数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 逆関数 - 逆正弦関数(逆余弦関数とその導関数、値、arccos)

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)、3(逆正弦関数)、練習問題4.を取り組んでみる。

4. 
   $\begin{array}{l}x=\cos y\\ \arccos x=y\\ \frac{dx}{dy}=-\sin y\\ \\ \frac{dy}{dx}=-\frac{1}{\sin y}\\ {\sin }^{2}y+{\cos }^{2}y=1\\ {\sin }^{2}y=1-{\cos }^{2}y\\ \sin y=\sqrt{1-{\cos }^{2}y}\\ =\sqrt{1-x^2}\\ \frac{dy}{dx}=-\frac{1}{\sqrt{1-x^2}}\\ \\ g\text{'}\left(\frac{1}{2}\right)=-\frac{1}{\sqrt{1-\frac{1}{4}}}\\ =-\frac{1}{\sqrt{\frac{3}{4}}}\\ =-\frac{2}{\sqrt{3}}\\ \\ g\text{'}\left(\frac{1}{\sqrt{2}}\right)=-\frac{1}{\sqrt{1-\frac{1}{2}}}\\ =-\sqrt{2}\\ \\ \arccos \frac{1}{2}=\frac{\pi }{3}\\ \arccos \frac{1}{\sqrt{2}}=\frac{\pi }{4}\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, acos, Derivative, sqrt, Rational

x = symbols('x')
g = acos(x)
g1 = Derivative(g, x, 1)
for f in [g, g1]:
    pprint(f)
    print()

p = [(g1, Rational(1, 2)),
     (g1, 1 / sqrt(2)),
     (g, Rational(1, 2)),
     (g, 1 / sqrt(2))]

pprint(p)
for f, x0 in p:
    pprint(f.subs({x: x0}).doit())
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample4.py
acos(x)

d          
──(acos(x))
dx         

⎡⎛d               ⎞  ⎛d            √2⎞                  ⎛         √2⎞⎤
⎢⎜──(acos(x)), 1/2⎟, ⎜──(acos(x)), ──⎟, (acos(x), 1/2), ⎜acos(x), ──⎟⎥
⎣⎝dx              ⎠  ⎝dx           2 ⎠                  ⎝         2 ⎠⎦
-2⋅√3 
──────
  3   

-√2

π
─
3

π
─
4

$

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="sample4.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],
    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.cos(x),
    g = (x) => Math.acos(x),
    h = (x) => -1 / Math.sqrt(1 - x ** 2);

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 = [],
        lines = [],        
        fns = [[(x) => x, 'red'],
               [f, 'green'],
               [g, 'blue'],
               [h, 'orange']],
        fns1 = [],
        fns2 = [];

    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 =