## 2017年12月1日金曜日

### 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(2乗、2乗乗、符号、導関数、二階微分、極値、変曲点、関数の凹凸、グラフの描画)

1. $\begin{array}{}f\text{'}\left(x\right)\\ =2x{e}^{-{x}^{2}}+{x}^{2}{e}^{-{x}^{2}}\left(-2x\right)\\ =2x{e}^{-{x}^{2}}-2{x}^{3}{e}^{-{x}^{2}}\\ =2x{e}^{-{x}^{2}}\left(1-{x}^{2}\right)\end{array}$
$\begin{array}{}f\text{'}\text{'}\left(x\right)\\ =2{e}^{-{x}^{2}}+2x{e}^{-{x}^{2}}\left(-2x\right)-6{x}^{2}{e}^{-{x}^{2}}-2{x}^{3}{e}^{-{x}^{2}}\left(-2x\right)\\ =2{e}^{-{x}^{2}}-4{x}^{2}{e}^{-{x}^{2}}-6{x}^{2}{e}^{-{x}^{2}}+4{x}^{4}{e}^{-{x}^{2}}\\ =2{e}^{-{x}^{2}}-10{x}^{2}{e}^{-{x}^{2}}+4{x}^{4}{e}^{-{x}^{2}}\\ =2{e}^{-{x}^{2}}\left(2{x}^{4}-5{x}^{2}+1\right)\end{array}$
$\begin{array}{}f\text{'}\left(x\right)=0\\ x=±1\end{array}$
$\begin{array}{}f\text{'}\text{'}\left(x\right)=0\\ 2{x}^{4}-5{x}^{2}+1=0\\ {x}^{2}=\frac{5±\sqrt{25-8}}{4}\\ =\frac{5±\sqrt{17}}{4}\\ x=±\frac{5±\sqrt{17}}{4}\end{array}$
$\begin{array}{}f\left(0\right)=0\\ f\left(±1\right)={e}^{-1}\\ \underset{x\to ±\infty }{\mathrm{lim}}f\left(x\right)=0\end{array}$

グラフの描画。

コード(Emacs)

Python 3

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

x = symbols('x')
f = x ** 2 * exp(-x ** 2)
f1 = Derivative(f, x, 1).doit()
f2 = Derivative(f, x, 2).doit()

for g in [f, f1, f2]:
for t in [g, solve(g)]:
pprint(t)
print()
print()

p = plot(f, exp(-1), show=False, legend=True)

for i, color in enumerate(['red', 'green']):
p[i].line_color = color

p.save('sample4.svg')


$./sample4.py 2 2 -x x ⋅ℯ [0] 2 2 3 -x -x - 2⋅x ⋅ℯ + 2⋅x⋅ℯ [-1, 0, 1] 2 ⎛ 4 2 ⎞ -x 2⋅⎝2⋅x - 5⋅x + 1⎠⋅ℯ ⎡ ___________ ___________ _________ _________⎤ ⎢ ╱ √17 5 ╱ √17 5 ╱ √17 5 ╱ √17 5 ⎥ ⎢- ╱ - ─── + ─ , ╱ - ─── + ─ , - ╱ ─── + ─ , ╱ ─── + ─ ⎥ ⎣ ╲╱ 4 4 ╲╱ 4 4 ╲╱ 4 4 ╲╱ 4 4 ⎦$


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">

<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="sample4.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) => x ** 2 * Math.exp(-(x ** 2));

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 = [[-1, y1, -1, y2, 'red'],
[1, y1, 1, y2, 'red'],
[-(5 + Math.sqrt(17)) / 4, y1, -(5 + Math.sqrt(17)) / 4, y2, 'blue'],
[(5 + Math.sqrt(17)) / 4, y1, (5 + Math.sqrt(17)) / 4, y2, 'blue'],
[-(5 - Math.sqrt(17)) / 4, y1, -(5 - Math.sqrt(17)) / 4, y2, 'blue'],
[(5 - Math.sqrt(17)) / 4, y1, (5 - Math.sqrt(17)) / 4, y2, 'blue'],
[x1, Math.exp(-1), x2, Math.exp(-1), '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();