数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 平均値の定理 - 最大点と最小点の定理(臨界点、極大、極小)

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)、1(最大点と最小点の定理)、練習問題1-13.を取り組んでみる。

1. 
   $\begin{array}{l}2x-2=0\\ x=1\end{array}$
2. 
   $\begin{array}{l}4x-3=0\\ x=\frac{3}{4}\end{array}$
3. 
   $\begin{array}{l}6x-1=0\\ x=\frac{1}{6}\end{array}$
4. 
   $\begin{array}{l}-2x+2=0\\ x=1\end{array}$
5. 
   $\begin{array}{l}-4x+3=0\\ x=\frac{3}{4}\end{array}$
6. 
   $\begin{array}{l}3x^2=0\\ x=0\end{array}$
7. 
   $\begin{array}{l}3x^2-3=0\\ x^2=1\\ x=\pm 1\end{array}$
8. 
   $\begin{array}{l}\cos x-\sin x=0\\ \tan x=1\\ x=\frac{\pi }{4}+n\pi \end{array}$
9. 
   $\begin{array}{l}-\sin x=0\\ \sin x=0\\ x=n\pi \end{array}$
10. 
    $\begin{array}{l}\cos x=0\\ x=\frac{\pi }{2}+n\pi \end{array}$
11. 
    

    正方形の1辺の長さを$x$ 、高さを$y$ 、箱の体積をVとする。

    $\begin{array}{l}C=x^2+4xy\\ V=x^{2}y\\ y=-\frac{x}{4}+\frac{C}{4x}\\ V=-\frac{x^3}{4}+\frac{C}{4}x=-\frac{x}{4}\left(x^2-C\right)\\ V\text{'}=-\frac{3}{4}{x}^2+\frac{C}{4}\\ -\frac{3}{4}{x}^2+\frac{C}{4}=0\\ -3x^2+C=0\\ x=\sqrt{\frac{C}{3}}\\ y=-\frac{1}{4}\sqrt{\frac{C}{3}}+\frac{C}{4\sqrt{\frac{C}{3}}}\\ =-\frac{\sqrt{C}}{4\sqrt{3}}+\frac{C\sqrt{3}}{4\sqrt{C}}\\ =\frac{-C+3C}{4\sqrt{3C}}\\ =\frac{2C}{4\sqrt{3C}}\\ =\frac{\sqrt{C}}{2\sqrt{3}}\end{array}$
12. 
    

    底面の半径を$r$ 、高さを$h$ とする。

    $\begin{array}{l}C=h·2r\pi +\pi r^2\\ V=\pi r^{2}h\\ \\ h=\frac{C-\pi r^2}{2\pi r}\\ V=\pi r^2\frac{C-\pi r^2}{2\pi r}=\frac{r}{2}\left(C-\pi r^2\right)=\frac{Cr-\pi r^3}{2}\\ V\text{'}=\frac{C-3\pi r^2}{2}\\ \\ \frac{C-3\pi r^2}{2}=0\\ r=\sqrt{\frac{C}{3\pi }}\\ h=\frac{C-\pi r^2}{2\pi r}\\ =\frac{C-\pi \frac{C}{3\pi }}{2\pi \sqrt{\frac{C}{3\pi }}}\\ =\frac{C-\frac{C}{3}}{2\sqrt{\frac{\pi C}{3}}}\\ =\frac{2\sqrt{C}}{6\sqrt{\frac{\pi }{3}}}\\ =\sqrt{\frac{C}{3\pi }}\end{array}$
13. 11. 
        $\begin{array}{l}C=2x^2+4xy\\ V=x^{2}y\\ y=\frac{C-2x^2}{4x}\\ V=\frac{x\left(C-2x^2\right)}{4}\\ V\text{'}=\frac{1}{4}\left(C-6x^2\right)\\ C-6x^2=0\\ x=\sqrt{\frac{C}{6}}\\ y=\frac{C-2\frac{C}{6}}{4\sqrt{\frac{C}{6}}}\\ =\frac{2}{3}C·\frac{1}{4}\sqrt{\frac{6}{C}}\\ =\sqrt{\frac{C}{6}}\end{array}$
    12. 
        $\begin{array}{l}C=h·2r\pi +2\pi r^2\\ V=\pi r^{2}h\\ h=\frac{C-2\pi r^2}{2\pi r}\\ V=\pi r^2\frac{C-2\pi r^2}{2\pi r}=\frac{Cr-2\pi r^3}{2}\\ V\text{'}=\frac{C-6\pi r^2}{2}\\ C-6\pi r^2=0\\ r=\sqrt{\frac{C}{6\pi }}\\ h=\frac{C-2\pi \frac{C}{6\pi }}{2\pi \sqrt{\frac{C}{6\pi }}}\\ =\frac{C-\frac{C}{3}}{2\sqrt{\frac{C\pi }{6}}}\\ =\frac{2C\sqrt{6}}{3·2\sqrt{C\pi }}\\ =\sqrt{\frac{2C}{3\pi }}\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sin, cos, solve, Derivative, plot

x = symbols('x')
fs = [x ** 2 - 2 * x + 5, sin(x) + cos(x)]
for i, f in enumerate(fs):
    d = Derivative(f, x, 1)
    f1 = d.doit()
    pprint(d)
    pprint(f1)
    pprint(solve(f1, x, dict=True))
    p = plot(f, f1, show=False, legend=True)
    for i, color in enumerate(['green', 'blue']):
        p[i].line_color = color
    p.save('sample1_{}.svg'.format(i))
    print()

入出力結果(Terminal, IPython)

$ ./sample1.py
d ⎛ 2          ⎞
──⎝x  - 2⋅x + 5⎠
dx              
2⋅x - 2
[{x: 1}]

d                  
──(sin(x) + cos(x))
dx                 
-sin(x) + cos(x)
⎡⎧   -3⋅π ⎫  ⎧   π⎫⎤
⎢⎨x: ─────⎬, ⎨x: ─⎬⎥
⎣⎩     4  ⎭  ⎩   4⎭⎦

$

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="x0">x0 = </label>
<input id="x0" type="number" step="0.01" 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="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_x0 = document.querySelector('#x0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
             input_x0],
    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.sin(x) + Math.cos(x),
    f1 = (x) => Math.cos(x) - Math.sin(x),
    g = (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),
        x0 = parseFloat(input_x0.value);
        
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;n
    }
    let points = [],
        lines = [];

    [[f, 'red'], [f1, 'green']]
        .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]);
                }
            }
        });
    [[g(x0), 'blue']]
        .forEach((o) => {
            let [f, color] = o;

            lines.push([x1, f(x1), x2, f(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(${padding}, 0)`)
        .call(yaxis);
    p(`f1(${x0}) = ${f1(x0)}`);
};

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

f1(1) = -0.30116867893975674

r = dx =
x1 = x2 =
y1 = y2 =
x0 =