## 2017年12月13日水曜日

### 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(底、対数関数の微分、導関数、最大値)

1. $\begin{array}{}xy={\mathrm{log}}_{a}x\\ x={a}^{xy}\\ \mathrm{log}x=\mathrm{log}{a}^{xy}\\ \mathrm{log}x=xy\mathrm{log}a\\ y=\frac{\mathrm{log}x}{x\mathrm{log}a}\end{array}$
$\begin{array}{}y\text{'}=\frac{\frac{1}{x}·x\mathrm{log}a-\left(\mathrm{log}x\right)\left(\mathrm{log}a\right)}{{x}^{2}{\left(\mathrm{log}a\right)}^{2}}\\ =\frac{\mathrm{log}a\left(1-\mathrm{log}x\right)}{{x}^{2}{\left(\mathrm{log}a\right)}^{2}}\\ =\frac{1-\mathrm{log}x}{{x}^{2}\mathrm{log}a}\\ \mathrm{log}x=1\\ x=e\\ x0\\ x>e\\ y\text{'}<0\end{array}$

x が e のとき最大値をもつことがわかる。

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

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()


$./sample15.py d ⎛ x⎞ ──⎝x ⎠ dx x x ⋅(log(x) + 1) -1 ℯ LambertW(zoo) ℯ 2 d ⎛ x⎞ ───⎝x ⎠ 2 dx x ⎛ 2 1⎞ x ⋅⎜(log(x) + 1) + ─⎟ ⎝ x⎠ <class 'NotImplementedError'> multiple generators [x, log(x)] No algorithms are implemented to solve equation (log(x) + 1)**2 + 1/x x lim x x─→0⁺ 1 x lim x x─→∞ ∞$


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,
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])
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();