## 2017年10月31日火曜日

### 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 - 指数関数(累乗、第n次導関数、帰納法)

1. n = 1のとき。

$\begin{array}{l}\frac{d}{dx}\left(x{e}^{x}\right)={e}^{x}+x{e}^{x}\\ =\left(x+1\right){e}^{x}\end{array}$

よって、n = 1の場合は成り立つ。

$\begin{array}{l}\frac{{d}^{n}}{{\left(dx\right)}^{n}}\left(x{e}^{x}\right)\\ =\frac{d}{dx}\left(\frac{{d}^{n-1}}{{\left(dx\right)}^{n-1}}\left(x{e}^{x}\right)\right)\\ =\frac{d}{dx}\left(\left(x+n-1\right){e}^{x}\right)\\ ={e}^{x}+\left(x+n-1\right){e}^{x}\\ =\left(x+n\right){e}^{x}\end{array}$

ゆえに、帰納法により任意の整数 n ≥ 1に対して成り立つ。(証明終)

2. n = 1のとき。

$\begin{array}{l}\frac{d}{dx}\left(x{e}^{-x}\right)\\ ={e}^{-x}+x{e}^{-x}\left(-1\right)\\ =\left(-1\right)\left(x-1\right){e}^{-x}\end{array}$

よって、n = 1 の場合は成り立つ。

$\begin{array}{l}\frac{{d}^{n}}{{\left(dx\right)}^{n}}\left(x{e}^{-x}\right)\\ =\frac{d}{dx}\left(\frac{{d}^{n-1}}{{\left(dx\right)}^{n-1}}\left(x{e}^{-x}\right)\right)\\ =\frac{d}{dx}\left({\left(-1\right)}^{n-1}\left(x-\left(n-1\right)\right){e}^{-x}\right)\\ ={\left(-1\right)}^{n-1}\left({e}^{-x}+\left(x-\left(n-1\right)\right){e}^{-x}\left(-1\right)\right)\\ ={\left(-1\right)}^{n}\left(-1+x-n+1\right){e}^{-x}\\ ={\left(-1\right)}^{n}\left(x-n\right){e}^{-x}\end{array}$

よって帰納法より、任意の整数 n ≥ 1 に対し成り立つ。(証明終)

3. n = 0のとき。

$\begin{array}{l}\frac{{d}^{0+1}}{{\left(dx\right)}^{0+1}}\left({x}^{0}\mathrm{log}x\right)\\ =\frac{d}{dx}\mathrm{log}x\\ =\frac{1}{x}\\ =\frac{0!}{x!}\end{array}$

第1次導関数について成り立つ。

$\begin{array}{l}\frac{{d}^{n+1}}{{\left(dx\right)}^{n+1}}\left({x}^{n}\mathrm{log}x\right)\\ =\frac{{d}^{n}}{{\left(dx\right)}^{n}}\left(\frac{d}{dx}\left({x}^{n}\mathrm{log}x\right)\right)\\ =\frac{{d}^{n}}{{\left(dx\right)}^{n}}\left(n{x}^{n-1}\mathrm{log}x+{x}^{n}\frac{1}{x}\right)\\ =\frac{{d}^{n}}{{\left(dx\right)}^{n}}\left(n{x}^{n-1}\mathrm{log}x+{x}^{n-1}\right)\\ =\left(n\frac{{d}^{n}}{{\left(dx\right)}^{n}}\left({x}^{n-1}\mathrm{log}x\right)+\frac{{d}^{n}}{{\left(dx\right)}^{n}}{x}^{n-1}\right)\\ =n·\frac{\left(n-1\right)!}{x}+0\\ =\frac{n!}{x}\end{array}$

よって帰納法より、任意の整数n ≥ 0 に対して成り立つ。(証明終)

コード(Emacs)

Python 3

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

print('6.')
x = symbols('x')
n = symbols('n', integer=True)
fs = [(x * exp(x), (x + n) * exp(x)),
(x * exp(-x), (-1) ** n * (x - n) * exp(-x)),
(x ** (n - 1) * log(x), factorial(n - 1) / x)]

for i, (f, fn) in enumerate(fs):
print(f'({chr(ord("a") + i)})')
for t in [f, fn]:
pprint(t)
print()
for n0 in range(1, 11):
print(Derivative(f.subs({n: n0}), x,
n0).doit().factor() == fn.subs({n: n0}))
print()

p = plot(*map(lambda x: x[0].subs({n: 1}), fs), show=False, legend=True)
for i, color in enumerate(['red', 'green', 'blue']):
p[i].line_color = color

p.save('sample6.svg')


$./sample6.py 6. (a) x x⋅ℯ x (n + x)⋅ℯ True True True True True True True True True True (b) -x x⋅ℯ n -x (-1) ⋅(-n + x)⋅ℯ True True True True True True True True True True (c) n - 1 x ⋅log(x) (n - 1)! ──────── x True True True True True True True True True True$


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="0.001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" step="0.1" value="0.1">

<label for="n0">n0 = </label>
<input id="n0" type="number" min="0" step="1" min="1" value="1">

<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="sample6.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_n0 = document.querySelector('#n0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_dx0, input_n0],
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),
n0 = parseInt(input_n0.value, 10);

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

let points = [],
lines = [],
f = (x) => (x + (n0 - 1)) * Math.exp(x),
f1 = (x) => (x + n0) * Math.exp(x),
g = (x0) => (x) => f1(x0) * (x - x0) + f(x0),
fns = [[f, 'red']],
fns1 = [],
fns2 = [[g, 'green']];

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();