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(正接の加法定理、扇形、半径、中心角、光源、照度、光の投射角、距離、比例、反比例、円錐、側面積、体積)

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25、26、27、28.を取り組んでみる。

$\begin{array}{l} ∠ O P A = α \\ \tan α = \frac{9}{x} \\ \tan (α + θ) = \frac{16}{x} \\ \tan (α + θ) = \frac{\tan α + \tan θ}{1 - \tan α \tan θ} \\ \frac{16}{x} = \frac{\frac{9}{x} + \tan θ}{1 - \frac{9}{x} \tan θ} \\ \frac{16}{x} - \frac{144}{x^{2}} \tan θ = \frac{9}{x} + \tan θ \\ \frac{7}{x} = (\frac{144}{x^{2}} + 1) \tan θ \\ \tan θ = \frac{7}{x (\frac{144}{x^{2}} + 1)} \\ = \frac{7 x}{144 + x^{2}} \\ \frac{d}{d x} \tan θ = 7 \frac{144 + x^{2} - x \cdot 2 x}{{(144 + x^{2})}^{2}} \\ = - 7 \frac{x^{2} - 144}{{(144 + x^{2})}^{2}} \\ x = 12 \end{array}$
半径の長さをr、中心角の大きさをθ、面積をSとする。
$\begin{array}{l} 2 r + 2 π r \frac{θ}{2 π} = 12 \\ 2 r + r θ = 12 \\ θ = \frac{12 - 2 r}{r} \\ S = π r^{2} \frac{θ}{2 π} \\ = \frac{θ r^{2}}{2} \\ = (6 - r) r \\ = - r^{2} + 6 r \\ \frac{d}{d r} S = - 2 r + 6 \\ = - 2 (r - 3) \\ r = 3 (m) \\ θ = 2 (r a d) \\ S = 9 (m^{2}) \end{array}$
光源の高さをhとする。
$\begin{array}{l} f (h) = \frac{\frac{h}{\sqrt{h^{2} + a^{2}}}}{(h^{2} + a^{2})} \\ = \frac{h}{{(h^{2} + a^{2})}^{\frac{3}{2}}} \\ f' (h) = \frac{{(h^{2} + a^{2})}^{\frac{3}{2}} - h \frac{3}{2} {(h^{2} + a^{2})}^{\frac{1}{2}} 2 h}{{(h^{2} + a^{2})}^{3}} \\ = \frac{{(h^{2} + a^{2})}^{\frac{1}{2}} (h^{2} + a^{2} - 3 h^{2})}{{(h^{2} + a^{2})}^{3}} \\ - 2 h^{2} + a^{2} = 0 \\ h = \frac{a}{\sqrt{2}} \end{array}$
$\begin{array}{l} S = π (h^{2} + r^{2}) \frac{2 π r}{2 π \sqrt{h^{2} + r^{2}}} \\ = π r {(h^{2} + r^{2})}^{\frac{1}{2}} \\ S^{2} = π^{2} r^{2} (h^{2} + r^{2}) \\ h^{2} = \frac{S^{2} - π^{2} r^{4}}{π r^{2}} \\ V = \frac{1}{3} π r^{2} h \\ V^{2} = \frac{1}{9} π^{2} r^{4} h^{2} \\ = \frac{1}{9} π r^{4} \frac{S^{2} - π^{2} r^{4}}{π r^{2}} \\ = \frac{1}{9} r^{2} (S^{2} - π^{2} r^{4}) \\ = \frac{1}{9} (r^{2} S^{2} - π^{2} r^{6}) \\ \frac{d}{d r} (r^{2} S^{2} - π^{2} r^{6}) = 2 S^{2} r - 6 π^{2} r^{5} \\ = 2 r (S^{2} - 3 π^{2} r^{4}) \\ S^{2} = 3 π^{2} r^{4} \\ h^{2} = \frac{S^{2} - π^{2} r^{4}}{π r^{2}} \\ = \frac{3 π^{2} r^{4} - π^{2} r^{4}}{π r^{2}} \\ = 2 r^{2} \\ r : h = 1 : \sqrt{2} \end{array}$

コード(Emacs)

Python 3

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

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

x = symbols('x')

fs = [(25, 7 * x / (144 + x ** 2), (0, 20)),
      (27, x / (x ** 2 + 10 ** 2) ** (3 / 2), (0, 10))]

for i, f, (x1, x2) in fs:
    d = Derivative(f, x, 1)
    pprint(d)
    f1 = d.doit()
    pprint(f1)
    pprint(solve(f1, x))
    p = plot(f, (x, x1, x2), show=False, legend=True)
    p.save('sample{0}.svg'.format(i))

入出力結果(Terminal, IPython)

$ ./sample25.py
d ⎛  7⋅x   ⎞
──⎜────────⎟
dx⎜ 2      ⎟
  ⎝x  + 144⎠
         2              
     14⋅x          7    
- ─────────── + ────────
            2    2      
  ⎛ 2      ⎞    x  + 144
  ⎝x  + 144⎠            
[-12, 12]
  ⎛            -1.5⎞
d ⎜  ⎛ 2      ⎞    ⎟
──⎝x⋅⎝x  + 100⎠    ⎠
dx                  
                   -2.5             -1.5
       2 ⎛ 2      ⎞       ⎛ 2      ⎞    
- 3.0⋅x ⋅⎝x  + 100⎠     + ⎝x  + 100⎠    
[-7.07106781186548, 7.07106781186548]
$

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="20">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="0.02">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" value="0.1">

<label for="a0">a = </label>
<input id="a0" type="number" min="0" value="5">

<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="sample25.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'),
    input_a0 = document.querySelector('#a0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_dx0, input_a0],
    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),
        dx0 = parseFloat(input_dx0.value),
        a0 = parseFloat(input_a0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],        
        lines = [],
        f = (x) => x / (x ** 2 + a0 ** 2) ** (3 / 2),
        f1 = (x) =>
        (x ** 2 + a0 ** 2) ** (1 / 2) * (x ** 2 + a0 ** 2 - 3 * x ** 2) /
        (x ** 2 + a0 ** 2) ** 3,
        g = (x0) => (x) => f1(x0) * (x - x0) + f(x0),
        fns = [[f, 'green']],
        fns1 = [],
        fns2 = [[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]);
    });
    fns2.forEach((o) => {
        let [fn, color] = o;
        
        for (let x = x1; x <= x2; x += dx0) {
            let g = fn(x);

            lines.push([x1, g(x1), x2, g(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.join('\n'));
    p(fns1.join('\n'));
};

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

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

Kamimura's blog

2017年6月27日火曜日

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