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(円の面積、座標、球体の体積、表面積)

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〈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63、64、65、66.を取り組んでみる。

1. $\begin{array}{l} f' (t) = 3 t^{2} - 6 t = 3 t (t - 2) \\ f'' (t) = 6 t - 6 = 6 (t - 1) \\ f' (4) = 24 \\ f'' (4) = 18 \end{array}$
2. $t = 2$
3. $t = 1$
$\begin{array}{l} \frac{d x}{d t} = 5 \\ \frac{d y}{d t} = \frac{d y}{d x} \frac{d x}{d t} = \frac{5}{x} \\ \frac{5}{5} = 1 (/ 秒) \end{array}$
$\begin{array}{l} S = π r^{2} \\ \frac{d S}{d t} = 10 π \\ r = \sqrt{\frac{S}{π}} \\ \frac{d r}{d t} = \frac{d r}{d S} \cdot \frac{d S}{d t} = \frac{1}{2} {(\frac{S}{π})}^{- \frac{1}{2}} \frac{1}{π} 10 π = 5 {(r^{2})}^{- \frac{1}{2}} = \frac{5}{r} \\ 2 c m / 秒 \\ 1 c m / 秒 \\ 0.5 c m / 秒 \end{array}$
1. $\begin{array}{l} V = \frac{4}{3} π r^{3} \\ \frac{d r}{d t} = 2 \\ \frac{d}{d t} V = \frac{d V}{d r} \cdot \frac{d r}{d t} = 4 π r^{2} \cdot 2 = 8 π r^{2} \\ 8 π 3^{2} = 72 π (c m^{3} / 秒) \end{array}$
2. $\begin{array}{l} S = 4 π r^{2} \\ \frac{d S}{d t} = 3 \\ \frac{d S}{d r} \cdot \frac{d r}{d t} = 8 π r \cdot \frac{d r}{d t} \\ 3 = 8 π r \frac{d r}{d t} \\ \frac{d r}{d t} = \frac{3}{8 r π} \\ \frac{d V}{d t} = \frac{d V}{d r} \cdot \frac{d r}{d t} = 4 π r^{2} \cdot \frac{3}{8 r π} = \frac{3}{2} r \\ \frac{3}{2} \cdot 4 = 6 (c m^{3} / 秒) \end{array}$

コード(Emacs)

Python 3

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

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

print('63.')
print('(1)')
t = symbols('t', nonnegative=True)
f = t ** 3 - 3 * t ** 2
f1 = Derivative(f, t, 1).doit()
f2 = Derivative(f, t, 2).doit()
for func in [f, f1, f2]:
    pprint(func.expand())
    pprint(func.factor())
    print()

for func in [f1, f2]:
    pprint(func.subs({t: 4}))
    print()

print('(2)')
pprint(solve(f1, t, dict=True))

print('(3)')
pprint(solve(f2, t, dict=True))

p = plot(f, f1, f2, show=False, legend=True)
for i, color in enumerate(['red', 'green', 'blue']):
    p[i].line_color = color

p.save('sample63.svg')

入出力結果(Terminal, IPython)

$ ./sample63.py
63.
(1)
 3      2
t  - 3⋅t 
 2        
t ⋅(t - 3)

   2      
3⋅t  - 6⋅t
3⋅t⋅(t - 2)

6⋅t - 6
6⋅(t - 1)

24

18

(2)
[{t: 0}, {t: 2}]
(3)
[{t: 1}]
$

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="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">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" value="0.1">

<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="sample63.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_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),
    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 = (t) => t ** 3 - 3 * t ** 2,
    f1 = (t) => 3 * t ** 2 - 6 * t,
    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 = [],
        lines = [];
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y, 'red']);
        }
    }
    for (let x = x1; x <= x2; x += dx0) {
        let h = g(x);

        lines.push([x1, h(x1), x2, h(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);
};

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

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

Kamimura's blog

2017年6月12日月曜日

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(円の面積、座標、球体の体積、表面積)

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