2018年10月23日火曜日

数学 - Python - JavaScript - 解析学 - 積分 - いくつかの計算練習 - スターリングの公式(定理の証明1-3、微分(合成関数、連鎖律、対数関数、累乗、分数関数))

1. $\begin{array}{}\phi \text{'}\left(x\right)\\ =\frac{d}{\mathrm{dx}}\left(\frac{1}{2}\mathrm{log}\frac{1+x}{1-x}-x\right)\\ =\frac{1}{2}·\frac{1-x}{1+x}·\frac{\left(1-x\right)-\left(1+x\right)\left(-1\right)}{{\left(1-x\right)}^{2}}-1\\ =\frac{1}{2}·\frac{1}{1+x}·\frac{2}{1-x}-1\\ =\frac{1}{1-{x}^{2}}-1\\ =\frac{1-\left(1-{x}^{2}\right)}{1-{x}^{2}}\\ =\frac{{x}^{2}}{1-{x}^{2}}\end{array}$

2. $\begin{array}{}\psi \text{'}\left(x\right)\\ =\frac{d}{\mathrm{dx}}\left(\phi \left(x\right)-\frac{{x}^{3}}{3\left(1-{x}^{2}\right)}\right)\\ =\frac{{x}^{2}}{1-{x}^{2}}-\frac{1}{3}·\frac{3{x}^{2}\left(1-{x}^{2}\right)-{x}^{3}\left(-2x\right)}{{\left(1-{x}^{2}\right)}^{2}}\\ =\frac{{x}^{2}}{1-{x}^{2}}-\frac{1}{3}·\frac{3{x}^{2}-{x}^{4}}{{\left(1-{x}^{2}\right)}^{2}}\\ =\frac{{x}^{2}·3\left(1-{x}^{2}\right)-3{x}^{2}+{x}^{4}}{3{\left(1-{x}^{2}\right)}^{2}}\\ =\frac{-2{x}^{4}}{3{\left(1-{x}^{2}\right)}^{2}}\end{array}$

3. $\begin{array}{}00\end{array}$

よって、

$\phi \left(x\right)>0\left(0

また、

$\begin{array}{}0

よって、

$\psi \left(x\right)<0\left(0

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Rational, log, Derivative, plot

print('1.')

x = symbols('x')
f = Rational(1, 2) * log((1 + x) / (1 - x)) - x
f1 = Derivative(f, x, 1).doit()
pprint(f1.simplify())

print('2.')
g = f - x ** 3 / (3 * (1 - x ** 2))
g1 = Derivative(g, x, 1).doit()

pprint(g1.factor())

print('3.')

p = plot(f, g, (x, 0, 1), ylim=(-10, 10), show=False, legend=True)
colors = ['red', 'green']

for i, color in enumerate(colors):
p[i].line_color = color
p.save('sample1.svg')


$./sample1.py 1. 2 -x ────── 2 x - 1 2. 4 -2⋅x ─────────────────── 2 2 3⋅(x - 1) ⋅(x + 1) 3.$


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="-2">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="2">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-2">
<label for="y2">y2 = </label>
<input id="y2" type="number" 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="sample1.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_n0 = document.querySelector('#n0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
p = (x) => pre0.textContent += x + '\n';

let f = (x) => 1 / 2 * Math.log((1 + x) / (1 - x)) - x,
fns = [[f, 'red'],
[(x) => f(x) - x ** 3 / (3 * (1 - x ** 2)), 'green']];

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 = [[0, y1, 0, y2, 'blue'],
[1, y1, 1, y2, 'brown']];

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

points.push([x, y, 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].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();