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(光源、高さ、投射角、照度、円錐、貯水槽)

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

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)、補充問題26、27、28.を取り組んでみる。

$\begin{array}{l} f (x) = \sum_{i = 1}^{n} {(a_{i} - x)}^{2} \\ f' (x) = - \sum_{i = 1}^{n} 2 (a_{i} - x) \\ = 2 n x - 2 \sum_{i = 1}^{n} a_{i} \\ 2 n x - 2 \sum_{i = 1}^{n} a_{i} = 0 \\ x = \frac{\sum_{i = 1}^{n} a_{i}}{n} \end{array}$
光源の高さをh、投射角をΘとする。
$\begin{array}{l} \cos θ = \frac{h}{\sqrt{h^{2} + 25^{2}}} \\ f (h) = \frac{h}{\sqrt{h^{2} + 25^{2}}} \cdot \frac{1}{h^{2} + 25^{2}} \\ = \frac{h}{{(h^{2} + 25^{2})}^{\frac{3}{2}}} \\ f' (x) = \frac{{(h^{2} + 25^{2})}^{\frac{3}{2}} - h \frac{3}{2} {(h^{2} + 25^{2})}^{\frac{1}{2}} 2 h}{{(h^{2} + 25^{2})}^{3}} \\ = \frac{{(h^{2} + 25^{2})}^{\frac{3}{2}} - 3 h^{2} {(h^{2} + 25^{2})}^{\frac{1}{2}}}{{(h^{2} + 25^{2})}^{3}} \\ {(h^{2} + 25^{2})}^{\frac{3}{2}} - 3 h^{2} {(h^{2} + 25^{2})}^{\frac{1}{2}} = 0 \\ (h^{2} + 25^{2}) - 3 h^{2} = 0 \\ h^{2} = \frac{25^{2}}{2} \\ h = \frac{25}{\sqrt{2}} (m) \end{array}$
等脚の間の角をΘとする。
$\begin{array}{l} 0 < θ < π \\ 0 < \frac{θ}{2} < \frac{π}{2} \\ f (θ) = \frac{1}{3} π {(60 \sin \frac{θ}{2})}^{2} (60 \cos \frac{θ}{2}) \\ f' (θ) = \frac{60^{3}}{3} π (2 (\sin \frac{θ}{2}) (\cos \frac{θ}{2}) \cdot \frac{1}{2} (\cos \frac{θ}{2}) - {(\sin \frac{θ}{2})}^{2} (\sin \frac{θ}{2}) \frac{1}{2}) \\ 2 (\sin \frac{θ}{2}) (\cos \frac{θ}{2}) \cdot \frac{1}{2} (\cos \frac{θ}{2}) - {(\sin \frac{θ}{2})}^{2} (\sin \frac{θ}{2}) \frac{1}{2} = 0 \\ (\sin \frac{θ}{2}) (2 {(\cos \frac{θ}{2})}^{2} - {(\sin \frac{θ}{2})}^{2}) = 0 \\ 2 {(\cos \frac{θ}{2})}^{2} - {(\sin \frac{θ}{2})}^{2} = 0 \\ 2 (1 - {(\sin \frac{θ}{2})}^{2}) - {(\sin \frac{θ}{2})}^{2} = 0 \\ {(\sin \frac{θ}{2})}^{2} = \frac{2}{3} \\ \sin \frac{θ}{2} = \sqrt{\frac{2}{3}} \\ \arcsin \sqrt{\frac{2}{3}} = \frac{θ}{2} \\ θ = 2 \arcsin \sqrt{\frac{2}{3}} \end{array}$
$\begin{array}{l} \tan \frac{θ}{2} = \sqrt{2} \\ θ = 2 \arctan \sqrt{2} \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Derivative, solve, pi, sin, cos, Rational, Pow, asin, sqrt, atan, plot

print('28.')
x = symbols('x', positive=True)
f = Pow(sin(x / 2), 2) * cos(x / 2)
d = Derivative(f, x, 1)
pprint(d)
f1 = d.doit()
pprint(f1)

s = solve(f1, x)
pprint(s)

for x0 in s:
    if x0 < pi:
        pprint(x0)
        print(float(x0))
        print(float(2 * asin(sqrt(Rational(2, 3)))))
        print(float(2 * atan(sqrt(2))))
        p = plot(f, f.subs({x: x0}), (x, 0, pi), show=False)
        p.save('sample28.svg')

入出力結果(Terminal, IPython)

$ ./sample26.py
28.
d ⎛   2⎛x⎞    ⎛x⎞⎞
──⎜sin ⎜─⎟⋅cos⎜─⎟⎟
dx⎝    ⎝2⎠    ⎝2⎠⎠
     3⎛x⎞                 
  sin ⎜─⎟                 
      ⎝2⎠      ⎛x⎞    2⎛x⎞
- ─────── + sin⎜─⎟⋅cos ⎜─⎟
     2         ⎝2⎠     ⎝2⎠
⎡      ⎛  _________⎞        ⎛  ________⎞⎤
⎣4⋅atan⎝╲╱ -√3 + 2 ⎠, 4⋅atan⎝╲╱ √3 + 2 ⎠⎦
      ⎛  _________⎞
4⋅atan⎝╲╱ -√3 + 2 ⎠
1.9106332362490186
1.9106332362490186
1.9106332362490186
$

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="25">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="0.001">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" value="0.5">

<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="sample26.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 / (x ** 2 + 25 ** 2) ** (3 / 2),
    f1 = (x) =>
    ((x ** 2 + 25 ** 2) ** (3 / 2) -
     x * 3 / 2 * (x ** 2 + 25 ** 2) ** (1 / 2) * 2 * x) /
    (x ** 2 + 25 ** 2) ** 3,
    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;
    }    

    let points = [],
        x3 = 25  / Math.sqrt(2),
        y3 = f(x3),
        lines = [[x3, y1, x3, y2, 'red'],
                 [x1, y3, x2, y3, 'red']],
        fns = [[f, 'green']],
        fns1 = [],
        fns2 = [[g, 'orange']];

    console.log(f(x3));
    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]);
                }
            }
        });                 

    fns2
        .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]);
            }
        });
    
    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);

    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

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

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

Kamimura's blog

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2017年7月5日水曜日

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(光源、高さ、投射角、照度、円錐、貯水槽)

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