数学 - Python - JavaScript - 関数の変化をとらえる - 関数の極限と微分法 – いろいろな関数の導関数 - 速度・加速度、その他の変化率(体積、円錐形、直円錐)

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〈4〉数列の極限，順列/順列・組合せ/確率/関数の極限と微分法(松坂和夫(著)、岩波書店)の第17章(関数の変化をとらえる - 関数の極限と微分法)、17.5(いろいろな関数の導関数)、速度・加速度、その他の変化率、問67、68、69、70.を取り組んでみる。

$\begin{array}{l} \frac{d y}{d t} = v \\ \frac{d x}{d t} = \frac{d x}{d y} \cdot \frac{d y}{d t} \\ = \frac{1}{\frac{d y}{d x}} \cdot v \\ = \frac{{(a - x)}^{2}}{a^{2}} v \\ {[\frac{d x}{d t}]}_{x = \frac{a}{2}} = \frac{{(a - \frac{a}{2})}^{2}}{a^{2}} v = \frac{v}{4} (c m / 秒) \end{array}$
$\begin{array}{l} \frac{h}{r} = \frac{100}{50} \\ r = \frac{1}{2} h \\ V = \frac{1}{3} π r^{2} h = \frac{1}{3} π \cdot \frac{1}{4} h^{3} = \frac{π h^{3}}{12} \\ \frac{d V}{d t} = 200 \\ \frac{d V}{d t} = \frac{d V}{d h} \cdot \frac{d h}{d t} \\ 200 = \frac{π}{4} h^{2} \cdot \frac{d h}{d t} \\ \frac{d h}{d t} = \frac{800}{π h^{2}} \\ {[\frac{800}{π h^{2}}]}_{h = 50} = \frac{800}{2500 π} = \frac{8}{25 π} (c m^{3} / 秒) \end{array}$
1. $\begin{array}{l} \frac{d θ}{d t} = \frac{1}{20} \\ \tan θ = \frac{y}{10} \\ y = 10 \tan θ \\ \frac{d y}{d t} = \frac{d y}{d θ} \cdot \frac{d θ}{d t} = 10 \cdot \frac{1}{\cos^{2} θ} \frac{1}{20} = \frac{1}{2 \cos^{2} θ} \\ {[\frac{d y}{d t}]}_{θ = \frac{π}{6}} = \frac{2}{3} \\ {[\frac{d y}{d t}]}_{θ = \frac{π}{4}} = 1 \\ {[\frac{d y}{d t}]}_{θ = \frac{π}{3}} = 2 \end{array}$
2. $\begin{array}{l} \frac{d y}{d t} = 1 \\ 1 = \frac{d y}{d θ} \cdot \frac{d θ}{d t} \\ 1 = \frac{10}{\cos^{2} θ} \cdot \frac{d θ}{d t} \\ \frac{d θ}{d t} = \frac{\cos^{2} θ}{10} \\ {[\frac{d θ}{d t}]}_{θ = \frac{π}{6}} = \frac{3}{40} \\ {[\frac{d θ}{d t}]}_{θ = \frac{π}{4}} = \frac{1}{20} \\ {[\frac{d θ}{d t}]}_{θ = \frac{π}{3}} = \frac{1}{40} \end{array}$
$\begin{array}{l} V = π r^{2} h \\ h = \frac{V}{π r^{2}} \\ \frac{d h}{d t} = \frac{d h}{d r} \cdot \frac{d r}{d t} \\ = - 2 \frac{V}{π} r^{- 3} \frac{d r}{d t} \\ \frac{d r}{d t} = \frac{d h}{d t} \cdot \frac{π}{- 2 V} r^{3} \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, pi, cos

print('67.')
a, x, v = symbols('a x v')
expr = (a - x) ** 2 / a ** 2 * v

pprint(expr.subs({x: a / 2}))

print('68.')
h = symbols('h')
expr = 800 / (pi * h ** 2)
pprint(expr.subs({h: 50}))

print('69.')
print('(1)')
theta = symbols('θ')
expr = 1 / (2 * cos(theta) ** 2)
thetas = [pi / 6, pi / 4, pi / 3]
for theta0 in thetas:
    pprint(expr.subs({theta: theta0}))

print('(2)')
expr = cos(theta) ** 2 / 10
thetas = [pi / 6, pi / 4, pi / 3]
for theta0 in thetas:
    pprint(expr.subs({theta: theta0}))

入出力結果(Terminal, IPython)

$ ./sample67.py
67.
v
─
4
68.
 8  
────
25⋅π
69.
(1)
2/3
1
2
(2)
3/40
1/20
1/40
$

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="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" 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="sample67.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 f = (x) => 1 / (2 * Math.cos(x) ** 2),
    g = (x) => Math.cos(x) ** 2 / 10;

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;
    }

    let points = [],
        fns = [[f, 'green'], [g, 'blue']],
        lines = [];

    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);
};

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

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

Kamimura's blog

2017年6月13日火曜日

数学 - Python - JavaScript - 関数の変化をとらえる - 関数の極限と微分法 – いろいろな関数の導関数 - 速度・加速度、その他の変化率(体積、円錐形、直円錐)

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