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(最大値、最小値、最大点、最小点、累乗、有理数)

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13-22.を取り組んでみる。

13. 
    $\begin{array}{l}f\text{'}\left(x\right)=2x-2=2\left(x-1\right)\\ f\left(1\right)=1-2-8=-9\\ f\left(0\right)=-8\\ f\left(4\right)=16-8-8=0\end{array}$
    1. 
       最大値: 、最小値: $-9$
    2. 
       最大値:$0$ 、最小値: $-9$
    3. 
       増加区間: $x\ge 1$ 、減少区間: $x\le 1$
14. 
    $\begin{array}{l}f\text{'}\left(x\right)=2x-2=2\left(x-1\right)\\ f\left(1\right)=1-2+1=0\\ f\left(-1\right)=1+2+1=4\\ f\left(4\right)=16-8+1=9\end{array}$
    1. 
       最大値: 、最小値:$0$
    2. 
       最大値: $9$ 、最小値: $0$
    3. 
       増加区間: $x\ge 1$ 、減少区間: $x\le 1$
15. 
    $\begin{array}{l}f\text{'}\left(x\right)=-4-2x=-2\left(x+2\right)\\ f\left(-2\right)=4+8-4=8\\ f\left(-1\right)=4+4-1=7\\ f\left(4\right)=4-16-16=-28\end{array}$
    1. 
       最大値: $7$ 、最小値:
    2. 
       最大値: $8$ 、最小値: $-28$
    3. 
       増加区間: $x\le -2$ 、減少区間: $x\ge -2$
16. 
    $\begin{array}{l}f\text{'}\left(x\right)=1-2x\\ f\left(\frac{1}{2}\right)=\frac{1}{2}-\frac{1}{4}=\frac{1}{4}\\ f\left(-1\right)=-1-1=-2\\ f\left(2\right)=2-4=-2\end{array}$
    1. 
       最大点: $\left(\frac{1}{2},\frac{1}{4}\right)$ 、最小点:
    2. 
       最大点: $\left(\frac{1}{2},\frac{1}{4}\right)$ 、最小点: $\begin{array}{l}\left(-1,-2\right)\\ \left(2,-2\right)\end{array}$
    3. 
       増加区間: $x\le \frac{1}{2}$ 、減少区間: $x\ge \frac{1}{2}$
17. 
    $\begin{array}{l}f\text{'}\left(x\right)=3-3x^2=-3\left(x^2-1\right)\\ f\left(-1\right)=-3+1=-2\\ f\left(1\right)=3-1=2\\ f\left(-2\right)=-6+8=2\\ f\left(\sqrt{3}\right)=3\sqrt{3}-3\sqrt{3}=0\end{array}$
    1. 
       最大点: 、最小点:
    2. 
       最大点: $\begin{array}{l}\left(-2,2\right)\\ \left(1,2\right)\end{array}$ 、最小点: $\left(-1,2\right)$
    3. 
       増加区間: $-1\le x\le 1$ 、減少区間: $x\le -1,1\le x$
18. 
    $\begin{array}{l}f\text{'}\left(x\right)=5{\left(x-4\right)}^4\\ f\left(4\right)=0\\ f\left(3\right)=-1\\ f\left(6\right)=32\end{array}$
    1. 
       最大点: 、最小点:
    2. 
       最大点: $\left(6,32\right)$ 、最小点: $\left(3,-1\right)$
    3. 
       増加区間: $\left(-\infty ,\infty \right)$ 、減少区間: $\varphi$
19. 
    $\begin{array}{l}f\text{'}\left(x\right)=4x^3-4x=4x\left(x^2-1\right)\\ f\left(-1\right)=1-2=-1\\ f\left(1\right)=1-2=-1\\ f\left(-2\right)=16-8=8\\ f\left(1\right)=1-2=-1\end{array}$
    1. 
       最大点: 、最小点: $\begin{array}{l}\left(-1,-1\right)\\ \left(1,-1\right)\end{array}$
    2. 
       最大点: $\left(-2,8\right)$ 、最小点: $\begin{array}{l}\left(-1,-1\right)\\ \left(1,-1\right)\end{array}$
    3. 
       増加区間: $-1\le x\le 0,1\le x$ 、減少区間: $x\le -1,0\le x\le 1$
20. 
    $\begin{array}{l}f\text{'}\left(x\right)=\frac{1}{3}{\left(x-1\right)}^{-\frac{2}{3}}+\frac{1}{2}·\frac{2}{3}{\left(x+1\right)}^{-\frac{1}{3}}\\ =\frac{1}{3}\left({\left(x-1\right)}^{-\frac{2}{3}}+{\left(x+1\right)}^{-\frac{1}{3}}\right)>0\\ f\left(-2\right)={\left(-3\right)}^{\frac{1}{3}}+\frac{1}{2}{\left(-1\right)}^{\frac{2}{3}}={\left(-3\right)}^{\frac{1}{3}}+\frac{1}{2}>0\\ f\left(7\right)=6^{\frac{1}{3}}+\frac{7^{\frac{2}{3}}}{2}>f\left(-2\right)\end{array}$
    1. 
       最大点: 、最小点:
    2. 
       最大点: $\left(7,6^{\frac{1}{3}}+\frac{7^{\frac{2}{3}}}{2}\right)$ 、最小点: $\left(-2,{\left(-3\right)}^{\frac{1}{3}}+\frac{1}{2}\right)$
    3. 
       増加区間: $\left(-\infty ,\infty \right)$ 、減少区間: $\varphi$
21. 
    $\begin{array}{l}f\text{'}\left(x\right)=\frac{2}{5}{x}^{-\frac{3}{5}}\\ f\left(0\right)=1\\ f\left(-1\right)=1+1=2\\ f\left(1\right)=1+1=2\end{array}$
    1. 
       最大点: 、最小点: $\left(0,1\right)$
    2. 
       最大点: $\left(\pm 1,2\right)$ 、最小点: $\left(0,1\right)$
    3. 
       増加区間: $x\ge 0$ 、減少区間: $x\le 0$
22. 
    $\begin{array}{l}f\text{'}\left(x\right)=\frac{1}{2}{\left(x^2+1\right)}^{-\frac{1}{2}}2=x{\left(x^2+1\right)}^{-\frac{1}{2}}\\ f\left(0\right)=1\\ f\left(\sqrt{8}\right)=3\end{array}$
    1. 
       最大点: 、最小点: $\left(0,1\right)$
    2. 
       最大点: $\left(\sqrt{8},3\right)$ 、最小点: $\left(0,1\right)$
    3. 
       増加区間: $x\ge 0$ 、減少区間: $x\le 0$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sqrt, Derivative, solve, plot, Rational, Pow

x = symbols('x', nonnegative=True)
fs = [Pow(x - 1, Rational(1, 3)) + Rational(1, 2) * Pow(x + 1, Rational(2, 3)),
      Pow(x, Rational(2, 5)) + 1,
      sqrt(Pow(x, 2) + 1)]

p = plot(*fs, show=False, legend=True)

for i, (f, color) in enumerate(zip(fs, ['red', 'green', 'blue'])):
    print('{0}.'.format(20 + i))
    d = Derivative(f, x)
    pprint(d)
    f1 = d.doit()
    pprint(solve(f1, x))
    p[i].line_color = color

p.save('sample13.svg')

入出力結果(Terminal, IPython)

$ ./sample13.py
20.
  ⎛                   2/3⎞
d ⎜3 _______   (x + 1)   ⎟
──⎜╲╱ x - 1  + ──────────⎟
dx⎝                2     ⎠
[]
21.
d ⎛ 2/5    ⎞
──⎝x    + 1⎠
dx          
[]
22.
  ⎛   ________⎞
d ⎜  ╱  2     ⎟
──⎝╲╱  x  + 1 ⎠
dx             
[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="0.005">
<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="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">

<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="sample13.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 factorial = (n) => range(1, n + 1).reduce((prev, next) => prev * next, 1);

let f = (x) => (x - 1) ** (1 / 3) + 1 / 2 * ((x + 1) ** 2) ** (1 / 3),
    g = (x) => (x ** 2) ** (1 / 5) + 1,
    h = (x) => Math.sqrt(x ** 2 + 1);

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;n
    }
    let points = [],
        lines = [],
        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 =