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(円形の池、岸辺、泳ぎ、歩行、速度、時間)

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34.を取り組んでみる。

円形の池の中心をO、PRの長さをx、∠ROQのなす角をΘとする。
$\begin{array}{l} 0 \leq x \leq 1 \\ 0 \leq θ \leq π \\ R Q = \sqrt{1^{2} - x^{2}} = \sqrt{1 - x^{2}} \\ R Q = \frac{1}{2} (\sin \frac{θ}{2}) \cdot 2 = \sin \frac{θ}{2} \\ \sqrt{1 - x^{2}} = \sin \frac{θ}{2} \\ \arcsin \sqrt{1 - x^{2}} = \frac{θ}{2} \\ θ = 2 \arcsin \sqrt{1 - x^{2}} \\ f (x) = \frac{x}{2} + \frac{π \cdot \frac{θ}{2 π}}{4} \\ = \frac{x}{2} + \frac{θ}{8} \\ = \frac{x}{2} + \frac{\arcsin \sqrt{1 - x^{2}}}{4} \\ f' (x) = \frac{1}{2} + \frac{1}{4} \frac{1}{\sqrt{1 - (1 - x^{2})}} \cdot \frac{1}{2} {(1 - x^{2})}^{- \frac{1}{2}} (- 2 x) \\ = \frac{1}{2} - \frac{1}{4 \sqrt{1 - x^{2}}} \\ \frac{1}{2} - \frac{1}{4 \sqrt{1 - x^{2}}} = 0 \\ 2 \sqrt{1 - x^{2}} - 1 = 0 \\ \sqrt{1 - x^{2}} = \frac{1}{2} \\ 1 - x^{2} = \frac{1}{4} \\ x^{2} = \frac{3}{4} \\ x = \frac{\sqrt{3}}{2} \\ f' (\frac{\sqrt{3}}{2}) = 0 \\ 0 \leq x < \frac{\sqrt{3}}{2} \\ f' (x) > 0 \\ \frac{\sqrt{3}}{2} < x \leq 1 \\ f' (x) < 0 \end{array}$
よって最小の時間、最大の時間はそれぞれ以下のようになる。
$\begin{array}{l} f (0) = \frac{\arcsin 1}{4} = \frac{\frac{π}{2}}{4} = \frac{π}{8} \\ f (1) = \frac{1}{2} > \frac{π}{8} \\ \frac{π}{8} \\ f (\frac{\sqrt{3}}{2}) = \frac{\frac{\sqrt{3}}{2}}{2} + \frac{\arcsin \sqrt{1 - {(\frac{\sqrt{3}}{2})}^{2}}}{4} \\ = \frac{\sqrt{3} + \arcsin \frac{1}{2}}{4} \\ = \frac{\sqrt{3} + \frac{π}{6}}{4} \\ = \frac{6 \sqrt{3} + π}{24} \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Derivative, solve, asin, sqrt, pi, plot

print('34.')
x = symbols('x', positive=True)
f = x / 2 + asin(sqrt(1 - x ** 2)) / 4
d = Derivative(f, x, 1)
f1 = d.doit()
pprint(d)
pprint(f1)
s = solve(f1)
pprint(s)

pprint(f.subs({x: s[0]}))
print(pi / 8 < 1 / 2)
p = plot(f, (x, 0, 1), show=False, legend=True)
p.save('sample34.svg')

入出力結果(Terminal, IPython)

$ ./sample34.py
34.
  ⎛        ⎛   __________⎞⎞
  ⎜        ⎜  ╱    2     ⎟⎟
d ⎜x   asin⎝╲╱  - x  + 1 ⎠⎟
──⎜─ + ───────────────────⎟
dx⎝2            4         ⎠
1          1       
─ - ───────────────
2        __________
        ╱    2     
    4⋅╲╱  - x  + 1 
⎡√3⎤
⎢──⎥
⎣2 ⎦
π    √3
── + ──
24   4 
True
$

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="1">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1">
<br>
<label for="x0">x0 = </label>
<input id="x0" type="number" min="0" max="1" step="0.01" 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="sample34.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_x0 = document.querySelector('#x0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              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 f = (x) => x / 2 + Math.asin(Math.sqrt(1 - x ** 2)) / 4,
    g = (x) => Math.sqrt(1 / 4 - (x - 1 / 2) ** 2);

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),
        x0 = parseFloat(input_x0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        x3 = Math.sqrt(3) / 2,
        y3 = g(x0),
        lines = [[0, y1, 0, y2, 'red'],
                 [x3, y1, x3, y2, 'red'],
                 [x0, y1, x0, y2, 'brown'],
                 [0, 0, x0, y3, 'blue'],
                 [x0, y3, 1, 0, 'blue']],
        fns = [[f, 'orange'],
               [g, 'green']],
        fns1 = [],
        fns2 = [];

    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 =
x0 =

Kamimura's blog

2017年7月7日金曜日

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(円形の池、岸辺、泳ぎ、歩行、速度、時間)

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