x\mathrm{log}x+x\\ =x\left(2\mathrm{log}x+1\right)\\ f\text{'}\text{'}\left(x\right)=2\mathrm{log}x+2x·\frac{1}{x}+1\\ =2\mathrm{log}x+3\\ f\text{'}\left(x\right)=0\\ 2\mathrm{log}x+1=0\\ \mathrm{log}x=-\frac{1}{2}\\ x={e}^{-\frac{1}{2}}\\ f\text{'}\text{'}\left(x\right)=0\\ 2\mathrm{log}x+3=0\\ \mathrm{log}x=-\frac{3}{2}\\ x={e}^{-\frac{3}{2}}\\ \underset{x\to +0}{\mathrm{lim}}f\left(x\right)=0\\ \underset{x\to \infty }{\mathrm{lim}}f\left(x\right)=\infty \end{array}$

曲線の描画。

3. $\begin{array}{}f\text{'}\left(x\right)={\left(\mathrm{log}x\right)}^{2}+{x}_{2}\left(\mathrm{log}x\right)·\frac{1}{x}\\ =\left(\mathrm{log}x\right)\left(\mathrm{log}x+2\right)\\ f\text{'}\text{'}\left(x\right)=\frac{1}{x}\left(\mathrm{log}x+2\right)+\left(\mathrm{log}x\right)\frac{1}{x}\\ =\frac{2}{x}\left(\mathrm{log}x+1\right)\\ f\text{'}\left(x\right)=0\\ \mathrm{log}x=0\\ x=1\\ \mathrm{log}x=-2\\ x={e}^{-2}\\ f\text{'}\text{'}\left(x\right)=0\\ \mathrm{log}x=-1\\ x={e}^{-1}\\ \underset{x\to +0}{\mathrm{lim}}f\left(x\right)=0\\ \underset{x\to \infty }{\mathrm{lim}}f\left(x\right)=\infty \end{array}$

曲線の描画。

4. $\begin{array}{}f\text{'}\left(x\right)=\frac{\mathrm{log}x-x·\frac{1}{x}}{{\left(\mathrm{log}x\right)}^{2}}\\ =\frac{\mathrm{log}x-1}{{\left(\mathrm{log}x\right)}^{2}}\\ f\text{'}\text{'}\left(x\right)=\frac{\frac{1}{x}·{\left(\mathrm{log}x\right)}^{2}-\left(\mathrm{log}x-1\right)·2\left(\mathrm{log}x\right)·\frac{1}{x}}{{\left(\mathrm{log}x\right)}^{4}}\\ =\frac{\mathrm{log}x-2\mathrm{log}x+2}{x{\left(\mathrm{log}x\right)}^{3}}\\ =\frac{2-16gx}{x{\left(\mathrm{log}x\right)}^{3}}\\ f\text{'}\left(x\right)=0\\ \mathrm{log}x=1\\ x=e\\ f\text{'}\text{'}\left(x\right)=0\\ \mathrm{log}x=2\\ x={e}^{2}\\ \underset{x\to +1}{\mathrm{lim}}f\left(x\right)=\infty \\ \underset{x\to \infty }{\mathrm{lim}}f\left(x\right)=\infty \end{array}$

曲線の描画。

コード(Emacs)

Python 3

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

x = symbols('x')
fs = [(x * log(x), 0),
(x ** 2 * log(x), 0),
(x * (log(x)) ** 2, 0),
(x / log(x), 1)]

for i, (f, x0) in enumerate(fs):
print(f'({chr(ord("a") + i)})')
for n in range(1, 3):
Dn = Derivative(f, x, n)
fn = Dn.doit()
for t in [Dn, fn, solve(fn)]:
pprint(t)
print()
print()
for x1 in [x0, oo]:
l = Limit(f, x, x1)
for t in [l, l.doit()]:
pprint(t)
print()
print()
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="-1">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-1">
<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="sample18.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 fa = (x) => x * Math.log(x),
fb = (x) => x ** 2 * Math.log(x),
fc = (x) => x * Math.log(x) ** 2,
fd = (x) => x / Math.log(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 / Math.E, y1, 1 / Math.E, y2, 'red'],
[x1, -1 / Math.E, x2, -1 / Math.E, 'red'],
[1 / Math.sqrt(Math.E), y1, 1 / Math.sqrt(Math.E), y2, 'green'],
[x1, -1 / (2 * Math.E), x2, -1 / (2 * Math.E), 'green'],
[Math.exp(-2), y1, Math.exp(-2), y2, 'blue'],
[x1, 4 * Math.exp(-2), x2, 4 * Math.exp(-2), 'blue'],
[Math.E, y1, Math.E, y2, 'orange'],
[x1, Math.E, x2, Math.E, 'orange']],

fns = [[fa, 'red'],
[fb, 'green'],
[fc, 'blue'],
[fd, '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])
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();