数学 - Python - JavaScript - 関数の変化をとらえる - 関数の極限と微分法 – 関数の連続性 - 区間における連続(整関数、分数関数、無理関数、指数関数、三角関数)

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)、17.2(関数の連続性)、区間における連続、問14、15、16.を取り組んでみる。

14. 1. 
       $\left(-\infty ,-1\right),\left(-1,\frac{1}{2}\right),\left(\frac{1}{2},\infty \right)$
    2. 
       $\left(-\infty ,2\right),\left(2,\infty \right)$
    3. 
       $\begin{array}{l}\frac{D}{4}=1-2<0\\ \left(-\infty ,\infty \right)\end{array}$
    4. 
       $\left(-\infty ,-3\right),\left(-3,0\right),\left(0,3\right),\left(3,\infty \right)$
15. 1. 
       $\begin{array}{l}n<2x<\left(n+1\right)\pi \\ \left(\frac{n\pi }{2},\frac{\left(n+1\right)\pi }{2}\right)\end{array}$
    2. 
       $\begin{array}{l}-\frac{\pi }{2}+n\pi <\frac{x}{2}<\frac{\pi }{2}+n\pi \\ \left(\left(2n-1\right)\pi ,\left(2n+1\right)\pi \right)\end{array}$
    3. 
       $\begin{array}{l}{4}^x=2\\ x=\frac{1}{2}\\ \left(-\infty ,\frac{1}{2}\right),\left(\frac{1}{2},\infty \right)\end{array}$
    4. 
       $\left(-\infty ,0\right),\left(0,\infty \right)$
16. 1. 
       $\begin{array}{l}\left|x\right|<1\\ f\left(x\right)=-1\\ f\left(1\right)=0\\ \left|x\right|>1\\ f\left(x\right)=1\\ x=\pm 1\end{array}$
    2. 
       $x=-1$
    3. 
       $x=\frac{\pi }{2}+m\pi$
    4. 
       $x=\frac{\pi }{2}+m\pi$

コード(Emacs)

Python 3

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

from sympy import Symbol, sin, tan, Limit, S, pprint, pi

x = Symbol('x', real=True)

exprs = [(1 / sin(2 * x), 5 * pi / 2),
         (tan(x / 2), 5 * pi),
         (1 / (4 ** x - 2), 1 / 2),
         (2 ** (1 / x), 0)]
for i, (expr, x0) in enumerate(exprs, 1):
    print('({0})'.format(i))
    l1 = Limit(expr, x, x0, dir='+')
    pprint(l1)
    pprint(l1.doit())
    l2 = Limit(expr, x, x0, dir='-')
    pprint(l2)
    pprint(l2.doit())
    print(l1.doit() == l2.doit())

入出力結果(Terminal, IPython)

$ ./sample14.py
(1)
          1    
  lim  ────────
   5⋅π sin(2⋅x)
x─→───⁺        
    2          
-∞
          1    
  lim  ────────
   5⋅π sin(2⋅x)
x─→───⁻        
    2          
∞
False
(2)
          ⎛x⎞
  lim  tan⎜─⎟
x─→5⋅π⁺   ⎝2⎠
-∞
          ⎛x⎞
  lim  tan⎜─⎟
x─→5⋅π⁻   ⎝2⎠
∞
False
(3)
         1   
  lim  ──────
x─→0.5⁺ x    
       4  - 2
∞
         1   
  lim  ──────
x─→0.5⁻ x    
       4  - 2
-∞
False
(4)
     x ___
 lim ╲╱ 2 
x─→0⁺     
∞
     x ___
 lim ╲╱ 2 
x─→0⁻     
0
False
$

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="-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="-2">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="2">
<br>
<label for="n0">n = </label>
<input id="n0" type="number" value="10">

<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="sample14.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_n = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2, input_n],
    p = (x) => pre0.textContent += x + '\n';

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),
        n = parseInt(input_n.value, 10);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2 || n < 1) {
        return;
    }

    let points = [],
        f1 = (x) => (Math.abs(x) ** n - 1) / (Math.abs(x) ** n + 1),
        f2 = (x) => (x ** (2 * n - 1) + 1) / (x ** (2 * n) + 1),
        f3 = (x) => Math.abs(Math.sin(x)) ** n,
        f4 = (x) => Math.sin(x) / (1 + Math.sin(x) ** (2 * n))
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f1(x);
        if (-Infinity < y  && y < Infinity) {
            points.push([x, f1(x)]);
        }
    }
    let t1 = points.length;
    for (let x = x1; x <= x2; x += dx) {
        let y = f2(x);
        if (-Infinity < y  && y < Infinity) {
            points.push([x, f2(x)]);
        }
    }
    let t2 = points.length;

    for (let x = x1; x <= x2; x += dx) {
        let y = f3(x);
        if (-Infinity < y  && y < Infinity) {
            points.push([x, f3(x)]);
        }
    }
    let t3 = points.length;

    for (let x = x1; x <= x2; x += dx) {
        let y = f4(x);
        if (-Infinity < y  && y < Infinity) {
            points.push([x, f4(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' :
              i < t3 ? 'blue' : 'orange');


    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);
};

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

r = dx =
x1 = x2 =
y1 = y2 =
n =