数学 - Python - JavaScript - 関数の変化をとらえる - 関数の極限と微分法 – 導関数とその計算 - 曲線の接線の方程式(双曲線、x軸、y軸、三角形の面積の一定性)

数学読本〈4〉

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

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)、17.3(導関数とその計算)、曲線の接線の方程式、問35.を取り組んでみる。

35. 1. 
       $\begin{array}{l}y=\frac{k}{x}\\ y\text{'}=\frac{-k}{x^2}\\ P\left(x_0,\frac{k}{x_0}\right)\\ y-\frac{k}{x_0}=\frac{-k}{x_0{}^2}\left(x-x_0\right)\\ 0-\frac{k}{x_0}=\frac{-k}{x_0{}^2}\left(x-x_0\right)\\ x=2x_0\\ Q\left(2x_0,0\right)\\ y-\frac{k}{x_0}=\frac{-k}{x_0{}^2}\left(0-x_0\right)\\ y=\frac{2k}{x_0}\\ R\left(0,\frac{2k}{x_0}\right)\\ \\ \frac{1}{2}\left(2x_0+0,0+\frac{2k}{x_0}\right)=\left(x_0,\frac{k}{x_0}\right)=P\end{array}$
    2. 
       $\frac{1}{2}\left|2x_0\right|\left|\frac{2k}{x_0}\right|=2k$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Derivative, solve

x, x0 = symbols('x x0')
k = symbols('k', positive=True)
y = k / x
pprint(y)

px = x0
py = y.subs({x: x0})
print('P')
pprint(dict(x=px, y=py))

y0 = y.subs({x: x0})
y1 = Derivative(y, x)
pprint(y1)

y1 = y1.doit()
pprint(y1)

f = y1.subs({x: x0}) * (x - x0) + y0
pprint(f)

qx = solve(f, x)[0]
qy = 0

print('Q')
pprint(dict(x=qx, y=qy))

rx = 0
ry = f.subs({x: rx})
print('R')
pprint(dict(x=rx, y=ry))

print('(1)')
print(px == (qx + rx) / 2 and py == (qy + ry) / 2)

print('(2)')
pprint(abs(1 / 2 * qx * ry))

入出力結果(Terminal, IPython)

$ ./sample35.py
k
─
x
P
⎧          k ⎫
⎨x: x₀, y: ──⎬
⎩          x₀⎭
∂ ⎛k⎞
──⎜─⎟
∂x⎝x⎠
-k 
───
  2
 x 
k    k⋅(x - x₀)
── - ──────────
x₀        2    
        x₀     
Q
{x: 2⋅x₀, y: 0}
R
⎧         2⋅k⎫
⎨x: 0, y: ───⎬
⎩          x₀⎭
(1)
True
(2)
2.0⋅k
$

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">
<br>
<label for="k0">k = </label>
<input id="k0" type="number" min="0" value="5">
<label for="x0">x0 = </label>
<input id="x0" type="number" value="2">

<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="sample35.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_k = document.querySelector('#k0'),
    input_x0 = document.querySelector('#x0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_k, 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 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),
        k = parseFloat(input_k.value),
        x0 = parseFloat(input_x0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2 || k === 0 || x0 === 0) {
        return;
    }

    let points = [],
        f = (x) => k / x,
        f1 = (x) => -k / x ** 2,
        y0 = f(x0),
        g = (x) => f1(x0) * (x - x0) + y0;
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }    

    let lines = [[x1, g(x1), x2, g(x2)]];
    
    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, i) => i <= 1 ? 'black' : 'blue');
    
    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', 'red');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
    p(Math.abs(1 / 2 * 2 * x0 * 2 * k / x0));
};

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

r = dx =
x1 = x2 =
y1 = y2 =
k = x0 =