数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 正弦と余弦 - 2つの基本的な極限(sin(x) / x, tan(x) / x)

解析入門 原書第3版
原著

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
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

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4章(正弦と余弦)、5(2つの基本的な極限)、練習問題1-18.を取り組んでみる。

1. 
   $\underset{h\to 0}{\lim }\frac{\sin 2h}{h}=\underset{k\to 0}{\lim }2\frac{\sin k}{k}=2$
2. 
   $\underset{h\to 0}{\lim }\frac{\sin 3h}{h}=\underset{k\to 0}{\lim }3\frac{\sin k}{k}=3$
3. 
   $\underset{h\to 0}{\lim }\frac{\sin h}{3h}=\frac{1}{3}$
4. 
   $\underset{h\to 0}{\lim }\frac{\tan 2h}{\sin h}=\underset{h\to 0}{\lim }2\frac{\tan 2h}{2h}·\frac{h}{\sin h}=2$
5. 
   $\underset{h\to 0}{\lim }\frac{\cos 2h}{1+\sin h}=1$
6. 
   $\underset{h\to 0}{\lim }\frac{\sin h^2}{h}=\underset{h\to 0}{\lim }h·{\left(\frac{\sin h}{h}\right)}^2=0$
7. 
   $\underset{h\to 0}{\lim }\frac{\sin 2h}{3h}=\underset{h\to 0}{\lim }\frac{2h}{3}\frac{\sin 2h^2}{2h^2}=0$
8. 
   $\underset{h\to 0}{\lim }\frac{\sin h^3}{h^3}=1$
9. 
   $\underset{h\to 0}{\lim }\frac{\sin 2h^3}{h^3}=\underset{h\to 0}{\lim }2\frac{\sin 2h^3}{2h^3}=2$
10. 
    $\underset{h\to 0}{\lim }\frac{h\sin h}{\sin 2h^2}=\underset{h\to 0}{\lim }\frac{1}{2}\frac{\sin h}{h}\frac{2h^2}{\sin 2h^2}=\frac{1}{2}$
11. 
    $\underset{h\to 0}{\lim }\frac{2}{3}\frac{\sinh }{h}·\frac{\sin 2h}{2h}·\frac{3h}{\sin 3h}=\frac{2}{3}$
12. 
    $\underset{h\to 0}{\lim }\frac{\tan h}{h\cos h}=\underset{h\to 0}{\lim }\frac{\sin h}{h}·\frac{1}{{\cos }^{2}h}=1$
13. 
    $\underset{h\to 0}{\lim }\frac{{\tan }^{2}h}{h\sin h}=\underset{h\to 0}{\lim }\frac{\sin h}{h{\cos }^{2}h}=1$
14. 
    $\underset{h\to 0}{\lim }\frac{{\sin }^{2}h}{h\tan h}=\underset{h\to 0}{\lim }\frac{\sinh }{h}·\cos h=1$
15. 
    $\underset{h\to 0}{\lim }\frac{{\tan }^{3}h}{h^2\sin h\cos h}=\underset{h\to 0}{\lim }\frac{{\sin }^{2}h}{h^2{\cos }^{4}h}=1$
16. 
    $\underset{h\to 0}{\lim }\frac{\frac{{\sin }^{2}h}{\cos h}}{h^2}=\underset{h\to 0}{\lim }\frac{1}{\cos h}·\frac{{\sin }^{2}h}{h^2}=1$
17. 
    $\underset{h\to 0}{\lim }\frac{1-\cos h}{\sin h}=\underset{h\to 0}{\lim }\frac{1-{\cos }^{2}h}{\sin h\left(1+\cos h\right)}=\underset{h\to 0}{\lim }\frac{{\sin }^{2}h}{\sin h\left(1+\cos h\right)}=\underset{h\to 0}{\lim }\frac{\sin h}{1+\cos h}=0$
18. 
    $\underset{h\to 0}{\lim }\frac{1-{\cos }^{2}h}{\frac{{\sin }^{2}h}{{\cos }^{2}h}\left(1+{\cos }^{2}h\right)}=\underset{h\to 0}{\lim }\frac{{\cos }^{2}h{\sin }^{2}h}{{\sin }^{2}h\left(1+{\cos }^{2}h\right)}=\frac{1}{2}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, Limit, cos, tan

h = symbols('h')

for dir in ['+', '-']:
    l = Limit((1 - cos(h)) / tan(h) ** 2, h, 0, dir=dir)
    pprint(l)
    pprint(l.doit())

入出力結果(Terminal, IPython)

$ ./sample1.py
     -cos(h) + 1
 lim ───────────
h─→0⁺     2     
       tan (h)  
1/2
     -cos(h) + 1
 lim ───────────
h─→0⁻     2     
       tan (h)  
1/2
$

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">

<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) => Math.sin(x) / x,
    g = (x) => Math.tan(x) / x;

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;n
    }
    let points = [],
        lines = [];

    [[f, 'red'], [g, 'green'], [(x) => Math.sin(x), 'blue'],
     [(x) => Math.tan(x), 'orange'], [(x) => x, 'brown']]
        .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]);
                }
            }
        });
    
    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);
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =