数学 - Python - JavaScript - 解析学 - 各種の初等関数 - 三角関数(続き)、逆三角関数(連続性、微分可能性、導関数の連続性)

解析入門〈1〉

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〈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6、7.を取り組んでみる。

6. 
   $\begin{array}{l}x\ne 0\hfill \\ -1\le \sin \frac{1}{x}\le 1\hfill \\ x>0\hfill \\ -x\le x\sin \frac{1}{x}\le x\hfill \\ x<0\hfill \\ x\le x\sin \frac{1}{x}\le -x\hfill \\ \underset{x\to 0}{\lim }\pm x=0\hfill \\ \underset{x\to 0}{\lim }x\sin \frac{1}{x}=0\hfill \\ よってfは0において連続。\hfill \\ \hfill \\ \underset{h\to 0}{\lim }\frac{(0+h)\sin \frac{1}{0+h}-0}{h}\hfill \\ =\underset{h\to 0}{\lim }\frac{h\sin \frac{1}{h}}{h}\hfill \\ =\underset{h\to 0}{\lim }\sin \frac{1}{h}\hfill \\ =\underset{x\to \infty }{\lim }\sin x\hfill \\ 収束しないので、fは0において微分可能でない。\hfill \end{array}$
7. 
   $\begin{array}{l}x\ne 0\\ f\text{'}\left(x\right)=2x\sin \frac{1}{x}+x^2\frac{-1}{x^2}\cos \frac{1}{x}\\ =2x\sin \frac{1}{x}-\cos \frac{1}{x}\\ x=0\\ f\text{'}\left(x\right)=0\\ \\ x\ne 0\\ -1\le \sin \frac{1}{x}\le 1\\ -x^2\le x^2\sin \frac{1}{x}\le x^2\\ \underset{x\to 0}{\lim }\pm x^2=0\\ \underset{x\to 0}{\lim }{x}^2\sin \frac{1}{x}=0\\ f\left(x\right)=0\\ fは全区間で微分可能で連続。\\ \\ \underset{h\to 0}{\lim }\frac{f\text{'}\left(0+h\right)-f\text{'}\left(0\right)}{h}\\ =\underset{h\to 0}{\lim }\frac{\left(2\left(0+h\right)\sin \frac{1}{0+h}-\cos \frac{1}{0+h}\right)-0}{h}\\ =\underset{h\to 0}{\lim }\frac{2h\sin \frac{1}{h}-\cos \frac{1}{h}}{h}\\ =\underset{h\to 0}{\lim }\left(2\sin \frac{1}{h}-\cos \frac{1}{h}\right)\\ =\underset{h\to 0}{\lim }\left(\sqrt{5}\left(\frac{2}{\sqrt{5}}\sin \frac{1}{h}-\frac{1}{\sqrt{5}}\cos \frac{1}{h}\right)\right)\\ \cos \theta =\frac{2}{\sqrt{5}},\sin \theta =\frac{1}{\sqrt{5}}\\ =\underset{h\to 0}{\lim }\left(\sqrt{5}\left(\sin \frac{1}{h}\cos \theta -\cos \frac{1}{h}\sin \theta \right)\right)\\ =\underset{h\to 0}{\lim }\left(\sqrt{5}\left(\sin \left(\frac{1}{h}-\theta \right)\right)\right)\\ =\sqrt{5}\underset{x\to \infty }{\lim }\sin x\\ 収束しないので、f\text{'}は0で連続でない。\end{array}$

コード(Emacs)

Python 3

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

from sympy import Symbol, sin, Limit, pprint, Derivative

print('6')
x = Symbol('x', nonzero=True)
expr = x * sin(1 / x)

pprint(expr)
l = Limit(expr, x, 0, dir='+')
pprint(l)
pprint(l.doit())
l = Limit(expr, x, 0, dir='-')
pprint(l)
pprint(l.doit())

h = Symbol('h', positive=True)
expr = ((0 + h) * sin(1 / (0 + h)) - 0) / h
pprint(expr)
l = Limit(expr, h, 0)
pprint(l)
pprint(l.doit())

print('7.')
x = Symbol('x', nonzero=True)
expr = x ** 2 * sin(1 / x)
pprint(expr)
l = Limit(expr, x, 0, dir='+')
pprint(l)
pprint(l.doit())
l = Limit(expr, x, 0, dir='-')
pprint(l)
pprint(l.doit())

d = Derivative(expr, x)
pprint(d)
f1 = d.doit()
pprint(f1)

l = Limit(f1, x, 0)
pprint(l)
pprint(l.doit())

入出力結果(Terminal, IPython)

$ ./sample6.py
6
     ⎛1⎞
x⋅sin⎜─⎟
     ⎝x⎠
          ⎛1⎞
 lim x⋅sin⎜─⎟
x─→0⁺     ⎝x⎠
0
          ⎛1⎞
 lim x⋅sin⎜─⎟
x─→0⁻     ⎝x⎠
0
   ⎛1⎞
sin⎜─⎟
   ⎝h⎠
        ⎛1⎞
 lim sin⎜─⎟
h─→0⁺   ⎝h⎠
<-1, 1>
7.
 2    ⎛1⎞
x ⋅sin⎜─⎟
      ⎝x⎠
      2    ⎛1⎞
 lim x ⋅sin⎜─⎟
x─→0⁺      ⎝x⎠
0
      2    ⎛1⎞
 lim x ⋅sin⎜─⎟
x─→0⁻      ⎝x⎠
0
d ⎛ 2    ⎛1⎞⎞
──⎜x ⋅sin⎜─⎟⎟
dx⎝      ⎝x⎠⎠
       ⎛1⎞      ⎛1⎞
2⋅x⋅sin⎜─⎟ - cos⎜─⎟
       ⎝x⎠      ⎝x⎠
     ⎛       ⎛1⎞      ⎛1⎞⎞
 lim ⎜2⋅x⋅sin⎜─⎟ - cos⎜─⎟⎟
x─→0⁺⎝       ⎝x⎠      ⎝x⎠⎠
<-1, 1>
$

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="0.0001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-1.4">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="1.4">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-1.4">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1.4">

<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="sample6.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';

let f6 = (x) => x === 0 ? 0 : x * Math.sin(1 / x),
    f71 = (x) => x === 0 ? 0 : x ** 2 * Math.sin(1 / x),
    f72 = (x) => x === 0 ? 0 : 2 * x * Math.sin(1 / x) - Math.cos(1 / 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;
    }

    let points = [];
    
    for (let x = x1; x <= x2; x += dx) {
        points.push([x, f6(x)]);
    }
    let t1 = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        points.push([x, f71(x)]);
    }
    let t2 = points.length;
    for (let x = x1; x <= x2; x += dx) {
        points.push([x, f72(x)]);
    }
    
    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, i) =>
              i < t1 ? 'red':
              i < t2 ? 'green' : 'blue');
    
    svg.append('g')
        .attr('transform', `translate(0, ${yscale(0)})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${xscale(0)}, 0)`)
        .call(yaxis);
    p(f6(0));
    p(f71(0));
    p(f72(0));
}

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

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