数学 - Python - SymPy - JavaScript - D3.js - 関数の変化をとらえる - 関数の極限と微分法 – 関数と極限 - 極限値の計算(定数を定める)

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)、17.1(関数と極限)、極限値の計算、問4.を取り組んでみる。

4. 1. 
      $\begin{array}{l}\underset{x\to 1}{\lim }\left(x^2+ax+b\right)\\ =\underset{x\to 1}{\lim }\left(x-1\right)\frac{x^2+ax+b}{x-1}\\ =0·5\\ =0\\ \\ 1+a+b=0\\ b=-1-a\\ \\ \underset{x\to 1}{\lim }\frac{x^2+ax-1-a}{x-1}\\ =\underset{x\to 1}{\lim }\frac{\left(x-1\right)\left(x+a+1\right)}{x-1}\\ =\underset{x\to 1}{\lim }\left(x+a+1\right)\\ =2+a\\ =5\\ a=3\\ b=-1-3=-4\end{array}$
   2. 
      $\begin{array}{l}\underset{x\to 3}{\lim }\left(x^2+ax+6\right)\\ =\underset{x\to 3}{\lim }\left(x^2-\left(3+b\right)x+3b\right)\frac{x^2+ax+6}{x^2-\left(3+b\right)x+3b}\\ =\underset{x\to 3}{\lim }\left(9-9-3b+3b\right)\frac{2}{5}\\ =\underset{x\to 3}{\lim }0·\frac{2}{5}\\ =0\\ \\ 9+3a+6=0\\ a=-5\\ \\ =\underset{x\to 3}{\lim }\frac{x^2-5x+6}{x^2-\left(3+b\right)x+3b}\\ =\underset{x\to 3}{\lim }\frac{\left(x-2\right)\left(x-3\right)}{\left(x-3\right)\left(x-b\right)}\\ =\frac{1}{3-b}\\ =\frac{2}{5}\\ 2\left(3-b\right)=5\\ b=\frac{1}{2}\end{array}$
   3. 
      $\begin{array}{l}\underset{x\to 2}{\lim }\left(x^3+ax+b\right)\\ =\underset{x\to 2}{\lim }\frac{x^3+ax+b}{x^2-x-2}\left(x^2-x-2\right)\\ =3·0\\ =0\\ \\ 8+2a+b=0\\ b=-2a-8\\ \\ \underset{x\to 2}{\lim }\frac{x^2-x-2}{x^3+ax-2a-8}\\ =\underset{x\to 2}{\lim }\frac{\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x^2+2x+a+4\right)}\\ =\underset{x\to 2}{\lim }\frac{x+1}{x^2+2x+a+4}\\ =\frac{3}{4+4+a+4}\\ =\frac{3}{12+a}\\ =\frac{1}{3}\\ 12+a=9\\ a=-3\\ b=-2\end{array}$
   4. 
      $\begin{array}{l}\underset{x\to 1}{\lim }\left(a\sqrt{x+1}-b\right)\\ =\underset{x\to 1}{\lim }\left(x-1\right)\frac{a\sqrt{x+1}-b}{x-1}\\ =0·\sqrt{2}\\ =0\\ \\ a\sqrt{2}-b=0\\ b=a\sqrt{2}\\ \\ \underset{x\to 1}{\lim }\frac{a\sqrt{x+1}-b}{x-1}\\ =a\underset{x\to 1}{\lim }\frac{\sqrt{x+1}-\sqrt{2}}{x-1}\\ =a\underset{x\to 1}{\lim }\frac{x+1-2}{\left(x-1\right)\left(\sqrt{x+1}+\sqrt{2}\right)}\\ =a\underset{x\to 1}{\lim }\frac{1}{\sqrt{x+1}+\sqrt{2}}\\ =\frac{a}{2\sqrt{2}}\\ =\sqrt{2}\\ a=4\\ b=4\sqrt{2}\end{array}$

コード(Emacs)

Python 3

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

from sympy import Symbol, Limit, sqrt, pprint

x = Symbol('x')

for i, t in enumerate([
        ((x ** 2 + 3 * x - 4) / (x - 1), x, 1),
        ((x ** 2 + -5 * x + 6) / (x ** 2 - (3 + 1/2) * x + 3 * 1/2), x, 3),
        ((x ** 2 - x - 2) / (x ** 3 - 3 * x - 2), x, 2),
        ((4 * sqrt(x + 1) - 4 * sqrt(2)) / (x - 1), x, 1)
                      ]):
    print('({})'.format(i + 1))
    pprint(Limit(*t).doit())

入出力結果(Terminal, IPython)

$ ./sample4.py
(1)
5
(2)
0.400000000000000
(3)
1/3
(4)
√2
$

コード(Emacs)

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="0">
<label for="x2">x2 = </label>
<input id="x2" 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="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_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    inputs = [input_x1, input_x2],
    p = (x) => pre0.textContent += x + '\n';

let f1 = (x) => (x ** 2 + 3 * x -4) / (x - 1),
    f2 = (x) => (x ** 2 - 5 * x + 6) / (x ** 2 - 7 / 2 * x + 1.5),
    f3 = (x) => (x ** 2 - x - 2) / (x ** 3 - 3 * x - 2),
    f4 = (x) => (4 * Math.sqrt(x + 1) - 4 * Math.sqrt(2)) / (x - 1);

let draw = () => {
    pre0.textContent = '';

    let x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value);
    
    let points = [];

    for (let x = x1; x <= x2; x += 0.001) {
        if (x !== 1) {
            points.push([x, f1(x)]);
        }
    }
    let t1 = points.length;

    for (let x = x1; x <= x2; x += 0.001) {
        if (x !== 3) {
            points.push([x, f2(x)]);
        }
    }
    let t2 = points.length;
    
    for (let x = x1; x <= x2; x += 0.001) {
        if (x !== 2) {
            points.push([x, f3(x)]);
        }
    }
    let t3 = points.length;
    for (let x = x1; x <= x2; x += 0.001) {
        if (x !== 1) {
            points.push([x, f4(x)]);
        }
    }
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let ys = points.map((a) => a[1]);
    let yscale = d3.scaleLinear()
        .domain([x1, x2])
        .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', 1)
        .attr('fill', (d, i) =>
              i < t1 ? 'red' :
              i < t2 ? 'green' :
              i < t3 ? 'blue' : 'gray');

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

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

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

x1 = x2 =