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6章(曲線をえがくこと)、3(凸関数)、練習問題1-1、2、3.を取り組んでみる。
-
-
- 凹の区間は x < 3。
- 凸の区間は x > 3。
-
- 凹の区間。
- 凸の区間。
-
- 凹の区間。
- 凸の区間。
-
コード(Emacs)
Python 3
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from sympy import pprint, symbols, solve, Derivative, Limit, plot
x = symbols('x')
fs = [(x ** 2 + 2) / (x - 3),
(x - 3) / (x ** 2 + 1),
(x + 1) / (x ** 2 + 1)]
for i, f in enumerate(fs, 1):
print(f'{i}.')
d = Derivative(f, x, 2)
pprint(d)
f2 = d.doit()
pprint(f2)
pprint(solve(f2))
p = plot(f, show=False, legend=True)
p.save(f'sample1_{i}.svg')
入出力結果(Terminal, IPython)
$ ./sample1.py
1.
2⎛ 2 ⎞
d ⎜x + 2⎟
───⎜──────⎟
2⎝x - 3 ⎠
dx
⎛ 2 ⎞
⎜ 2⋅x x + 2 ⎟
2⋅⎜- ───── + 1 + ────────⎟
⎜ x - 3 2⎟
⎝ (x - 3) ⎠
──────────────────────────
x - 3
[]
2.
2
d ⎛x - 3 ⎞
───⎜──────⎟
2⎜ 2 ⎟
dx ⎝x + 1⎠
⎛ 2 ⎞
⎜4⋅x ⋅(x - 3) ⎟
2⋅⎜──────────── - 3⋅x + 3⎟
⎜ 2 ⎟
⎝ x + 1 ⎠
──────────────────────────
2
⎛ 2 ⎞
⎝x + 1⎠
⎡ ⎛ 1 √3⋅ⅈ⎞ 3 ___________ 10
⎢3 + ⎜- ─ - ────⎟⋅╲╱ 30 + 10⋅ⅈ + ──────────────────────────, 3 + ────────────
⎢ ⎝ 2 2 ⎠ ⎛ 1 √3⋅ⅈ⎞ 3 ___________ ⎛ 1 √3⋅ⅈ⎞
⎢ ⎜- ─ - ────⎟⋅╲╱ 30 + 10⋅ⅈ ⎜- ─ + ────⎟
⎣ ⎝ 2 2 ⎠ ⎝ 2 2 ⎠
10 ⎛ 1 √3⋅ⅈ⎞ 3 ___________ 10 3 ___________
────────────── + ⎜- ─ + ────⎟⋅╲╱ 30 + 10⋅ⅈ , 3 + ───────────── + ╲╱ 30 + 10⋅ⅈ
3 ___________ ⎝ 2 2 ⎠ 3 ___________
⋅╲╱ 30 + 10⋅ⅈ ╲╱ 30 + 10⋅ⅈ
⎤
⎥
⎥
⎥
⎦
3.
2
d ⎛x + 1 ⎞
───⎜──────⎟
2⎜ 2 ⎟
dx ⎝x + 1⎠
⎛ 2 ⎞
⎜4⋅x ⋅(x + 1) ⎟
2⋅⎜──────────── - 3⋅x - 1⎟
⎜ 2 ⎟
⎝ x + 1 ⎠
──────────────────────────
2
⎛ 2 ⎞
⎝x + 1⎠
[1, -2 - √3, -2 + √3]
$
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="-20"> <label for="x2">x2 = </label> <input id="x2" type="number" value="20"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="-20"> <label for="y2">y2 = </label> <input id="y2" type="number" value="20"> <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="sample1.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 f = (x) => (x ** 2 + 2) / (x - 3);
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 = [],
fns = [[f, '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);
[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