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(増加の範囲、減少の範囲、導関数、符号)

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1-10.を取り組んでみる。

1. 
   $f\text{'}\left(x\right)=3x^2\ge 0$

   増加である範囲: $\left(-\infty ,\infty \right)$

   減少である範囲: $\varphi$

2. 
   $\begin{array}{l}f\text{'}(x)=(x-2)(x-3)+(x-1)(x-3)+(x-1)(x-2)\hfill \\ =3x^2-12x+11\hfill \\ x=\frac{6\pm \sqrt{36-33}}{3}=2\pm \frac{1}{\sqrt{3}}\hfill \end{array}$

   増加の範囲: $x\le 2-\frac{1}{\sqrt{3}},2+\frac{1}{\sqrt{3}}\le x$

   減少の範囲: $2-\frac{1}{\sqrt{3}}\le x\le 2+\frac{1}{\sqrt{3}}$

3. 
   $f\text{'}\left(x\right)=2x-1$

   増加の範囲: $\frac{1}{2}\le x$

   減少の範囲: $x\le \frac{1}{2}$

4. 
   $f\text{'}\left(x\right)=\cos x-\sin x$

   増加の範囲: $2n\pi \le x\le \frac{\pi }{4}+2n\pi ,\frac{5}{4}\pi +2n\pi \le x\le 2\pi +2n\pi$

   減少の範囲: $\frac{\pi }{4}+2n\pi \le x\le \frac{5}{4}\pi +2n\pi$

5. 
   $f\text{'}\left(x\right)=2\cos 2x$

   増加の範囲: $0\le x\le \frac{\pi }{4},\frac{3}{4}\pi \le x\le \frac{5}{4}\pi ,\frac{7}{4}\pi \le x\le 2\pi$

   減少の範囲: $\frac{\pi }{4}\le x\le \frac{3}{4}\pi ,\frac{5}{4}\pi \le x\le \frac{7}{4}\pi$

6. 
   $\begin{array}{l}f\text{'}\left(x\right)=4x^3-6x=2x\left(2x^2-3\right)\\ x=0,\pm \sqrt{\frac{3}{2}}\end{array}$

   増加の範囲: $-\sqrt{\frac{3}{2}}\le x\le 0,\sqrt{\frac{3}{2}}\le x$

   減少の範囲: $x\le -\sqrt{\frac{3}{2}},0\le x\le \sqrt{\frac{3}{2}}$

7. 
   $f\text{'}\left(x\right)=3x^2+1>0$

   増加の範囲: $\left(-\infty ,\infty \right)$

   減少の範囲: $\varphi$

8. 
   $\begin{array}{l}f\text{'}\left(x\right)=-3x^2+2\\ x=\pm \sqrt{\frac{2}{3}}\end{array}$

   増加の範囲: $-\sqrt{\frac{2}{3}}\le x\le \sqrt{\frac{2}{3}}$

   減少の範囲: $x\le -\sqrt{\frac{2}{3}},\sqrt{\frac{2}{3}}\le x$

9. 
   $f\text{'}\left(x\right)=6x^2\ge 0$

   増加の範囲: $\left(-\infty ,\infty \right)$

   減少の範囲: $\varphi$

10. 
    $f\text{'}\left(x\right)=10x$

    増加の範囲: $0\le x$

    減少の範囲: $x\le 0$

コード(Emacs)

Python 3

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

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

x = symbols('x')
f = sin(x) + cos(x)
f1 = Derivative(f, x).doit()

pprint(solve(f1, x, dict=True))

p = plot(f, show=False)
p.save('sample1.svg')

入出力結果(Terminal, IPython)

$ ./sample1.py
⎡⎧   -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="-4">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="4">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-4">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="4">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0.1" 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) => x ** 4 - 3 * x ** 2 + 1,
    f1 = (x) => 4 * x ** 3 - 6 * 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),
        dx0 = parseFloat(input_dx0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;n
    }
    let points = [],
        lines = [],
        fns = [[f, 'red']],
        fns1 = [[g, 'blue']];

    fns1
        .forEach((o) => {
            let [f, color] = o;

            for (let x = x1; x <= x2; x += dx0) {
                let g = f(x);
                
                lines.push([x1, g(x1), x2, g(x2), color]);
            }
        });

    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);
    
};

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

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