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〈1〉(松坂 和夫(著)、岩波書店)の第5章(各種の初等関数)、5.4(三角関数(続き)、逆三角関数)、問題10.を取り組んでみる。
コード(Emacs)
Python 3
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from sympy import Symbol, symbols, sin, cos, factorial, summation, solve, pprint
import random
x = Symbol('x', positive=True)
n, i = symbols('n i', integer=True, positive=True)
expr1 = summation((-1) ** (i - 1) * x ** (2 * i - 1) / factorial(2 * i - 1),
(i, 1, n))
expr2 = summation((-1) ** (i - 1) * x ** (2 * i - 2) / factorial(2 * i - 2),
(i, 1, n))
pprint(expr1)
pprint(expr2)
try:
print(expr1.subs({n: 2 * n}) < sin(x) < expr1.subs({n: 2 * n - 1}))
print(expr2.subs({n: 2 * n}) < cos(x) < expr2.subs({n: 2 * n - 1}))
except Exception as err:
print(type(err), err)
入出力結果(Terminal, IPython)
$ ./sample10.py
⎛ │ 2 ⎞
n + 1 2⋅n + 2 ┌─ ⎜ 1 │ -x ⎟
(-1) ⋅x ⋅ ├─ ⎜ │ ────⎟
1╵ 2 ⎝n + 1, n + 3/2 │ 4 ⎠
─────────────────────────────────────────────── + sin(x)
x⋅(2⋅n + 1)!
⎛ │ 2 ⎞
n + 1 2⋅n + 2 ┌─ ⎜ 1 │ -x ⎟
(-1) ⋅x ⋅ ├─ ⎜ │ ────⎟
1╵ 2 ⎝n + 1/2, n + 1 │ 4 ⎠
─────────────────────────────────────────────── + cos(x)
2
x ⋅(2⋅n)!
<class 'TypeError'> cannot determine truth value of Relational
$
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="0.01"> <label for="x2">x2 = </label> <input id="x2" type="number" value="5"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="-1.5"> <label for="y2">y2 = </label> <input id="y2" type="number" value="1.5"> <br> <label for="n0">n = </label> <input id="n0" type="number" min="1" step="1" value="2"> <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="sample10.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_n = document.querySelector('#n0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_n],
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 factorial = (n) => range(1, n + 1).reduce((x, y) => x * y, 1);
let f = (n, x) =>
range(1, n + 1)
.map((i) => (-1) ** (i - 1) * x ** (2 * i - 1) / factorial(2 * i - 1))
.reduce((x, y) => x + y, 0),
g = (n, x) =>
range(1, n + 1)
.map((i) => (-1) ** (i - 1) * x ** (2 * i - 2) / factorial(2 * i - 2))
.reduce((x, y) => x + y, 0);
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),
n = parseInt(input_n.value, 10);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
f0 = (x) => f(2 * n, x),
f1 = (x) => f(2 * n - 1, x),
g0 = (x) => g(2 * n, x),
g1 = (x) => g(2 * n - 1, x);
for (let x = x1; x <= x2; x += dx) {
let y = f0(x);
if (Math.abs(y) < Infinity) {
points.push([x, y]);
}
}
let t1 = points.length;
for (let x = x1; x <= x2; x += dx) {
let y = f1(x);
if (Math.abs(y) < Infinity) {
points.push([x, y]);
}
}
let t2 = points.length;
for (let x = x1; x <= x2; x += dx) {
let y = g0(x);
if (Math.abs(y) < Infinity) {
points.push([x, y]);
}
}
let t3 = points.length;
for (let x = x1; x <= x2; x += dx) {
let y = g1(x);
if (Math.abs(y) < Infinity) {
points.push([x, y]);
}
}
let t4 = points.length;
for (let x = x1; x <= x2; x += dx) {
let y = Math.sin(x);
if (Math.abs(y) < Infinity) {
points.push([x, y]);
}
}
let t5 = points.length;
for (let x = x1; x <= x2; x += dx) {
let y = Math.cos(x);
if (Math.abs(y) < Infinity) {
points.push([x, y]);
}
}
let xscale = d3.scaleLinear()
.domain([x1, x2])
.range([padding, width - padding]);
let ys = points.map((a) => a[1]);
y1 = Math.min(y1, ...ys);
y2 = Math.max(y2, ...ys);
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, i) =>
i < t1 ? 'red':
i < t2 ? 'green' :
i < t3 ? 'blue' :
i < t4 ? 'orange' :
i < t5 ? 'brown': 'purple');
svg.selectAll('line')
.data([[x1, 0, x2, 0], [0, y1, 0, y2]])
.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', 'black');
svg.append('g')
.attr('transform', `translate(0, ${height - padding})`)
.call(xaxis);
svg.append('g')
.attr('transform', `translate(${padding}, 0)`)
.call(yaxis);}
inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();
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