数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 逆関数 - 逆正接関数(定義域(区間)、値域、導関数、計算)

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, Microsoft Edge, 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7章(逆関数)、4(逆正接関数)、練習問題3、4.を取り組んでみる。

3. 
   $\begin{array}{l}y=\tan x\left(\frac{\pi }{2}<x<\frac{3}{2}\pi \right)\\ x=\arctan y\left(y\in ℝ\right)\\ \\ \frac{dx}{dy}=\frac{1}{\frac{dy}{dx}}\\ =\frac{1}{\frac{1}{{\cos }^{2}x}}\\ =\frac{1}{\frac{{\sin }^{2}x+{\cos }^{2}x}{{\cos }^{2}x}}\\ =\frac{1}{{\tan }^{2}x+1}\\ =\frac{1}{y^2+1}\\ \\ \frac{d}{dx}\arctan x=\frac{1}{1+x^2}\end{array}$
4. 1. 
      $\begin{array}{l}\arctan \left(\tan \frac{3\pi }{4}\right)\\ =\arctan \left(-1\right)\\ =-\frac{\pi }{4}\end{array}$
   2. 
      $\begin{array}{l}\arctan \left(\tan 2\pi \right)\\ =\arctan \left(\tan 0\right)\\ =0\end{array}$
   3. 
      $\begin{array}{l}\arctan \left(\tan \frac{5}{6}\pi \right)\\ =\arctan \left(\tan \left(-\frac{\pi }{6}\right)\right)\\ =-\frac{\pi }{6}\end{array}$
   4. 
      $\begin{array}{l}\arctan \left(\tan \left(-\frac{5\pi }{6}\right)\right)\\ =\arctan \left(\tan \frac{\pi }{6}\right)\\ =\frac{\pi }{6}\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, tan, atan, Derivative, pi

print('3.')
x = symbols('x')
D = Derivative(atan(x), x, 1)
for t in [D, D.doit()]:
    pprint(t)
    print()

print('4.')
nums = [3 * pi / 4, 2 * pi, 5 * pi / 6, -5 * pi / 6]

for i, n in enumerate(nums):
    print(f'({chr(ord("a") + i)})')
    pprint(n)
    pprint(atan(tan(n)))
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample3.py
3.
d          
──(atan(x))
dx         

  1   
──────
 2    
x  + 1

4.
(a)
3⋅π
───
 4 
-π 
───
 4 

(b)
2⋅π
0

(c)
5⋅π
───
 6 
-π 
───
 6 

(d)
-5⋅π 
─────
  6  
π
─
6

$

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="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" step="0.0001" value="0.1">

<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="sample3.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_dx0 = document.querySelector('#dx0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_dx0],
    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.atan(x),
    f1 = (x) => 1 / (1 + x ** 2),
    g = (x0) => (x) => f1(x0) * (x - x0) + f(x0);

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),
        dx0 = parseFloat(input_dx0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [],        
        fns = [[f, 'red']],
        fns1 = [],
        fns2 = [[g, 'green']];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                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();

r = dx =
x1 = x2 =
y1 = y2 =
dx0 =