数学 - Python - JavaScript - 曲線の性質、最大・最小 - 微分法の応用 – 関数の増減の判定およびその応用 - 導関数の符号と関数の増減(極値、3次関数)

数学読本〈5〉

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

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第18章(曲線の性質、最大・最小 - 微分法の応用)、18.2(関数の増減の判定およびその応用)、導関数の符号と関数の増減、問2、3、4、5.を取り組んでみる。

2. 
   $\begin{array}{l}y\text{'}=2ax+b\\ 2ax+b=0\\ x=-\frac{b}{2a}\end{array}$

   x		-b/2a
   y'	-	0	+
   y	↘︎		↗︎

3. 
   $\begin{array}{l}y\text{'}=3x^2+2ax+b\\ 3x^2+2ax+b=0\\ \frac{D}{4}=a^2-3b\end{array}$
4. 1. 
      $\begin{array}{l}y\text{'}=12-3x^2\hfill \\ 12-3x^2=0\hfill \\ x=\pm 2\hfill \\ -24+8=-16\hfill \\ 24-8=16\hfill \end{array}$

      x		-2		2
      y'	-	0	+	0	-
      y	↘︎	-16	↗︎	16	↘︎

   2. 
      $\begin{array}{l}y\text{'}=x^2+2x\\ x^2+2x=0\\ x\left(x+2\right)=0\\ x=0,-2\\ 0\\ -\frac{8}{3}+4=\frac{4}{3}\end{array}$

      x		-2		0
      y'	+	0	-	0	+
      y	↗︎	4/3	↘︎	0	↗︎

   3. 
      $y\text{'}=3x^2+1\ge 1$

      増加関数で、極値はない。

   4. 
      $\begin{array}{l}y\text{'}=-2+2x-3x^2\\ 3x^2-2x+2=0\\ \frac{D}{4}=1-6=-5<0\\ y\text{'}<0\end{array}$

      減少関数で、極値はない。

   5. 
      $\begin{array}{l}y\text{'}=4x^3-4x\\ =4x\left(x^2-1\right)\\ =4x\left(x+1\right)\left(x-1\right)\\ \\ 1-2+1=0\\ 1\\ 1-2+1=0\end{array}$

      x		-1		0		1
      y'	-	0	+	0	-	0	+
      y	↘︎	0	↗︎	1	↘︎	0	↗︎

   6. 
      $\begin{array}{l}y\text{'}=4x^3+2x\\ =2x\left(x^2+1\right)\\ 0\end{array}$

      x		0
      y'	-	0	+
      y	↘︎	0	↗︎

   7. 
      $\begin{array}{l}y\text{'}=4x^3-12x^2\\ =4x^2\left(x-3\right)\\ 0\\ -27\end{array}$

      x		0		3
      y'	-	0	-	0	+
      y	↘︎	0	↘︎	-27	↗︎

   8. 
      $\begin{array}{l}y=\left|x^2\left(x-3\right)\right|\\ x<3\\ y=-x^3+3x^2\\ y\text{'}=-3x^2+6x\\ =-3x\left(x-2\right)\\ x\ge 3\\ y=x^3-3x^2\\ y\text{'}=3x^2-6x\\ =3x\left(x-2\right)\\ 0\\ -8+3·4=4\\ 27-27=0\end{array}$

      x		0		2		3
      y'	-	0	+	0	-		+
      y	↘︎	0	↗︎	4	↘︎	0	↗︎

   9. 
      $\begin{array}{l}y\text{'}=1-\frac{1}{x^2}\\ 1-\frac{1}{x^2}=0\\ x^2-1=0\\ x=\pm 1\\ -1-1=-2\\ 1+1=2\end{array}$

      x		-1		0		1
      y'	+	0	-		-	0	+
      y	↗︎	-2	↘︎		↘︎	2	↗︎

   10. 
       $\begin{array}{l}y=e^x+x{e}^x\\ e^x+x{e}^x=0\\ x=-1\\ e^0+0e^0=1\end{array}$

       x		-1
       y'	-	0	+
       y	↘︎	1	↗︎

   11. 
       $\begin{array}{l}y\text{'}=1+2\cos x\\ 1+2\cos x=0\\ \cos x=-\frac{1}{2}\\ x=\frac{2}{3}\pi ,\frac{4}{3}\pi \\ \frac{2}{3}\pi +2\frac{\sqrt{3}}{2}=\frac{2}{3}\pi +\sqrt{3}\\ \frac{4}{3}\pi +2·\left(-\frac{\sqrt{3}}{2}\right)=\frac{4}{3}\pi -\sqrt{3}\end{array}$

       x		2π/3		4π/3
       y'	+	0	-	0	+
       y	↗︎	2π/3 + √3	↘︎	4π/3 - √3	↗︎

5. 1. 
      $\begin{array}{l}y=a{x}^3+b{x}^2+cx+d\\ y\text{'}=3a{x}^2+2bx+c\\ \\ 3a-2b+c=0\\ -a+b-c+d=10\\ 12a+4b+c=0\\ 8a+4b+2c+d=-17\\ \\ c=-3a+2b\\ \\ -a+b+3a-2b+d=10\\ 12a+4b-3a+2b=0\\ 8a+4b-6a+4b+d=-17\\ \\ 2a-b+d=10\\ 3a+2b=0\\ 2a+8b+d=-17\\ \\ d=10-2a+b\\ \\ 3a+2b=0\\ 2a+8b+10-2a+b=-17\\ \\ 9b=-27\\ b=-3\\ a=2\\ d=10-4-3=3\\ c=-6-6=-12\\ \\ y=2x^3-3x^2-12x+3\end{array}$
   2. 
      $\begin{array}{l}c=0\\ d=-8\\ 12a+4b+c=0\\ 8a+4b+2c+d=0\\ \\ 12a+4b=0\\ 8a+4b-8=0\\ \\ 3a+b=0\\ 2a+b-2=0\\ b=-3a\\ 2a-3a-2=0\\ a=-2\\ b=6\\ y=-2x^3+6x^2-8\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sin, plot, pi, sqrt

print('4 (11).')
x = symbols('x')
f = x + 2 * sin(x)
g = 2 * pi / 3 + sqrt(3)
h = 4 * pi / 3 - sqrt(3)

p = plot(f, g, h, (x, 0, 2 * pi), show=False, legend=True)
for i, color in enumerate(['red', 'green', 'blue']):
    p[i].line_color = color

p.save('sample2.svg')

入出力結果(Terminal, IPython)

$ ./sample2.py
4 (11).
$

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="-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">
<br>
<label for="a0">a = </label>
<input id="a0" type="number" value="-2">
<label for="b0">b = </label>
<input id="b0" type="number" value="6">
<label for="c0">c = </label>
<input id="c0" type="number" value="0">
<label for="d0">d = </label>
<input id="d0" type="number" value="-8">
<br>
<label for="x0">x0 = </label>
<input id="x0" type="number" value="0">


<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="sample2.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_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),
    input_c0 = document.querySelector('#c0'),
    input_d0 = document.querySelector('#d0'),
    input_x0 = document.querySelector('#x0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_b0, input_c0, input_d0, 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 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),
        a = parseFloat(input_a0.value),
        b = parseFloat(input_b0.value),
        c = parseFloat(input_c0.value),
        d = parseFloat(input_d0.value),
        x0 = parseFloat(input_x0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        lines = [],
        f = (x) => a * x ** 3 + b * x ** 2 + c * x + d,
        f1 = (x) => 3 * a * x ** 2 + 2 * b * x + c,
        g = (x) => f1(x0) * (x - x0) + f(x0),
        fns = [[f, 'green']],
        fns1 = [[g, 'blue']];

    fns.forEach((o) => {
        let [fn, color] = o;
        for (let x = x1; x <= x2; x += dx) {
            let y = fn(x);
            if (Math.abs(y) < Infinity) {
                points.push([x, y, color]);
            }
        }
    });
    fns1.forEach((o) => {
        let [fn, color] = o;
        lines.push([x1, fn(x1), x2, fn(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(fns);
};

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

r = dx =
x1 = x2 =
y1 = y2 =
a = b = c = d =
x0 =