数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 平均値の定理 - 増加・減少関数(正弦、余弦、不等式、一般式、定数)

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5章(平均値の定理)、3(増加・減少関数)、練習問題11-12.を取り組んでみる。

11. 1. 
       $\begin{array}{l}{f}_1\text{'}\left(x\right)=1-\cos x\ge 0\\ f_1\left(0\right)=0-0=0\\ f_1\left(x\right)\ge 0\end{array}$

       増加関数。

    2. 
       $\begin{array}{l}{f}_2\text{'}\left(x\right)=x-\sin x\ge 0\\ f_2\left(0\right)=-1+0+1=0\\ f_2\left(x\right)\ge 0\end{array}$

       増加関数。

       $\begin{array}{l}{f}_3\text{'}\left(x\right)=-1+\frac{x^2}{2}+\cos x\ge 0\\ f_3\left(0\right)=0+0+0=0\\ f_3\left(x\right)\ge 0\end{array}$

       増加関数。

       $\begin{array}{l}{f}_4\text{'}\left(x\right)=-x+\frac{x^3}{3·2}+\sin x\ge 0\\ f_4\left(0\right)=1-0+0-1=0\\ f_4\left(x\right)\ge 0\end{array}$

       増加関数。

       $\begin{array}{l}{f}_5\text{'}\left(x\right)=1-\frac{x^2}{2}+\frac{x^4}{4·3·2}-\cos x\ge 0\\ f_5\left(0\right)=0-0+0-0=0\\ f_5\left(x\right)\ge 0\end{array}$

       増加関数。

    3. 
       $\begin{array}{l}{f}_n\left(x\right)={\displaystyle \sum _{i=1}^n{\left(-1\right)}^{i-1}\frac{x^{2i-1}}{\left(2i-1\right)!}}\\ f_{2k}\left(x\right)\le \sin x\le f_{2k+1}\left(x\right)\\ \\ g_n\left(x\right)={\displaystyle \sum _{i=0}^n{\left(-1\right)}^i\frac{x^{2i}}{\left(2i\right)!}}\\ g_{2k-1}\left(x\right)\le \cos x\le g_{2k}\left(x\right)\end{array}$
12. 
    $\begin{array}{l}\frac{d}{dx}\left(\frac{g\left(x\right)}{f\left(x\right)}\right)\\ =\frac{g\text{'}\left(x\right)f\left(x\right)-g\left(x\right)f\text{'}\left(x\right)}{{\left(f\left(x\right)\right)}^2}\\ =\frac{g\left(x\right)f\left(x\right)-g\left(x\right)f\text{'}\left(x\right)}{{\left(f\left(x\right)\right)}^2}\\ =0\\ \frac{g\left(x\right)}{f\left(x\right)}=C\\ g\left(x\right)=Cf\left(x\right)\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sin, cos, factorial, Derivative

print('11.')
x = symbols('x', nonnegative=True)
eqs = [x - sin(x),
       -1 + x ** 2 / 2 + cos(x),
       -x + x ** 3 / factorial(3) + sin(x),
       1 - x ** 2 / 2 + x ** 4 / factorial(4) - cos(x),
       x - x ** 3 / factorial(3) + x ** 5 / factorial(5) - sin(x)]

for i, eq in enumerate(eqs, 1):
    print('{0}.'.format(i))
    pprint(eq)
    pprint(Derivative(eq, x).doit())
    print(eq.subs({x: 0}) == 0)
    print()

入出力結果(Terminal, IPython)

$ ./sample11.py
11.
1.
x - sin(x)
-cos(x) + 1
True

2.
 2             
x              
── + cos(x) - 1
2              
x - sin(x)
True

3.
 3             
x              
── - x + sin(x)
6              
 2             
x              
── + cos(x) - 1
2              
True

4.
 4    2             
x    x              
── - ── - cos(x) + 1
24   2              
 3             
x              
── - x + sin(x)
6              
True

5.
  5    3             
 x    x              
─── - ── + x - sin(x)
120   6              
 4    2             
x    x              
── - ── - cos(x) + 1
24   2              
True

$

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="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<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="n0">n = </label>
<input id="n0" type="number" min="2" step="1" value="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="sample11.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_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_n0],
    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 factorial = (n) => range(1, n + 1).reduce((prev, next) => prev * next, 1);

let term = (i) => (x) =>
    (-1) ** (i - 1) * x ** (2 * i - 1) / factorial(2 * i - 1),
    fn = (n) => (x) =>
    range(1, n + 1).reduce((prev, i) => prev + term(i)(x), 0);

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),
        n0 = parseFloat(input_n0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;n
    }
    let points = [],
        lines = [],
        f = fn(2 * n0),
        g = fn(2 * n0 + 1),
        h = Math.sin,
        fns = [[f, 'red'], [g, 'green'], [h, 'blue']];

    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]);
                }
            }
        });                 
    
    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);

    p(fns.join('\n'));
};

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

r = dx =
x1 = x2 =
y1 = y2 =
n =