## 2017年7月3日月曜日

### 数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 平均値の定理 - 増加・減少関数(距離の和、角、余弦)

1. $\begin{array}{l}f\left(x\right)=\sqrt{{x}^{2}+{p}^{2}}+\sqrt{{\left(a-x\right)}^{2}+{q}^{2}}\\ f\text{'}\left(x\right)=\frac{1}{2}{\left({x}^{2}+{p}^{2}\right)}^{-\frac{1}{2}}2x-\frac{1}{2}{\left({\left(a-x\right)}^{2}+{q}^{2}\right)}^{-\frac{1}{2}}2\left(a-x\right)\\ ={\left({x}^{2}+{p}^{2}\right)}^{-\frac{1}{2}}x-{\left({\left(a-x\right)}^{2}+{q}^{2}\right)}^{-\frac{1}{2}}\left(a-x\right)\\ {\left({x}^{2}+{p}^{2}\right)}^{-\frac{1}{2}}x-{\left({\left(a-x\right)}^{2}+{q}^{2}\right)}^{-\frac{1}{2}}\left(a-x\right)=0\\ \frac{x}{\sqrt{{x}^{2}+{p}^{2}}}-\frac{a-x}{\sqrt{{\left(a-x\right)}^{2}+{q}^{2}}}=0\\ \mathrm{cos}{\theta }_{1}-\mathrm{cos}{\theta }_{2}=0\\ \mathrm{cos}{\theta }_{1}=\mathrm{cos}{\theta }_{2}\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, sqrt, Derivative, solve

a, p, q, x = symbols('a p q x', nonnegative=True)

f = sqrt(x ** 2 + p ** 2) + sqrt((a - x) ** 2 + q ** 2)
d = Derivative(f, x, 1)
pprint(d)
f1 = d.doit()
pprint(f1)
s = solve(f1, x)
pprint(s)
for x0 in s:
pprint(x0)
print()
y1 = (p / x0).factor()
y2 = (q / (a - x0)).factor()
pprint(y1)
print()
pprint(y2)
print(y1 == y2)

$./sample22.py ⎛ _________ _______________⎞ ∂ ⎜ ╱ 2 2 ╱ 2 2 ⎟ ──⎝╲╱ p + x + ╲╱ q + (a - x) ⎠ ∂x x -a + x ──────────── + ────────────────── _________ _______________ ╱ 2 2 ╱ 2 2 ╲╱ p + x ╲╱ q + (a - x) ⎡ a⋅p a⋅p ⎤ ⎢─────, ─────⎥ ⎣p - q p + q⎦ a⋅p ───── p - q p - q ───── a -(p - q) ───────── a False a⋅p ───── p + q p + q ───── a p + q ───── a True$

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="0.0001" 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="25">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="25">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" value="0.5">
<label for="p0">p = </label>
<input id="p0" type="number" min="0"  value="5">
<label for="q0">q = </label>
<input id="q0" type="number" min="0"  value="10">
<label for="a0">a = </label>
<input id="a0" type="number" min="0"  value="15">
<label for="x0">x = </label>
<input id="x0" type="number" min="0"  value="2">

<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'),
input_dx0 = document.querySelector('#dx0'),
input_p0 = document.querySelector('#p0'),
input_q0 = document.querySelector('#q0'),
input_a0 = document.querySelector('#a0'),
input_x0 = document.querySelector('#x0'),

inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_dx0, input_p0, input_q0, input_a0, input_x0],
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 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),
p0 = parseFloat(input_p0.value),
q0 = parseFloat(input_q0.value),
a0 = parseFloat(input_a0.value),
x0 = parseFloat(input_x0.value);

if (r === 0 || dx === 0 || x1 > x2 || y1 > y2 || x0 > a0) {
return;
}

let points = [],
x3 = a0 * p0 / (p0 + q0),
f =
(x) => Math.sqrt(x ** 2 + p0 ** 2) + Math.sqrt((a0 - x) ** 2 + q0 ** 2),
f1 =
(x) => x / Math.sqrt(x ** 2  + p0 ** 2) -
(a0 - x) / Math.sqrt((a0 - x) ** 2 + q0 ** 2),
g = (x0) => (x) => f1(x0) * (x - x0) + f(x0),
lines = [[0, 0, 0, p0, 'green'],
[a0, 0, a0, q0, 'green'],
[0, p0, x0, 0, 'green'],
[x0, 0, a0, q0, 'green'],
[x3, y1, x3, y2, 'red']],
fns = [[f, 'blue']],
fns1 = [],
fns2 = [[g, 'orange']];

fns
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = f(x);

if (Math.abs(y) < Infinity) {
points.push([x, y, 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();