数学 - Python - JavaScript - 曲線の性質、最大・最小 - 微分法の応用 – 媒介変数で表される曲線 - 媒介変数で表された関数の微分法(曲線、サイクロイド、接線の方程式)

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, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
参考書籍

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂和夫(著)、岩波書店)の第18章(曲線の性質、最大・最小 - 微分法の応用)、18.4(媒介変数で表される曲線)、媒介変数で表された関数の微分法、問46、47.を取り組んでみる。

dx dt =2t dy dt =3 t 2 dy dx = 3 t 2 2t = 3 2 t
1. $\begin{array}{l} y = \frac{3}{2} (x - 1^{2}) + 1^{3} \\ = \frac{3}{2} x - \frac{1}{2} \end{array}$
2. $\begin{array}{l} y = - 3 (x - 4) - 8 \\ = - 3 x + 4 \end{array}$
3. $\begin{array}{l} y = \frac{3}{4} (x - \frac{1}{4}) + \frac{1}{8} \\ = \frac{3}{4} x - \frac{1}{16} \end{array}$
dx dθ =a( 1−cosθ ) dy dθ =asinθ dy dx = asinθ a( 1−cosθ ) = sinθ 1−cosθ
1. $\begin{array}{l} y = \frac{\frac{\sqrt{3}}{2}}{1 - \frac{1}{2}} (x - a (\frac{π}{3} - \frac{\sqrt{3}}{2})) + a (1 - \frac{1}{2}) \\ = \sqrt{3} x + a (\frac{3}{2} - \frac{\sqrt{3}}{3} + \frac{1}{2}) \\ = \sqrt{3} x + a (2 - \frac{1}{\sqrt{3}}) \end{array}$
2. $\begin{array}{l} y = \frac{\sin π}{1 - \cos π} (x - a (π - \sin π)) + a (1 - \cos π) \\ = 2 a \end{array}$
3. $\begin{array}{l} y = \frac{\sin \frac{5}{4} π}{1 - \cos \frac{5}{4} π} (x - a (\frac{5}{4} π - \sin \frac{5}{4} π)) + a (1 - \cos \frac{5}{4} π) \\ = \frac{- \frac{1}{\sqrt{2}}}{1 + \frac{1}{\sqrt{2}}} (x - a (\frac{5}{4} π + \frac{1}{\sqrt{2}})) + a (1 + \frac{1}{\sqrt{2}}) \\ = - \frac{1}{\sqrt{2} + 1} (x - a (\frac{5}{4} π + \frac{1}{\sqrt{2}})) + a (1 + \frac{1}{\sqrt{2}}) \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Derivative, Rational, sin, cos, pi
import random

print('46.')

t = symbols('t')
x = t ** 2
y = t ** 3

pprint(x)
pprint(y)

m = Derivative(y, t, 1) / Derivative(x, t, 1)
pprint(m)
m = m.doit()
pprint(m)

ts = [1, -2, Rational(1, 2)]
for i, t0 in enumerate(ts, 1):
    print(f'({i})')
    f = (m * (symbols('x') - x) + y).subs({t: t0})
    pprint(f)
    print()

print('47.')
a = symbols('a')
Θ = symbols('Θ')
x = a * (Θ - sin(Θ))
y = a * (1 - cos(Θ))

pprint(x)
pprint(y)

m = Derivative(y, Θ, 1) / Derivative(x, Θ, 1)
pprint(m)
m = m.doit()
pprint(m)

Θs = [pi / 3, pi, 5 / 4 * pi]
for i, Θ0 in enumerate(Θs, 1):
    print(f'({i})')
    f = (m * (symbols('x') - x) + y).subs({Θ: Θ0})
    pprint(f)
    print()

入出力結果(Terminal, IPython)

$ ./sample46.py
46.
 2
t 
 3
t 
d ⎛ 3⎞
──⎝t ⎠
dt    
──────
d ⎛ 2⎞
──⎝t ⎠
dt    
3⋅t
───
 2 
(1)
3⋅x   1
─── - ─
 2    2

(2)
-3⋅x + 4

(3)
3⋅x   1 
─── - ──
 4    16

47.
a⋅(Θ - sin(Θ))
a⋅(-cos(Θ) + 1)
∂                  
──(a⋅(-cos(Θ) + 1))
∂Θ                 
───────────────────
 ∂                 
 ──(a⋅(Θ - sin(Θ)))
 ∂Θ                
   sin(Θ)  
───────────
-cos(Θ) + 1
(1)
a      ⎛    ⎛  √3   π⎞    ⎞
─ + √3⋅⎜- a⋅⎜- ── + ─⎟ + x⎟
2      ⎝    ⎝  2    3⎠    ⎠

(2)
2⋅a

(3)
                ⎛    ⎛√2         ⎞    ⎞
             √2⋅⎜- a⋅⎜── + 1.25⋅π⎟ + x⎟
  ⎛√2    ⎞      ⎝    ⎝2          ⎠    ⎠
a⋅⎜── + 1⎟ - ──────────────────────────
  ⎝2     ⎠             ⎛√2    ⎞        
                     2⋅⎜── + 1⎟        
                       ⎝2     ⎠        

$

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="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">

<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="sample46.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 fx = (t) => t ** 2,
    fy = (t) => t ** 3;

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 = [],        
        lines = [],
        fns = [],
        fns1 = [],
        fns2 = [];

    for (let x = x1; x <= x2; x += dx) {
        let x0 = fx(x),
            y0 = fy(x);

        if (Math.abs(x0) < Infinity && Math.abs(y0) < Infinity) {
            points.push([x0, y0, 'green']);
        }

    }
    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]);
            }
        }
    });
    fns1.forEach((o) => {
        let [fn, color] = o;
        
        lines.push([x1, fn(x1), x2, fn(x2), color]);
    });
    fns2.forEach((o) => {
        let [fn, color] = o;

        for (let x = x1; x <= x2; x += dx0) {
            let g = fn(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 =

Kamimura's blog

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2017年7月9日日曜日

数学 - Python - JavaScript - 曲線の性質、最大・最小 - 微分法の応用 – 媒介変数で表される曲線 - 媒介変数で表された関数の微分法(曲線、サイクロイド、接線の方程式)

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