数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 正弦と余弦 - グラフ(sine、累乗、逆数、微分係数、ニュートン商)

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4章(正弦と余弦)、2(グラフ)、練習問題9、10.を取り組んでみる。

9. 
   $\begin{array}{l}\frac{h^2\sin \frac{1}{h}-0}{h}\\ =h\sin \frac{1}{h}\\ \left|h\sin \frac{1}{h}\right|\le \left|h\right|\\ \underset{h\to 0}{\lim }\left|h\right|=0\\ \underset{h\to 0}{\lim }h\sin \frac{1}{h}=0\\ f\text{'}\left(0\right)=0\end{array}$
10. 
    $\begin{array}{l}\frac{h\sin \frac{1}{h}-0}{h}\\ =\sin \frac{1}{h}\\ =\sin \frac{1}{\frac{2}{n\pi }}\\ =\sin \frac{n}{2}\pi \\ n=1,2,3,4,5\\ 1,0,-1,0,1\cdots \\ n=4k+1,\sin \frac{n}{2}\pi =1\\ n=4k+2,\sin \frac{n}{2}\pi =0\\ n=4k+3,\sin \frac{n}{2}\pi =-1\\ n=4k,\sin \frac{n}{2}\pi =0\end{array}$

コード(Emacs)

Python 3

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

from sympy import symbols, sin, Limit, pprint, pi, plot

print('9')
x, h = symbols('x h')

f = x ** 2 * sin(1 / x)

l = Limit((f.subs({x: h}) - 0) / h, h, 0)
pprint(l)
pprint(l.doit())

print('10')
g = x * sin(1 / x)
g1 = (g.subs({x: h}) - 0) / h
l = Limit(g1, h, 0)
pprint(l)
pprint(l.doit())

for n in range(1, 10):
    print('n = {0}'.format(n))
    pprint(2 / (n * pi))
    pprint(g1.subs({h: 2 / (n * pi)}))
    print()

p = plot(f, g, legend=True, show=False)
p[0].line_color = 'g'
p[1].line_color = 'b'
p.save('sample9.svg')

入出力結果(Terminal, IPython)

$ ./sample9.py
9
          ⎛1⎞
 lim h⋅sin⎜─⎟
h─→0⁺     ⎝h⎠
0
10
        ⎛1⎞
 lim sin⎜─⎟
h─→0⁺   ⎝h⎠
<-1, 1>
n = 1
2
─
π
1

n = 2
1
─
π
0

n = 3
 2 
───
3⋅π
-1

n = 4
 1 
───
2⋅π
0

n = 5
 2 
───
5⋅π
1

n = 6
 1 
───
3⋅π
0

n = 7
 2 
───
7⋅π
-1

n = 8
 1 
───
4⋅π
0

n = 9
 2 
───
9⋅π
1

$

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="0.00001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-0.1">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="0.1">
<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="sample9.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 f = (x) => x ** 2 * Math.sin(1 / x),
    g = (x) => x * Math.sin(1 / x);

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 = [];
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        let y = g(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('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.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 < t ? 'green' : 'blue');
    
    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 =