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, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
- 参考書籍
解析入門〈2〉(松坂 和夫(著)、岩波書店)の第8章(積分の計算)、8.2(定積分の計算)、問題4.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, Integral
print('4.')
x, a, b = symbols('x a b')
a0, a1, a2, a3 = symbols('a0 a1 a2 a3', integer=True, nonnegative=True)
f = a0 + a1 * x + a2 * x ** 2 + a3 * x ** 3
pprint(f)
I = Integral(f, (x, a, b))
pprint(I)
g = I.doit()
pprint(g)
h = (b - a) / 6 * (f.subs({x: a}) + 4 *
f.subs({x: (a + b) / 2}) + f.subs({x: b}))
pprint(h)
print(g.expand() == h.expand())
入出力結果(Terminal, IPython)
$ ./sample4.py
4.
2 3
a₀ + a₁⋅x + a₂⋅x + a₃⋅x
b
⌠
⎮ ⎛ 2 3⎞
⎮ ⎝a₀ + a₁⋅x + a₂⋅x + a₃⋅x ⎠ dx
⌡
a
4 3 2 2 3 4
a ⋅a₃ a ⋅a₂ a ⋅a₁ a₁⋅b a₂⋅b a₃⋅b
- ───── - ───── - ───── - a⋅a₀ + a₀⋅b + ───── + ───── + ─────
4 3 2 2 3 4
⎛
⎛ a b⎞ ⎜ 3 2 ⎛a b⎞ 2 ⎛a
⎜- ─ + ─⎟⋅⎜a ⋅a₃ + a ⋅a₂ + a⋅a₁ + 6⋅a₀ + a₁⋅b + 4⋅a₁⋅⎜─ + ─⎟ + a₂⋅b + 4⋅a₂⋅⎜─
⎝ 6 6⎠ ⎝ ⎝2 2⎠ ⎝2
2 3⎞
b⎞ 3 ⎛a b⎞ ⎟
+ ─⎟ + a₃⋅b + 4⋅a₃⋅⎜─ + ─⎟ ⎟
2⎠ ⎝2 2⎠ ⎠
True
$
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="a01">a = </label> <input id="a01" type="number" value="-1"> <label for="b01">b = </label> <input id="b01" type="number" value="1"> <label for="a0">a0 = </label> <input id="a0" type="number" value="1"> <label for="a1">a1 = </label> <input id="a1" type="number" value="1"> <label for="a2">a2 = </label> <input id="a2" type="number" value="1"> <label for="a3">a3 = </label> <input id="a3" type="number" value="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="sample4.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_a01 = document.querySelector('#a01'),
input_b01 = document.querySelector('#b01'),
input_a0 = document.querySelector('#a0'),
input_a1 = document.querySelector('#a1'),
input_a2 = document.querySelector('#a2'),
input_a3 = document.querySelector('#a3'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_a0, input_a1, input_a2, input_a3],
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 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),
a = parseFloat(input_a01.value),
b = parseFloat(input_b01.value),
a0 = parseFloat(input_a0.value),
a1 = parseFloat(input_a1.value),
a2 = parseFloat(input_a2.value),
a3 = parseFloat(input_a3.value);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
f = (x) => a0 + a1 * x + a2 * x ** 2 + a3 * x ** 3,
g = (x) => (b - a) / 6 * (f(a) + 4 * f((a + b) / 2) + f(b)),
lines = [[a, y1, a, y2, 'red'],
[b, y1, b, y2, 'red']],
fns = [[f, 'green'],
[g, '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]);
}
}
});
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('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.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.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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