数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 正弦と余弦 - 導関数(直角三角形、ラジアン、減少、フェリスの輪(Ferris wheel)、高さ、速さ、気球、観測者、上昇速度)

解析入門 原書第3版
原著

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

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4章(正弦と余弦)、4(導関数)、練習問題17、18、19.を取り組んでみる。

17. 1. 
       $\begin{array}{l}\frac{d\theta }{dt}=-\frac{1}{30}\\ \tan \theta =\frac{y}{x}\\ y=x\tan \theta \\ \frac{dy}{d\theta }·\frac{d\theta }{dt}=\frac{x}{{\cos }^2\theta }\frac{-1}{30}\\ \frac{12}{{\cos }^2\frac{\pi }{3}}·\frac{-1}{30}=-\frac{12·4}{30}=-\frac{8}{5}\end{array}$
    2. 
       $\begin{array}{l}\sin \theta =\frac{10\sqrt{2}}{z}\\ z=\frac{10\sqrt{2}}{\sin \theta }\\ \frac{dz}{d\theta }·\frac{d\theta }{dt}=10\sqrt{2}\frac{-\cos \theta }{{\sin }^2\theta }·\frac{-1}{30}\\ \frac{\sqrt{2}}{3}·\frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}}=\frac{2}{3}\end{array}$
    3. 
       $\begin{array}{l}{x}^2+y^2=4\\ y=\sqrt{4-x^2}\\ \sin \theta =\frac{y}{2},\cos \theta =\frac{x}{2}\\ x=2\cos \theta \\ \frac{dx}{dt}=\frac{dx}{d\theta }·\frac{d\theta }{dt}\\ =-2\sin \theta ·\frac{-1}{30}\\ =\frac{y}{30}\\ =\frac{\sqrt{4-x^2}}{30}\\ \\ \frac{\sqrt{4-1}}{30}=\frac{\sqrt{3}}{30}\end{array}$
18. 
    $\begin{array}{l}\frac{d\theta }{dt}=\pi \\ h=\frac{50}{2}\sin \theta +30=25\sin \theta +30\\ \frac{dh}{dt}=\frac{dh}{d\theta }\frac{d\theta }{dt}=25\pi \cos \theta \\ 25\sin \theta +30=42.5\\ \sin \theta =\frac{1}{2}\\ \theta =\frac{1}{3}\pi ,\frac{2}{3}\pi \\ 25\pi \cos \frac{1}{3}\pi =25\pi \frac{\sqrt{3}}{2}=12.5\sqrt{3}\pi \left(フィート/分\right)\end{array}$
19. 1. 
       $\begin{array}{l}\frac{d\theta }{dt}=0.3\\ h=300\tan \theta \\ \frac{dh}{dt}=\frac{dh}{d\theta }\frac{d\theta }{dt}=300·\frac{1}{{\cos }^2\theta }·0.3=\frac{90}{{\cos }^2\theta }\\ 180\left(フィート/秒\right)\end{array}$
    2. 
       $360\left(フィート/秒\right)$
    3. 
       $\frac{90}{0.04}=2250\left(フィート/秒\right)$
    4. 
       $\begin{array}{l}\cos \theta =\sqrt{1-0.09}=\sqrt{0.91}\\ \frac{90}{0.91}=\frac{9000}{91}\left(フィート/秒\right)\end{array}$
    5. 
       $\begin{array}{l}\frac{{\sin }^2\theta }{{\cos }^2\theta }=16\\ 1-{\cos }^2\theta =16{\cos }^2\theta \\ {\cos }^2\theta =\frac{1}{17}\\ 1530\left(フィート/秒\right)\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, tan, plot

print('19.')
theta, x = symbols('θ x')

h = 300 * tan(theta)
fs = [-h.subs({theta: 0.3 * t}) * x / 300 + h.subs({theta: 0.3 * t})
      for t in range(5)]

p = plot(*fs, (x, 0, 300), show=False)
p.save('sample17.svg')

入出力結果(Terminal, IPython)

$ ./sample17.py
19.
$

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="300">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5000">

<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="sample17.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'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    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 h = (theta) => 300 * Math.tan(theta),
    dt = 0.03,
    len = Math.floor((Math.PI / 2 / dt)),
    fns = range(0, len)
    .map((t) => (x) => - h(dt * t) * x / 300 + h(dt * t))

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);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;n
    }
    let i = -1;
    setInterval(() => {
        i += 1;
        if (i === len) {
            i = 0;
        }
        let points = [],
            lines = [],
            fn = fns[i];

        lines.push([x1, fn(x1), x2, fn(x2), 'green']);
        
        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);
    }, 500);
};

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

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