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)

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

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。

$\begin{array}{l} f' (x) = 3 x^{2} - 12 \\ = 3 (x^{2} - 4) \\ = 3 (x + 2) (x - 2) \\ f (- 2) = - 8 + 24 + 10 = 26 \\ f (2) = 8 - 24 + 10 = - 6 \\ f (- 3) = - 27 + 36 + 10 = 19 \\ f (5) = 125 - 60 + 10 = 75 \end{array}$
最大値 75、最小値 -6。
$\begin{array}{l} y = 1 - x \\ 0 \leq x \leq 1 \\ f (x) = x^{3} + 2 {(1 - x)}^{3} \\ f' (x) = 3 x^{2} + 6 {(1 - x)}^{2} (- 1) \\ = 3 x^{2} - 6 (x^{2} - 2 x + 1) \\ = - 3 (- x^{2} + 2 (x^{2} - 2 x + 1)) \\ = - 3 (x^{2} - 4 x + 2) \\ x^{2} - 4 x + 2 = 0 \\ x = 2 \pm \sqrt{4 - 2} = 2 \pm \sqrt{2} \\ x = 2 - \sqrt{2} \\ f (0) = 2 \\ f (1) = 1 \\ f (2 - \sqrt{2}) = {(2 - \sqrt{2})}^{3} + 2 {(1 - (2 - \sqrt{2}))}^{3} \\ = 8 + 3 \cdot 2 \cdot 2 + 3 \cdot 4 \cdot (- \sqrt{2}) - 2 \sqrt{2} + 2 {(\sqrt{2} - 1)}^{3} \\ = 8 + 12 - 12 \sqrt{2} - 2 \sqrt{2} + 2 (2 \sqrt{2} + 3 \sqrt{2} + 3 \cdot 2 (- 1) - 1) \\ = 20 - 14 \sqrt{2} + 4 \sqrt{2} + 6 \sqrt{2} - 12 - 2 \\ = 6 - 4 \sqrt{2} \end{array}$
最大値 2、最小値 $6 - 4 \sqrt{2}$
$\begin{array}{l} S = h \cdot 2 π r + 2 π r^{2} \\ h = \frac{S - 2 π r^{2}}{2 π r} \\ V = h π r^{2} \\ = \frac{S - 2 π r^{2}}{2 π r} \cdot π r^{2} \\ = \frac{1}{2} r (S - 2 π r^{2}) \\ \frac{d}{d r} V = \frac{1}{2} (S - 6 π r^{2}) \\ 6 π r^{2} = S \\ r = \sqrt{\frac{S}{6 π}} \\ h = \frac{S - 2 π \cdot \frac{S}{6 π}}{2 π \frac{S}{6 π}} \\ = \frac{\frac{2 S}{3}}{\frac{1 S}{3}} \\ = 2 \\ r : h = 1 : 2 \end{array}$

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();

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

Kamimura's blog

2017年6月23日金曜日

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)

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