2017年6月10日土曜日

開発環境

証明と SymPy による正解の確認をしてみた。(あと、SymPy(matplotlib)によるグラフの描画と、D3.jsによる任意の回数の微分によるグラフの描画も確認できるようにしてみた。)

f 0 (x)= e x 2 f (1) (x)= e x 2 · 1 2 = 1 2 e x 2 f (2) (x)= 1 2 e x · 1 2 = 1 2 2 e x 2

予想: f (n) = 1 2 n e x 2

予想が当たっているかどうか。

f ( n ) ( x )= d dx ( 1 2 n1 e 1 2 x ) = 1 2 n1 · e 1 2 x · 1 2 = 1 2 n e 1 2 x

予想の帰納法による証明完了。

lim n f (n) (x)= 1 2 e x 2 =0

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, exp, Derivative, Limit, S, plot

x = symbols('x')
n = symbols('n', integer=True, positive=True)

f = exp(x / 2)
pprint(f)
print()

fn = Derivative(f, x, n)
pprint(fn)
print()

# グラフの描画用
fns = []

for n0 in range(1, 11):
    print('{0}回微分'.format(n0))
    fn0 = Derivative(f, x, n0)
    pprint(fn0)
    fn0 = fn0.doit()
    fns.append(fn0)
    pprint(fn0)
    print()

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

p = plot(*fns, (x, 0, 5), show=False, legend=True)

for i, _ in enumerate(p):
    p[i].line_color = '#{:0<6}'.format(hex(i * 100)[2:])

p.save('sample6.svg')

入出力結果(Terminal, IPython)

$ ./sample6.py
 x
 ─
 2
ℯ 

     ⎛ x⎞
   2 ⎜ ─⎟
  d  ⎜ 2⎟
─────⎝ℯ ⎠
dn dx    

1回微分
  ⎛ x⎞
  ⎜ ─⎟
d ⎜ 2⎟
──⎝ℯ ⎠
dx    
 x
 ─
 2
ℯ 
──
2 

2回微分
   ⎛ x⎞
  2⎜ ─⎟
 d ⎜ 2⎟
───⎝ℯ ⎠
  2    
dx     
 x
 ─
 2
ℯ 
──
4 

3回微分
   ⎛ x⎞
  3⎜ ─⎟
 d ⎜ 2⎟
───⎝ℯ ⎠
  3    
dx     
 x
 ─
 2
ℯ 
──
8 

4回微分
   ⎛ x⎞
  4⎜ ─⎟
 d ⎜ 2⎟
───⎝ℯ ⎠
  4    
dx     
 x
 ─
 2
ℯ 
──
16

5回微分
   ⎛ x⎞
  5⎜ ─⎟
 d ⎜ 2⎟
───⎝ℯ ⎠
  5    
dx     
 x
 ─
 2
ℯ 
──
32

6回微分
   ⎛ x⎞
  6⎜ ─⎟
 d ⎜ 2⎟
───⎝ℯ ⎠
  6    
dx     
 x
 ─
 2
ℯ 
──
64

7回微分
   ⎛ x⎞
  7⎜ ─⎟
 d ⎜ 2⎟
───⎝ℯ ⎠
  7    
dx     
  x
  ─
  2
 ℯ 
───
128

8回微分
   ⎛ x⎞
  8⎜ ─⎟
 d ⎜ 2⎟
───⎝ℯ ⎠
  8    
dx     
  x
  ─
  2
 ℯ 
───
256

9回微分
   ⎛ x⎞
  9⎜ ─⎟
 d ⎜ 2⎟
───⎝ℯ ⎠
  9    
dx     
  x
  ─
  2
 ℯ 
───
512

10回微分
    ⎛ x⎞
 10 ⎜ ─⎟
d   ⎜ 2⎟
────⎝ℯ ⎠
  10    
dx      
  x 
  ─ 
  2 
 ℯ  
────
1024

         ⎛ x⎞
       2 ⎜ ─⎟
      d  ⎜ 2⎟
lim ─────⎝ℯ ⎠
n─→∞dn dx    
0
$

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="n0">n = </label>
<input id="n0" type="number" min="0" step="1" value="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="sample6.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_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_n0],
    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(1 / 2 * x);    

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),
        n = parseInt(input_n0.value, 10);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;n
    }

    let points = [],
        lines = [],
        fn = (x) => 1 / 2 ** n * Math.exp(1 / 2 * x);
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t1 = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        let y = fn(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    
    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', '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, i) =>
              i < t1 ? 'green' : 'blue');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    [f, g, h].forEach((fn) => p(fn(Math.random())));
};

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








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