数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 曲線をえがくこと - xが大きくなるときの様子(正の無限大、負の無限大、商の極限、三角関数、正弦、余弦)

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

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)、1(xが大きくなるときの様子)、練習問題1-10.を取り組んでみる。

$\begin{array}{l} \frac{\frac{2}{x} - \frac{1}{x^{3}}}{1 - \frac{1}{x^{4}}} \\ \lim_{x \to \infty} Q (x) = 0 \\ \lim_{x \to - \infty} Q (x) = 0 \end{array}$
$\begin{array}{l} \lim_{x \to \infty} Q (x) = 0 \\ \lim_{x \to - \infty} Q (x) = 0 \end{array}$
$\begin{array}{l} \lim_{x \to \infty} Q (x) = 0 \\ \lim_{x \to - \infty} Q (x) = 0 \end{array}$
$\begin{array}{l} \lim_{x \to \infty} Q (x) = \frac{1}{π} \\ \lim_{x \to - \infty} Q (x) = \frac{1}{π} \end{array}$
$\begin{array}{l} \lim_{x \to \infty} Q (x) = 0 \\ \lim_{x \to - \infty} Q (x) = 0 \end{array}$
$\begin{array}{l} \lim_{x \to \infty} Q (x) = \infty \\ \lim_{x \to - \infty} Q (x) = - \infty \end{array}$
$\begin{array}{l} \lim_{x \to \infty} Q (x) = - \infty \\ \lim_{x \to - \infty} Q (x) = \infty \end{array}$
$\begin{array}{l} \lim_{x \to \infty} Q (x) = - \frac{1}{2} \\ \lim_{x \to - \infty} Q (x) = - \frac{1}{2} \end{array}$
$\begin{array}{l} \lim_{x \to \infty} Q (x) = - \infty \\ \lim_{x \to - \infty} Q (x) = \infty \end{array}$
$\begin{array}{l} \lim_{x \to \infty} Q (x) = 0 \\ \lim_{x \to - \infty} Q (x) = 0 \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Limit, S, sin, cos, pi

x = symbols('x')

qs = [(2 * x ** 3 - x) / (x ** 4 - 1),
      sin(x) / x,
      cos(x) / x,
      (x ** 2 + 1) / (pi * x ** 2 - 1),
      sin(4 * x) / x ** 3,
      (5 * x ** 4 - x ** 3 + 3 * x + 2) / (x ** 3 - 1),
      (- x ** 2 + 1) / (x + 5),
      (2 * x ** 4 - 1) / (- 4 * x ** 4 + x ** 2),
      (2 * x ** 4 - 1) / (-4 * x ** 3 + x ** 2),
      (2 * x ** 4 - 1) / (-4 * x ** 5 + x ** 2)]

for i, q in enumerate(qs, 1):
    print(f'{i}.')
    for x0 in [S.Infinity, -S.Infinity]:
        f = Limit(q, x, x0)
        pprint(f)
        pprint(f.doit())
    print()

入出力結果(Terminal, IPython)

$ ./sample1.py
1.
       3    
    2⋅x  - x
lim ────────
x─→∞  4     
     x  - 1 
0
        3    
     2⋅x  - x
 lim ────────
x─→-∞  4     
      x  - 1 
0

2.
    sin(x)
lim ──────
x─→∞  x   
0
     sin(x)
 lim ──────
x─→-∞  x   
0

3.
    cos(x)
lim ──────
x─→∞  x   
0
     cos(x)
 lim ──────
x─→-∞  x   
0

4.
      2     
     x  + 1 
lim ────────
x─→∞   2    
    π⋅x  - 1
1
─
π
       2     
      x  + 1 
 lim ────────
x─→-∞   2    
     π⋅x  - 1
1
─
π

5.
    sin(4⋅x)
lim ────────
x─→∞    3   
       x    
0
     sin(4⋅x)
 lim ────────
x─→-∞    3   
        x    
0

6.
       4    3          
    5⋅x  - x  + 3⋅x + 2
lim ───────────────────
x─→∞        3          
           x  - 1      
∞
        4    3          
     5⋅x  - x  + 3⋅x + 2
 lim ───────────────────
x─→-∞        3          
            x  - 1      
-∞

7.
       2    
    - x  + 1
lim ────────
x─→∞ x + 5  
-∞
        2    
     - x  + 1
 lim ────────
x─→-∞ x + 5  
∞

8.
         4     
      2⋅x  - 1 
lim ───────────
x─→∞     4    2
    - 4⋅x  + x 
-1/2
          4     
       2⋅x  - 1 
 lim ───────────
x─→-∞     4    2
     - 4⋅x  + x 
-1/2

9.
         4     
      2⋅x  - 1 
lim ───────────
x─→∞     3    2
    - 4⋅x  + x 
-∞
          4     
       2⋅x  - 1 
 lim ───────────
x─→-∞     3    2
     - 4⋅x  + x 
∞

10.
         4     
      2⋅x  - 1 
lim ───────────
x─→∞     5    2
    - 4⋅x  + x 
0
          4     
       2⋅x  - 1 
 lim ───────────
x─→-∞     5    2
     - 4⋅x  + x 
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="0.01">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-100">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="100">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-0.1">
<label for="y2">y2 = </label>
<input id="y2" type="number" 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="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'),
    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.sin(x) / x,
    f2 = (x) => Math.cos(x) / x,
    f3 = (x) => Math.sin(4 * x) / x ** 3;

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']],
        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 =

Kamimura's blog

2017年7月12日水曜日

数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 曲線をえがくこと - xが大きくなるときの様子(正の無限大、負の無限大、商の極限、三角関数、正弦、余弦)

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