数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 曲線をえがくこと - 曲線をえがくこと(7つの性質、有理式)

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, 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6章(曲線をえがくこと)、2(曲線をえがくこと)、練習問題1、2.を取り組んでみる。

1. 1. 
      

      x軸との交点無し。

      y軸との交点は$\left(0,-\frac{2}{3}\right)$

   2. 
      

      臨界点

      $\begin{array}{l}f\left(x\right)=\frac{2x\left(x-3\right)-\left(x^2+2\right)}{{\left(x-3\right)}^2}\\ =\frac{x^2-6x-2}{{\left(x-3\right)}^2}\\ x^2-6x-2=0\\ x=3\pm \sqrt{9+2}=3\pm \sqrt{11}\end{array}$
   3. 
      

      増加する範囲。

      $x\le 3-\sqrt{11},3+\sqrt{11}\le x$
   4. 
      

      減少する範囲。

      $3-\sqrt{11}\le x<3,3<x\le 3+\sqrt{11}$
   5. 
      

      極大点。

      $3-\sqrt{11}$

      極小点。

      $3+\sqrt{11}$
   6. 
      

      $\underset{x\to \infty }{\lim }f(x)=\infty$

      $\underset{x\to -\infty }{\lim }f\left(x\right)=-\infty$

   7. 
      x=3
2. 1. 
      

      x軸との交点(3, 0)、y軸との交点(0, -3)

   2. 
      $\begin{array}{l}f\text{'}\left(x\right)=\frac{x^2+1-\left(x-3\right)2x}{{\left(x^2+1\right)}^2}\\ =\frac{-x^2+6x+1}{{\left(x^2+1\right)}^2}\\ -x^2+6x+1=0\\ x^2-6x-1=0\\ x=3\pm \sqrt{9+1}=3\pm \sqrt{10}\end{array}$
   3. 
      $3-\sqrt{10}\le x\le 3+\sqrt{10}$
   4. 
      $x\le 3-\sqrt{10},3+\sqrt{10}\le x$
   5. 
      

      極大点。

      $3+\sqrt{10}$

      極小点。

      $3-\sqrt{10}$
   6. 
      

      $\underset{x\to \infty }{\lim }f\left(x\right)=0$

      $\underset{x\to -\infty }{\lim }f\left(x\right)=0$

   7. 
      

      無し。

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, solve, Derivative, Limit, S

x = symbols('x')
fs = [(x ** 2 + 2) / (x - 3),
      (x - 3) / (x ** 2 + 1)]

for i, f in enumerate(fs, 1):
    print(f'{i}.')
    pprint(f)
    pprint(solve(f))
    pprint(f.subs({x: 0}))
    d = Derivative(f, x, 1)
    pprint(d)
    f1 = d.doit()
    pprint(f1)
    pprint(solve(f1))
    for x0 in [S.Infinity, -S.Infinity]:
        l = Limit(f, x, x0)
        pprint(l)
        pprint(l.doit())
    print()

入出力結果(Terminal, IPython)

$ ./sample1.py
1.
 2    
x  + 2
──────
x - 3 
[-√2⋅ⅈ, √2⋅ⅈ]
-2/3
  ⎛ 2    ⎞
d ⎜x  + 2⎟
──⎜──────⎟
dx⎝x - 3 ⎠
          2     
 2⋅x     x  + 2 
───── - ────────
x - 3          2
        (x - 3) 
[3 + √11, -√11 + 3]
     2    
    x  + 2
lim ──────
x─→∞x - 3 
∞
      2    
     x  + 2
 lim ──────
x─→-∞x - 3 
-∞

2.
x - 3 
──────
 2    
x  + 1
[3]
-3
d ⎛x - 3 ⎞
──⎜──────⎟
dx⎜ 2    ⎟
  ⎝x  + 1⎠
  2⋅x⋅(x - 3)     1   
- ─────────── + ──────
           2     2    
   ⎛ 2    ⎞     x  + 1
   ⎝x  + 1⎠           
[3 + √10, -√10 + 3]
    x - 3 
lim ──────
x─→∞ 2    
    x  + 1
0
     x - 3 
 lim ──────
x─→-∞ 2    
     x  + 1
0

$

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="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-20">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="20">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-20">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="20">

<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="sample1.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('#n0'),
    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) => (x ** 2 + 2) / (x - 3),
    f2 = (x) => (x - 3) / (x ** 2 + 1);

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 = [[3, y1, 3, y2, 'red']],
        f = (x) =>
        range(0, n0 + 1)
        .map((i) => a0 * (n0 + 1 - i) * x ** i)
        .reduce((prev, next) => prev + next),
        fns = [[f1, 'green'], [f2, 'orange']],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                if (Math.abs(y) < Infinity) {
                    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 =