数学 - Python - JavaScript - 関数の変化をとらえる - 関数の極限と微分法 – 導関数とその計算 - 微分可能性と連続性(左側微分可能、右側微分可能)

数学読本〈4〉

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

数学読本〈4〉数列の極限，順列/順列・組合せ/確率/関数の極限と微分法(松坂 和夫(著)、岩波書店)の第17章(関数の変化をとらえる - 関数の極限と微分法)、17.3(導関数とその計算)、微分可能性と連続性、問23.を取り組んでみる。

23. 1. 
       $\begin{array}{l}\frac{\left|{\left(0+h\right)}^2-\left(0+h\right)\right|-\left|0^2-0\right|}{h}\\ =\frac{\left|h^2-h\right|}{h}\\ \underset{h\to 0+}{\lim }\frac{\left|h^2-h\right|}{h}\\ =\underset{h\to 0+}{\lim }\left|h-1\right|=1\\ \underset{h\to 0-}{\lim }-\left|h-1\right|=-1\end{array}$
    2. 
       $\begin{array}{l}\frac{\left(0+h\right)\left|h\right|}{h}\\ =\left|h\right|\\ \underset{h\to \text{0}}{\lim }\left|h\right|=0\end{array}$

コード(Emacs)

Python 3

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

from sympy import symbols, sqrt, Limit, pprint

x, h = symbols('x h')

exprs = [abs(x ** 2 - x), x * abs(x)]

for i, expr in enumerate(exprs, 1):
    print('({0})'.format(i))
    expr = (expr.subs({x: 0 + h}) - expr.subs({x: 0})) / h
    pprint(expr)
    for dir in ['+', '-']:
        l = Limit(expr, h, 0, dir=dir)
        pprint(l)
        pprint(l.doit())

入出力結果(Terminal, IPython)

$ ./sample23.py
(1)
│ 2    │
│h  - h│
────────
   h    
     │ 2    │
     │h  - h│
 lim ────────
h─→0⁺   h    
1
     │ 2    │
     │h  - h│
 lim ────────
h─→0⁻   h    
-1
(2)
│h│
 lim │h│
h─→0⁺   
0
 lim │h│
h─→0⁻   
0
$

コード(Emacs)

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" value="0.01">
<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">
<br>
<label for="h0">h = </label>
<input id="h0" type="number" step="0.001" 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="sample23.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_h = document.querySelector('#h0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_h],
    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 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),
        h = parseFloat(input_h.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2 || h === 0) {
        return;
    }

    let points = [],
        f = (fn, x) => (fn(x + h) - fn(x)) / h,
        f1 = (x) => Math.abs(x ** 2 - x),
        f2 = (x) => x * Math.abs(x),
        fns = [f1, f2];
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f1(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t1 = points.length;
    for (let x = x1; x <= x2; x += dx) {
        let y = f2(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    
    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' : 'green');

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]])
        .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', 'black');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    p(
        fns.map((fn, i) => `(${i + 1}) ${f(fn, 0)}`)
            .join('\n')
    );
};

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

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