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、一般化)

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8.を取り組んでみる。

8. 1. 
      $\begin{array}{l}-1\le \sin \frac{1}{x^n}\le 1\\ -\left|x^m\right|\le x^m\sin \frac{1}{x^n}\le \left|x^m\right|\\ \underset{x\to 0}{\lim }\pm \left|x^m\right|=0\\ \underset{x\to 0}{\lim }{x}^m\sin \frac{1}{x^n}=0\\ fは0において連続。\end{array}$
   2. 
      $\begin{array}{l}\underset{h\to 0}{\lim }\frac{f\left(h\right)-f\left(0\right)}{h}\\ =\underset{h\to 0}{\lim }{h}^{m-1}\sin \frac{1}{h^n}\\ m=1\\ \underset{h\to 0}{\lim }\frac{f\left(h\right)-f\left(0\right)}{h}\\ =\sin 1\ne 0\\ fは0において微分可能ではない。\\ m>1\\ \left|h^{m-1}\sin \frac{1}{h^n}\right|\le \left|h^{m-1}\right|\\ \underset{h\to 0}{\lim }\left|h^{m-1}\right|=0\\ \underset{h\to 0}{\lim }\frac{f\left(h\right)-f\left(0\right)}{h}=0\\ fは0において微分可能。\end{array}$
   3. 
      $\begin{array}{l}f\text{'}\left(x\right)=m{x}^{m-1}\sin \frac{1}{x^n}+x^m·\frac{-n{x}^{n-1}}{x^{2n}}\cos \frac{1}{x^n}\\ =m{x}^{m-1}\sin \frac{1}{x^n}-n{x}^{m-n-1}\cos \frac{1}{x^n}\\ \cos \theta =\frac{m{x}^{n-1}}{\sqrt{m^2{x}^{2\left(m-1\right)}+n^2{x}^{2\left(m-n-1\right)}}}\\ \sin \theta =\frac{n{x}^{m-n-1}}{\sqrt{m^2{x}^{2\left(m-1\right)}+n^2{x}^{2\left(m-n-1\right)}}}\\ f\text{'}\left(x\right)=\sqrt{m^2{x}^{2\left(m-1\right)}+n^2{x}^{2\left(m-n-1\right)}}\sin \left(\frac{1}{x^n}-\theta \right)\\ m-n-1>0\\ m>n+1\end{array}$
   4. 
      $\begin{array}{l}\underset{h\to 0}{\lim }\frac{f\text{'}\left(h\right)-f\text{'}\left(0\right)}{h}\\ =\underset{h\to 0}{\lim }\frac{m{h}^{m-1}\sin \frac{1}{h^n}-n{h}^{m-n-1}\cos \frac{1}{h^n}}{h}\\ =\underset{h\to 0}{\lim }\left(m{h}^{m-2}\sin \frac{1}{h^n}-n{h}^{m-n-2}\cos \frac{1}{h^n}\right)\\ m-n-2>0\\ m>n+2\end{array}$
   5. 
      $\begin{array}{l}f\text{'}\text{'}\left(x\right)\\ =m\left(m-1\right)x^{m-2}\sin \frac{1}{x^n}+m{x}^{m-1}\frac{-n{x}^{n-1}}{x^{2n}}\cos \frac{1}{x^n}-n\left(m-n-1\right)x^{m-n-2}\cos \frac{1}{x^n}-n{x}^{m-n-1}\left(-\sin \frac{1}{x^n}\right)\frac{-n{x}^{n-1}}{x^{2n}}\\ =m\left(m-1\right)x^{m-2}\sin \frac{1}{x^n}-mn{x}^{m-n-2}\cos \frac{1}{x^n}-n\left(m-n-1\right)x^{m-n-2}\cos \frac{1}{x^n}-n^2{x}^{m-2n-2}\sin \frac{1}{x^n}\\ m-2n-2>0\\ m>2n+2\end{array}$

コード(Emacs)

Python 3

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

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

print('8.')
x = Symbol('x', nonzero=True)
m, n = symbols('m n', integer=True, positive=True)
f = x ** m * sin(1 / x ** n)

for i in range(3):
    fi = Derivative(f, x, i)
    pprint(fi)
    pprint(fi.doit())

入出力結果(Terminal, IPython)

$ ./sample8.py
8.
 m    ⎛ -n⎞
x ⋅sin⎝x  ⎠
 m    ⎛ -n⎞
x ⋅sin⎝x  ⎠
d ⎛ m    ⎛ -n⎞⎞
──⎝x ⋅sin⎝x  ⎠⎠
dx             
   m    ⎛ -n⎞      m  -n    ⎛ -n⎞
m⋅x ⋅sin⎝x  ⎠   n⋅x ⋅x  ⋅cos⎝x  ⎠
───────────── - ─────────────────
      x                 x        
  2             
 d ⎛ m    ⎛ -n⎞⎞
───⎝x ⋅sin⎝x  ⎠⎠
  2             
dx              
 m ⎛ 2    ⎛ -n⎞          -n    ⎛ -n⎞        ⎛ -n⎞    2  -n    ⎛ -n⎞    2  -2⋅n 
x ⋅⎝m ⋅sin⎝x  ⎠ - 2⋅m⋅n⋅x  ⋅cos⎝x  ⎠ - m⋅sin⎝x  ⎠ + n ⋅x  ⋅cos⎝x  ⎠ - n ⋅x    ⋅
───────────────────────────────────────────────────────────────────────────────
                                                     2                         
                                                    x                          

   ⎛ -n⎞      -n    ⎛ -n⎞⎞
sin⎝x  ⎠ + n⋅x  ⋅cos⎝x  ⎠⎠
──────────────────────────
                          
                          
$

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">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="1">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-1">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1">
<br>
<label for="m0">m = </label>
<input id="m0" type="number" min="1" step="1" value="1">
<label for="n0">n = </label>
<input id="n0" type="number" min="1" step="1" value="1">

<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="sample8.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_m = document.querySelector('#m0'),
    input_n = document.querySelector('#n0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2, input_m, input_n],
    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),
        m = parseInt(input_m.value, 10),
        n = parseInt(input_n.value, 10);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        f = (x) => x ** m * Math.sin(x ** -n),
        f1 = (x) => (m * x ** m * Math.sin(x ** -n) - n * x ** (m - n) * Math.cos(x ** -n)) / x,
        f2 = (x) =>
        x ** m *
        (m ** 2 * Math.sin(x ** -n) -
         2 * m * n * x ** -n * Math.cos(x ** -n) -
         m * Math.sin(x ** - n) +
         n ** 2 * x ** -n * Math.cos(x ** -n) -
         n ** 2 * x ** (-2 * n) * Math.sin(x ** -n) +
         n * x ** -n * Math.cos(x ** -n)) / x ** 2;
    
    for (let x = x1; x <= x2; x += dx) {
        if (x !== 0) {
            points.push([x, f(x)]);
        }
    }
    let t1 = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        if (x !== 0) {
            points.push([x, f1(x)]);
        }
    }
    let t2 = points.length;

    for (let x = x1; x <= x2; x += dx) {
        if (x !== 0) {
            points.push([x, f2(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 =
m = n =