2017年6月10日土曜日

開発環境

e(ネイピア数、オイラー数)の不思議を指数関数とその各点における接線のグラフによって視覚的に体感してみる。

一応、SymPyにより、何回(無限回)微分しても e x は変わらないことを確認。

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, E, Pow, Derivative, Limit, S, exp

x = symbols('x')
n = symbols('n', integer=True, positive=True)
f = exp(x)
pprint(f)
fn = Derivative(f, x, n)
pprint(fn)

for n0 in range(0, 11):
    print('{0}回微分'.format(n0))
    fn0 = Derivative(f, x, n0)
    pprint(fn0)
    pprint(fn0.doit())
    print()

l = Limit(f, n, S.Infinity)
pprint(l)
pprint(l.doit())

入出力結果(Terminal, IPython)

$ ./sample7.py
 x
ℯ 
   2     
  d  ⎛ x⎞
─────⎝ℯ ⎠
dn dx    
0回微分
 x
ℯ 
 x
ℯ 

1回微分
d ⎛ x⎞
──⎝ℯ ⎠
dx    
 x
ℯ 

2回微分
  2    
 d ⎛ x⎞
───⎝ℯ ⎠
  2    
dx     
 x
ℯ 

3回微分
  3    
 d ⎛ x⎞
───⎝ℯ ⎠
  3    
dx     
 x
ℯ 

4回微分
  4    
 d ⎛ x⎞
───⎝ℯ ⎠
  4    
dx     
 x
ℯ 

5回微分
  5    
 d ⎛ x⎞
───⎝ℯ ⎠
  5    
dx     
 x
ℯ 

6回微分
  6    
 d ⎛ x⎞
───⎝ℯ ⎠
  6    
dx     
 x
ℯ 

7回微分
  7    
 d ⎛ x⎞
───⎝ℯ ⎠
  7    
dx     
 x
ℯ 

8回微分
  8    
 d ⎛ x⎞
───⎝ℯ ⎠
  8    
dx     
 x
ℯ 

9回微分
  9    
 d ⎛ x⎞
───⎝ℯ ⎠
  9    
dx     
 x
ℯ 

10回微分
 10     
d   ⎛ x⎞
────⎝ℯ ⎠
  10    
dx      
 x
ℯ 

     x
lim ℯ 
n─→∞  
 x
ℯ
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" 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">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" step="0.01" 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="sample7.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_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 = (x) => Math.exp(x),
    fn = (x) => Math.exp(x),    // e^x (eはネイピア数(オイラー数)は微分しても変わらない
    g = (x0) => (x) => fn(x0) * (x - x0) + f(x0); // x0 における接線


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;n
    }

    let points = [],
        lines = [];

    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    
    for (let x = x1; x <= x2; x += dx0) {
        let g0 = g(x);

        lines.push([x1, g0(x1), x2, g0(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', 'red');
    
    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();








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