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
オイラーの贈物―人類の至宝eiπ=-1を学ぶ (吉田 武(著)、東海大学出版会)の第Ⅰ部(基礎理論(Basic Theory))、2章(方程式と関数)、2.6(関数の最大値・最小値)、2.1.6(接線の傾きと極値)、問題3.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, solve, sqrt
x, a, b, m, n = symbols('x a b m n')
f = m * x + n
fs = [sqrt(a * x + b),
1 / (a * x + b)]
for i, f0 in enumerate(fs, 1):
print(f'[{i}]')
pprint(f0)
ms = solve(f - f0, m)
for m0 in ms:
pprint(m0)
pprint(m0.expand())
pprint(m0.factor())
print()
入出力結果(Terminal, IPython)
$ ./sample3.py
[1]
_________
╲╱ a⋅x + b
_________
-n + ╲╱ a⋅x + b
────────────────
x
_________
n ╲╱ a⋅x + b
- ─ + ───────────
x x
⎛ _________⎞
-⎝n - ╲╱ a⋅x + b ⎠
───────────────────
x
[2]
1
───────
a⋅x + b
-a⋅n⋅x - b⋅n + 1
────────────────
x⋅(a⋅x + b)
a⋅n⋅x b⋅n 1
- ────────── - ────────── + ──────────
2 2 2
a⋅x + b⋅x a⋅x + b⋅x a⋅x + b⋅x
-(a⋅n⋅x + b⋅n - 1)
───────────────────
x⋅(a⋅x + b)
$
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="-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">a = </label> <input id="dx0" type="number" min="0" value="0.1"> <label for="a0">a = </label> <input id="a0" type="number" min="0" value="2"> <label for="b0">b = </label> <input id="b0" type="number" min="0" value="3"> <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="sample3.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'),
input_a0 = document.querySelector('#a0'),
input_b0 = document.querySelector('#b0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_dx0, input_a0, input_b0],
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 / (x ** 2 + 25 ** 2) ** (3 / 2),
f1 = (x) =>
((x ** 2 + 25 ** 2) ** (3 / 2) -
x * 3 / 2 * (x ** 2 + 25 ** 2) ** (1 / 2) * 2 * x) /
(x ** 2 + 25 ** 2) ** 3,
g = (x0) => (x) => f1(x0) * (x - x0) + f(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),
a0 = parseFloat(input_a0.value),
b0 = parseFloat(input_b0.value);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
lines = [],
f = (x) => Math.sqrt(a0 * x + b0),
mf = (x) => a0 / (2 * Math.sqrt(a0 * x + b0)),
g = (x) => 1 / (a0 * x + b0),
mg = (x) => - a0 / (a0 * x + b0) ** 2,
hf = (x0) => (x) => mf(x0) * (x - x0) + f(x0),
hg = (x0) => (x) => mg(x0) * (x - x0) + g(x0),
fns = [[f, 'red'],
[g, 'green']],
fns1 = [],
fns2 = [[hf, 'blue'],
[hg, 'orange']];
fns
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = f(x);
if (Math.abs(y) < Infinity) {
points.push([x, y, 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])
.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();
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