数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 正弦と余弦 - 導関数(正接(tangent)、正割(secant)、余接(cotangent)、sine、cosine、曲線の傾き)

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)、4(導関数)、練習問題1-15.を取り組んでみる。

1. 1. 
      $\begin{array}{l}\frac{d}{dx}\tan x=\frac{d}{dx}\left(\frac{\sin x}{\cos x}\right)\\ =\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}\\ =\frac{1}{{\cos }^{2}x}\\ ={\sec }^{2}x\end{array}$
   2. 
      $\begin{array}{l}\frac{d}{dx}\cot x=\frac{d}{dx}\left(\frac{1}{\tan x}\right)\\ =\frac{-{\sec }^{2}x}{{\tan }^{2}x}\\ =-\frac{\frac{1}{{\cos }^{2}x}}{\frac{{\sin }^{2}x}{{\cos }^{2}x}}\\ =-\frac{1}{{\sin }^{2}x}\end{array}$
2. 
   $3\cos \left(3x\right)$
3. 
   $-5\sin \left(5x\right)$
4. 
   $\left(8x+1\right)\cos \left(4x^2+x\right)$
5. 
   $\frac{3x^2}{\cos \left(x^3-5\right)}$
6. 
   $\frac{4x^3-3x^2}{\cos \left(x^4-x^3\right)}$
7. 
   $\frac{\cos x}{{\cos }^2\left(\sin x\right)}$
8. 
   $\frac{\cos \left(\tan x\right)}{{\cos }^{2}x}$
9. 
   $-\frac{\sin \left(\tan x\right)}{{\cos }^{2}x}$
10. 
    $\begin{array}{l}y\text{'}=\cos x\\ \cos \pi =-1\end{array}$
11. 
    $\begin{array}{l}y\text{'}=-3\sin \left(3x\right)\\ -3\sin \pi =0\end{array}$
12. 
    $\begin{array}{l}y\text{'}=\cos x\\ \cos \frac{\pi }{6}=\frac{\sqrt{3}}{2}\end{array}$
13. 
    $\begin{array}{l}y\text{'}=\cos x-\sin x\\ \cos \frac{3}{4}\pi -\sin \frac{3}{4}\pi =-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}=-\sqrt{2}\end{array}$
14. 
    $\begin{array}{l}y\text{'}=\frac{1}{{\cos }^{2}x}\\ 2\end{array}$
15. 
    $\begin{array}{l}y\text{'}=\frac{-\cos x}{{\sin }^{2}x}\\ -\frac{\frac{\sqrt{3}}{2}}{\frac{1}{4}}=-2\sqrt{3}\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, tan, cot, Derivative

print('1.')
x = symbols('x')
for i, f in enumerate([tan, cot]):
    print('({})'.format(chr(ord('a') + i)))
    d = Derivative(f(x), x, 1)
    pprint(d)
    f1 = d.doit()
    pprint(f1)
    print()

入出力結果(Terminal, IPython)

$ ./sample1.py
1.
(a)
d         
──(tan(x))
dx        
   2       
tan (x) + 1

(b)
d         
──(cot(x))
dx        
     2       
- cot (x) - 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="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" 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'),
    input_dx0 = document.querySelector('#dx0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
             input_dx0],
    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) => Math.tan(x),
    f1 = (x) => 1 / Math.cos(x) ** 2,
    g = (x) => 1 / Math.tan(x),
    g1 = (x) => -1 / Math.sin(x) ** 2,
    h = (f, f1, x0) => (x) => f1(x0) * (x - x0) + f(x0);

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),
        dx0 = parseFloat(input_dx0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;n
    }

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

        if (Math.abs(y) < Infinity) {
            points.push([x, y, 'red']);
        }
    }
    for (let x = x1; x <= x2; x += dx) {
        let y = g(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y, 'green']);
        }
    }
    for (let x = x1; x <= x2; x += dx0) {
        let fh = h(f, f1, x);

        lines.push([x1, fh(x1), x2, fh(x2), 'blue']);
    }
    for (let x = x1; x <= x2; x += dx0) {
        let gh = h(g, g1, x);

        lines.push([x1, gh(x1), x2, gh(x2), 'purple']);
    }
    
    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);
};

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

r = dx =
x1 = x2 =
y1 = y2 =
dx0 =