2018年1月9日火曜日

数学 - Python - JavaScript - 三角関数、グラフの描画、微分、導関数、二階微分、極値、変曲点、極限( @fmathsecond )ラジオ2さんのツイートより

ということで。極値点、変曲点、極限等を求めてグラフを描いてみた。

$\begin{array}{}f\left(x\right)=\frac{\mathrm{sin}x}{\sqrt{2+2\mathrm{cos}x}}=\frac{1}{\sqrt{2}}·\frac{\mathrm{sin}x}{\sqrt{1+\mathrm{cos}x}}\\ f\text{'}\left(x\right)=\frac{1}{\sqrt{2}}·\frac{\mathrm{cos}x\sqrt{1+\mathrm{cos}x}-\mathrm{sin}x·\frac{1}{2}{\left(1+\mathrm{cos}x\right)}^{-\frac{1}{2}}·\left(-\mathrm{sin}x\right)}{1+\mathrm{cos}x}\\ =\frac{1}{\sqrt{2}}\frac{\mathrm{cos}x\sqrt{1+\mathrm{cos}x}+{\mathrm{sin}}^{2}x·\frac{1}{2\sqrt{1+\mathrm{cos}x}}}{1+\mathrm{cos}x}\\ =\frac{2\mathrm{cos}x\left(1+\mathrm{cos}x\right)+{\mathrm{sin}}^{2}x}{2\sqrt{2}{\left(1+\mathrm{cos}x\right)}^{\frac{3}{2}}}\\ =\frac{2\mathrm{cos}x+2{\mathrm{cos}}^{2}x+{\mathrm{sin}}^{2}x}{2\sqrt{2}{\left(1+\mathrm{cos}x\right)}^{\frac{3}{2}}}\\ =\frac{{\mathrm{cos}}^{2}x+2\mathrm{cos}x+1}{2\sqrt{2}{\left(1+\mathrm{cos}x\right)}^{\frac{3}{2}}}\\ =\frac{{\left(\mathrm{cos}x+1\right)}^{2}}{2\sqrt{2}{\left(1+\mathrm{cos}x\right)}^{\frac{3}{2}}}\\ =\frac{\sqrt{\mathrm{cos}x+1}}{2\sqrt{2}}\\ f\text{'}\text{'}\left(x\right)=\frac{{\left(\mathrm{cos}x+1\right)}^{-\frac{1}{2}}\left(-\mathrm{sin}x\right)}{4\sqrt{2}}\\ =\frac{-\mathrm{sin}x}{4\sqrt{2}\sqrt{\mathrm{cos}x+1}}\end{array}$
$\begin{array}{}x\ne \pi +2n\pi \left(n\in \text{ℤ}\right)\\ f\text{'}\left(x\right)>0\\ 2n\pi \le x<\pi +2n\pi \\ f\text{'}\text{'}\left(x\right)\le 0\\ -\pi +2n\pi
$\begin{array}{}\underset{x\to \pi -0}{\mathrm{lim}}f\left(x\right)\\ =\frac{1}{\sqrt{2}}\underset{x\to \pi -0}{\mathrm{lim}}\frac{\sqrt{{\mathrm{sin}}^{2}x}}{\sqrt{1+\mathrm{cos}x}}\\ =\frac{1}{\sqrt{2}}\underset{x\to \pi -0}{\mathrm{lim}}\sqrt{\frac{1-{\mathrm{cos}}^{2}x}{1+\mathrm{cos}x}}\\ =\frac{1}{\sqrt{2}}\underset{x\to \pi -0}{\mathrm{lim}}\sqrt{1-\mathrm{cos}x}\\ =\frac{1}{\sqrt{2}}·\sqrt{2}\\ =1\end{array}$
$\begin{array}{}\underset{x\to \pi +0}{\mathrm{lim}}f\left(x\right)\\ =\frac{1}{\sqrt{2}}\underset{x\to \pi +0}{\mathrm{lim}}\frac{-\sqrt{{\mathrm{sin}}^{2}x}}{\sqrt{1+\mathrm{cos}x}}\\ =-1\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sin, cos, sqrt, Derivative, pi, Limit, plot

x = symbols('x')
f = sin(x) / sqrt(2 + 2 * cos(x))

for n in range(1, 3):
Dn = Derivative(f, x, n)
for t in [Dn, Dn.doit()]:
pprint(t)
print()
print()

for dir in ['+', '-']:
l = Limit(f, x, pi)
try:
for t in [l, l.doit()]:
pprint(t)
print()
print()
except Exception as err:
print(type(err), err)

p = plot(f, -1, 1, show=False, legend=True)
for i, color in enumerate(['red', 'green', 'blue']):
p[i].line_color = color

p.save('sample.svg')


$./sample.py d ⎛ sin(x) ⎞ ──⎜────────────────⎟ dx⎜ ______________⎟ ⎝╲╱ 2⋅cos(x) + 2 ⎠ 2 cos(x) sin (x) ──────────────── + ───────────────── ______________ 3/2 ╲╱ 2⋅cos(x) + 2 (2⋅cos(x) + 2) 2 d ⎛ sin(x) ⎞ ───⎜────────────────⎟ 2⎜ ______________⎟ dx ⎝╲╱ 2⋅cos(x) + 2 ⎠ ⎛ 2 ⎞ ⎜ 6⋅cos(x) 3⋅sin (x) ⎟ √2⋅⎜-4 + ────────── + ─────────────⎟⋅sin(x) ⎜ cos(x) + 1 2⎟ ⎝ (cos(x) + 1) ⎠ ─────────────────────────────────────────── ____________ 8⋅╲╱ cos(x) + 1 <class 'NotImplementedError'> <class 'NotImplementedError'>$


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="0.005">
<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="sample.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) => Math.sin(x) / Math.sqrt(2 + 2 * Math.cos(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 = [[x1, -1, x2, -1, 'red'],
[x1, 1, x2, 1, 'blue'],
[-Math.PI, y1, -Math.PI, y2, 'orange'],
[Math.PI, y1, Math.PI, y2, 'orange']],
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();