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 - 最大・最小問題(関数の最大値、最小値、直円柱の全表面積、体積、半径と高さの比率)

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)、最大・最小問題、問18、19、20.を取り組んでみる。

18. 
    $\begin{array}{l}f\text{'}\left(x\right)=3x^2-12\\ =3\left(x^2-4\right)\\ =3\left(x+2\right)\left(x-2\right)\\ f\left(-2\right)=-8+24+10=26\\ f\left(2\right)=8-24+10=-6\\ f\left(-3\right)=-27+36+10=19\\ f\left(5\right)=125-60+10=75\end{array}$

    最大値 75、最小値 -6。

19. 
    $\begin{array}{l}y=1-x\\ 0\le x\le 1\\ f\left(x\right)=x^3+2{\left(1-x\right)}^3\\ f\text{'}\left(x\right)=3x^2+6{\left(1-x\right)}^2\left(-1\right)\\ =3x^2-6\left(x^2-2x+1\right)\\ =-3\left(-x^2+2\left(x^2-2x+1\right)\right)\\ =-3\left(x^2-4x+2\right)\\ x^2-4x+2=0\\ x=2\pm \sqrt{4-2}=2\pm \sqrt{2}\\ x=2-\sqrt{2}\\ f\left(0\right)=2\\ f\left(1\right)=1\\ f\left(2-\sqrt{2}\right)={\left(2-\sqrt{2}\right)}^3+2{\left(1-\left(2-\sqrt{2}\right)\right)}^3\\ =8+3·2·2+3·4·\left(-\sqrt{2}\right)-2\sqrt{2}+2{\left(\sqrt{2}-1\right)}^3\\ =8+12-12\sqrt{2}-2\sqrt{2}+2\left(2\sqrt{2}+3\sqrt{2}+3·2\left(-1\right)-1\right)\\ =20-14\sqrt{2}+4\sqrt{2}+6\sqrt{2}-12-2\\ =6-4\sqrt{2}\end{array}$

    最大値 2、最小値 $6-4\sqrt{2}$

20. 
    $\begin{array}{l}S=h·2\pi r+2\pi r^2\\ h=\frac{S-2\pi r^2}{2\pi r}\\ V=h\pi r^2\\ =\frac{S-2\pi r^2}{2\pi r}·\pi r^2\\ =\frac{1}{2}r\left(S-2\pi r^2\right)\\ \frac{d}{dr}V=\frac{1}{2}\left(S-6\pi r^2\right)\\ 6\pi r^2=S\\ r=\sqrt{\frac{S}{6\pi }}\\ h=\frac{S-2\pi ·\frac{S}{6\pi }}{2\pi \frac{S}{6\pi }}\\ =\frac{\frac{2S}{3}}{\frac{1S}{3}}\\ =2\\ r:h=1:2\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, plot

x = symbols('x')
fs = [(x ** 3 - 12 * x + 10, (-3, 5)),
      (x ** 3 + 2 * (1 - x) ** 3, (0, 1))]

for i, (f, (x1, x2)) in enumerate(fs, 18):
    print('{0}.'.format(i))
    pprint(f)
    p = plot(f, (x, x1, x2), show=False)
    p.save('sample{0}.svg'.format(i))
    print()

入出力結果(Terminal, IPython)

$ ./sample18.py
18.
 3            
x  - 12⋅x + 10

19.
 3             3
x  + 2⋅(-x + 1) 

$

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="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">
<br>

<label for="s0">S = </label>
<input id="s0" type="number" min="0" 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="sample18.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_s0 = document.querySelector('#s0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
             input_s0],
    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),
        s0 = parseFloat(input_s0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],        
        lines = [],
        h = (x) => (s0 - 2 * Math.PI * x ** 2) / (2 * Math.PI * r),
        v = (x) => h(x) * Math.PI * x ** 2,
        f = (x) => h(x) / x,
        fns = [[(x) => x, 'red'], [h, 'green'], [v, 'blue'], [f, 'brown']];

    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]);
            }
        }
    });
    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();

(x) => x,red
(x) => (s0 - 2 * Math.PI * x ** 2) / (2 * Math.PI * r),green
(x) => h(x) * Math.PI * x ** 2,blue
(x) => h(x) / x,brown

r = dx =
x1 = x2 =
y1 = y2 =
S =