学習環境
- Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
- Windows 10 Pro (OS)
- 数式入力ソフト(TeX, MathML): MathType
- MathML対応ブラウザ: Firefox、Safari
- MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax
- 参考書籍
数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第18章(曲線の性質、最大・最小 - 微分法の応用)、18.3(曲線の凹凸、曲線をえがくこと)、グラフをえがくこと、問36.を取り組んでみる。
コード(Emacs)
Python 3
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from sympy import pprint, symbols, exp, log, Derivative, solve
print('36.')
x = symbols('x')
fs = [3 * x ** 4 + 4 * x ** 3,
-x - 4 / x,
exp(-x ** 2),
x * exp(-x),
x ** 2 / (x ** 2 + 1),
log(x) / x]
for i, f in enumerate(fs, 1):
print('({0})'.format(i))
pprint(f)
pprint(solve(f, x))
print()
for n in range(1, 3):
d = Derivative(f, x, n)
fn = d.doit()
pprint(d)
pprint(fn)
pprint(solve(fn, x))
print()
print()
入出力結果(Terminal, IPython)
=$ ./sample36.py
36.
(1)
4 3
3⋅x + 4⋅x
[-4/3, 0]
d ⎛ 4 3⎞
──⎝3⋅x + 4⋅x ⎠
dx
3 2
12⋅x + 12⋅x
[-1, 0]
2
d ⎛ 4 3⎞
───⎝3⋅x + 4⋅x ⎠
2
dx
12⋅x⋅(3⋅x + 2)
[-2/3, 0]
(2)
4
-x - ─
x
[-2⋅ⅈ, 2⋅ⅈ]
d ⎛ 4⎞
──⎜-x - ─⎟
dx⎝ x⎠
4
-1 + ──
2
x
[-2, 2]
2
d ⎛ 4⎞
───⎜-x - ─⎟
2⎝ x⎠
dx
-8
───
3
x
[]
(3)
2
-x
ℯ
[]
⎛ 2⎞
d ⎜ -x ⎟
──⎝ℯ ⎠
dx
2
-x
-2⋅x⋅ℯ
[0]
2⎛ 2⎞
d ⎜ -x ⎟
───⎝ℯ ⎠
2
dx
2
⎛ 2 ⎞ -x
2⋅⎝2⋅x - 1⎠⋅ℯ
⎡-√2 √2⎤
⎢────, ──⎥
⎣ 2 2 ⎦
(4)
-x
x⋅ℯ
[0]
d ⎛ -x⎞
──⎝x⋅ℯ ⎠
dx
-x -x
- x⋅ℯ + ℯ
[1]
2
d ⎛ -x⎞
───⎝x⋅ℯ ⎠
2
dx
-x
(x - 2)⋅ℯ
[2]
(5)
2
x
──────
2
x + 1
[0]
⎛ 2 ⎞
d ⎜ x ⎟
──⎜──────⎟
dx⎜ 2 ⎟
⎝x + 1⎠
3
2⋅x 2⋅x
- ───────── + ──────
2 2
⎛ 2 ⎞ x + 1
⎝x + 1⎠
[0]
2⎛ 2 ⎞
d ⎜ x ⎟
───⎜──────⎟
2⎜ 2 ⎟
dx ⎝x + 1⎠
⎛ 4 2 ⎞
⎜ 4⋅x 5⋅x ⎟
2⋅⎜───────── - ────── + 1⎟
⎜ 2 2 ⎟
⎜⎛ 2 ⎞ x + 1 ⎟
⎝⎝x + 1⎠ ⎠
──────────────────────────
2
x + 1
⎡-√3 √3⎤
⎢────, ──⎥
⎣ 3 3 ⎦
(6)
log(x)
──────
x
[1]
d ⎛log(x)⎞
──⎜──────⎟
dx⎝ x ⎠
log(x) 1
- ────── + ──
2 2
x x
[ℯ]
2
d ⎛log(x)⎞
───⎜──────⎟
2⎝ x ⎠
dx
2⋅log(x) - 3
────────────
3
x
⎡ 3/2⎤
⎣ℯ ⎦
$
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="15"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="-0.05"> <label for="y2">y2 = </label> <input id="y2" type="number" value="0.4"> <br> <label for="dx0">dx0 = </label> <input id="dx0" type="number" min="0" value="0.1"> <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="sample36.js"></script>
JavaScript
let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
padding = 50,
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'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_dx0],
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.log(x) / x,
f1 = (x) => (1 - Math.log(x)) / x ** 2,
f2 = (x) => (-3 + 2 * Math.log(x)) / x ** 3,
g = (x0) => (x) => f1(x0) * (x - x0) + f(x0),
h = (x0) => (x) => f2(x0) * (x - x0) + f1(x0);
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);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
lines = [],
fns = [[f, 'red'], [f1, 'blue']],
fns1 = [],
fns2 = [[g, 'green'], [h, 'orange']];
fns.forEach((o) => {
let [fn, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = fn(x);
if (Math.abs(y) < Infinity) {
points.push([x, y, color]);
}
}
});
fns1.forEach((o) => {
let [fn, color] = o;
lines.push([x1, fn(x1), x2, fn(x2), color]);
});
fns2.forEach((o) => {
let [fn, color] = o;
for (let x = x1; x <= x2; x += dx0) {
let g = fn(x);
lines.push([x1, g(x1), x2, g(x2), color]);
}
});
let xscale = d3.scaleLinear()
.domain([x1, x2])
.range([padding, width - padding]);
let yscale = d3.scaleLinear()
.domain([y1, y2])
.range([height - padding, padding]);
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();
0 コメント:
コメントを投稿