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)、18.2(関数の増減の判定およびその応用)、最大・最小問題、問29.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, Derivative, solve, sqrt, plot
x = symbols('x')
fs = [1 / 2 * x * (2 / (x - 1) + 2),
sqrt(x ** 2 + (2 / (x - 1) + 2) ** 2)]
for i, f in enumerate(fs, 1):
d = Derivative(f, x, 1)
pprint(d)
f1 = d.doit()
pprint(f1)
pprint(solve(f1, x))
p = plot(f, (x, 1.5, 5), show=False, legend=True)
p.save('sample29_{0}.svg'.format(i))
print()
入出力結果(Terminal, IPython)
$ ./sample29.py
d ⎛ ⎛ 2 ⎞⎞
──⎜0.5⋅x⋅⎜2 + ─────⎟⎟
dx⎝ ⎝ x - 1⎠⎠
1.0⋅x 1.0
- ──────── + 1.0 + ─────
2 x - 1
(x - 1)
[0.0, 2.0]
⎛ ___________________⎞
⎜ ╱ 2 ⎟
d ⎜ ╱ 2 ⎛ 2 ⎞ ⎟
──⎜ ╱ x + ⎜2 + ─────⎟ ⎟
dx⎝╲╱ ⎝ x - 1⎠ ⎠
⎛ 2 ⎞
2⋅⎜2 + ─────⎟
⎝ x - 1⎠
x - ─────────────
2
(x - 1)
────────────────────────
___________________
╱ 2
╱ 2 ⎛ 2 ⎞
╱ x + ⎜2 + ─────⎟
╲╱ ⎝ x - 1⎠
⎡ 2/3 2/3 ⎛ 1 √3⋅ⅈ⎞ 2/3 ⎛ 1 √3⋅ⅈ⎞⎤
⎢1 + 2 , 1 + 2 ⋅⎜- ─ - ────⎟, 1 + 2 ⋅⎜- ─ + ────⎟⎥
⎣ ⎝ 2 2 ⎠ ⎝ 2 2 ⎠⎦
$
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="1.1"> <label for="x2">x2 = </label> <input id="x2" type="number" value="5"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="2"> <label for="y2">y2 = </label> <input id="y2" type="number" value="10"> <br> <label for="dx0">dx0 = </label> <input id="dx0" type="number" min="0" value="0.1"> <label for="x0">x = </label> <input id="x0" type="number" min="1" 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="sample29.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_x0 = document.querySelector('#x0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_dx0, input_x0],
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) => 1 / 2 * x * ( 2 / (x - 1) + 2),
f1 = (x) => x * (x - 2) / (x - 1) ** 2,
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),
x0 = parseFloat(input_x0.value);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
lines = [],
fns = [[f, 'green']],
fns1 = [],
fns2 = [[g, 'blue']];
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);
p(fns.join('\n'));
p(fns1.join('\n'));
};
inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();
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