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
解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第2部(微分と基本的な関数)、第5章(平均値の定理)、3(増加・減少関数)、補充問題20.を取り組んでみる。
直円柱の高さをh、両端の円の半径をrとする。
コード(Emacs)
Python 3
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from sympy import pprint, symbols, pi, Pow, Derivative, solve
V, r = symbols('V r', positive=True)
h = (3 * V - 4 * pi * Pow(r, 3)) / (3 * pi * Pow(r, 2))
f = 2 * pi * r * h + 2 * 4 * pi * Pow(r, 2)
d = Derivative(f, r, 1)
f1 = d.doit()
rs = solve(f1, r)
pprint(rs)
for r0 in rs:
pprint(r0)
pprint(h.subs({r: r0}))
入出力結果(Terminal, IPython)
$ ./sample20.py
⎡ 2/3 3 ___ 3 ___⎤
⎢2 ⋅╲╱ 3 ⋅╲╱ V ⎥
⎢────────────────⎥
⎢ 3 ___ ⎥
⎣ 4⋅╲╱ π ⎦
2/3 3 ___ 3 ___
2 ⋅╲╱ 3 ⋅╲╱ V
────────────────
3 ___
4⋅╲╱ π
2/3 3 ___ 3 ___
2 ⋅╲╱ 3 ⋅╲╱ V
────────────────
3 ___
╲╱ π
$
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="25"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="0"> <label for="y2">y2 = </label> <input id="y2" type="number" value="25"> <br> <label for="v0">v0 = </label> <input id="v0" type="number" min="0" step="1" value="1000"> <label for="r1">r1 = </label> <input id="r1" type="number" min="0" step="0.1" 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="sample20.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_v0 = document.querySelector('#v0'),
input_r1 = document.querySelector('#r1'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_v0, input_r1],
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.sqrt(x ** 2 + 64 / 9 * x ** 2 / (2 * x - 9) ** 2),
f1 = (x) =>
1 / 2 * (x ** 2 + 64 / 9 * x ** 2 / (2 * x - 9 ) ** 2) ** (-1 / 2) *
(2 * x + 64 / 9 *
(2 * x * (2 * x - 9) ** 2 - x ** 2 * 2 * (2 * x - 9) * 2) /
(2 * x - 9) ** 4),
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),
v0 = parseFloat(input_v0.value),
r1 = parseFloat(input_r1.value);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
h0 = (v0 - 4 / 3 * Math.PI * r1 ** 3) / (Math.PI * r1 ** 2),
x3 = 2 * r1,
x4 = 1 / 2 * (3 * v0 / (2 * Math.PI)) ** (1 / 3),
y3 = r1 + h0,
lines = [[0, r1, 0, y3, 'green'],
[x3, r1, x3, y3, 'green'],
[r1, y1, r1, y2, 'red'],
[x4, y1, x4, y2, 'blue']],
f = (x) => Math.sqrt(r1 ** 2 - (x - r1) ** 2) + y3,
g = (x) => -Math.sqrt(r1 ** 2 - (x - r1) ** 2) + r1,
fns = [[f, 'green'],
[g, 'green']],
fns1 = [],
fns2 = [];
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);
p(fns.join('\n'));
};
inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();
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