数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 逆関数 - 逆正弦関数(三角関数(正割、余割)、導関数、合成関数の微分法)

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

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11、12、13、14.を取り組んでみる。

$\begin{array}{l} \frac{1}{\sqrt{1 - {(x^{2} - 1)}^{2}}} \cdot 2 x \\ = \frac{2 x}{\sqrt{1 - {(x^{2} - 1)}^{2}}} \end{array}$
$\begin{array}{l} - \frac{1}{\sqrt{1 - {(2 x + 5)}^{2}}} \cdot 2 \\ = - \frac{2}{\sqrt{1 - {(2 x + 5)}^{2}}} \end{array}$
$\begin{array}{l} \frac{- 1}{{(\arcsin x)}^{2}} \cdot \frac{1}{\sqrt{1 - x^{2}}} \\ = - \frac{1}{{(\arcsin x)}^{2} \sqrt{1 - x^{2}}} \end{array}$
$\begin{array}{l} \frac{- 2}{{(\arccos 2 x)}^{2}} \cdot (- \frac{1}{\sqrt{1 - {(2 x)}^{2}}}) \cdot 2 \\ = \frac{4}{{(\arccos 2 x)}^{2} \sqrt{1 - 4 x^{2}}} \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, asin, acos, Derivative, plot

x = symbols('x')
fs = [asin(x ** 2 - 1),
      acos(2 * x + 5),
      1 / asin(x),
      2 / acos(2 * x)]

for i, f in enumerate(fs, 11):
    print(f'({i})')
    f1 = Derivative(f, x, 1)
    for t in [f1, f1.doit()]:
        pprint(t)
        print()
    print()
    p = plot(f, show=False, legend=True)
    p.save(f'sample{i}.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample11.py
(11)
d ⎛    ⎛ 2    ⎞⎞
──⎝asin⎝x  - 1⎠⎠
dx              

         2⋅x         
─────────────────────
    _________________
   ╱           2     
  ╱    ⎛ 2    ⎞      
╲╱   - ⎝x  - 1⎠  + 1 


(12)
d                
──(acos(2⋅x + 5))
dx               

         -2          
─────────────────────
   __________________
  ╱            2     
╲╱  - (2⋅x + 5)  + 1 


(13)
d ⎛   1   ⎞
──⎜───────⎟
dx⎝asin(x)⎠

         -1           
──────────────────────
   __________         
  ╱    2          2   
╲╱  - x  + 1 ⋅asin (x)


(14)
d ⎛    2    ⎞
──⎜─────────⎟
dx⎝acos(2⋅x)⎠

            4             
──────────────────────────
   ____________           
  ╱      2          2     
╲╱  - 4⋅x  + 1 ⋅acos (2⋅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="0.001">
<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="sample11.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 f1 = (x) => Math.asin(x ** 2 - 1),
    f2 = (x) => Math.acos(2 * x + 5),
    f3 = (x) => 1 / Math.asin(x),
    f4 = (x) => 2 / Math.acos(2 * 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 = [],
        lines = [],        
        fns = [[f1, 'red'],
               [f2, 'green'],
               [f3, 'blue'],
               [f4, '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 =

Kamimura's blog

2017年9月28日木曜日

数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 逆関数 - 逆正弦関数(三角関数(正割、余割)、導関数、合成関数の微分法)

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