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 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7章(逆関数)、4(逆正接関数)、練習問題17、18、19、20、21.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, asin, acos, atan, Derivative, sqrt, Rational
x = symbols('x')
fs = [(asin(x), 1 / sqrt(2)),
(acos(x), 1 / sqrt(2)),
(atan(2 * x), sqrt(3) / 2),
(atan(x), -1),
(asin(x), -Rational(1, 2))]
for i, (f, x0) in enumerate(fs, 17):
print(f'{i}.')
D = Derivative(f, x, 1)
f1 = D.doit()
g = f1.subs({x: x0}) * (x - x0) + f.subs({x: x0})
for t in [f, D, f1, g]:
pprint(t)
print()
print()
入出力結果(Terminal, Jupyter(IPython))
$ ./sample17.py
17.
asin(x)
d
──(asin(x))
dx
1
─────────────
__________
╱ 2
╲╱ - x + 1
⎛ √2⎞ π
√2⋅⎜x - ──⎟ + ─
⎝ 2 ⎠ 4
18.
acos(x)
d
──(acos(x))
dx
-1
─────────────
__________
╱ 2
╲╱ - x + 1
⎛ √2⎞ π
- √2⋅⎜x - ──⎟ + ─
⎝ 2 ⎠ 4
19.
atan(2⋅x)
d
──(atan(2⋅x))
dx
2
────────
2
4⋅x + 1
x √3 π
─ - ── + ─
2 4 3
20.
atan(x)
d
──(atan(x))
dx
1
──────
2
x + 1
x π 1
─ - ─ + ─
2 4 2
21.
asin(x)
d
──(asin(x))
dx
1
─────────────
__________
╱ 2
╲╱ - x + 1
2⋅√3⋅(x + 1/2) π
────────────── - ─
3 6
$
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="0.01"> <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"> <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="sample17.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'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
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 f1 = (x) => Math.asin(x),
g1 = (x) => Math.sqrt(2) * x - 1 + Math.PI / 4,
f2 = (x) => Math.acos(x),
g2 = (x) => -Math.sqrt(2) * x + 1 + Math.PI / 4,
f3 = (x) => Math.atan(2 * x),
g3 = (x) => 1 / 2 * x - Math.sqrt(3) / 4 + Math.PI / 3,
f4 = (x) => Math.atan(x),
g4 = (x) => 1 / 2 * x + 1 / 2 - Math.PI / 4,
f5 = (x) => Math.asin(x),
g5 = (x) => 2 / Math.sqrt(3) * x + 1 / Math.sqrt(3) - Math.PI / 6;
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);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
lines = [[1 / Math.sqrt(2), y1, 1 / Math.sqrt(2), y2, 'red'],
[Math.sqrt(3) / 2, y1, Math.sqrt(3) / 2, y2, 'blue'],
[-1, y1, -1, y2, 'orange'],
[-1 / 2, y1, -1 / 2, y2, 'skyblue']],
fns = [[f1, 'red'],
[g1, 'red'],
[f2, 'green'],
[g2, 'green'],
[f3, 'blue'],
[g3, 'blue'],
[f4, 'orange'],
[g4, 'orange'],
[f5, 'skyblue'],
[g5, 'skyblue']],
fns1 = [],
fns2 = [];
fns
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = f(x);
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();
0 コメント:
コメントを投稿