2017年12月11日月曜日

学習環境

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第2部(微分と基本的な関数)、第8章(指数関数と対数関数)、4(大きさの程度)、練習問題14.を取り組んでみる。


  1. x が0ではない場合の導関数は、常に

    f n x = a x m + k x l e 1 x 2

    という 形になり、

    lim x 0 f n x = 0

    となるので、0においてさらに高次の微分係数をもっていて、その値は0である。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, exp, Derivative, Limit

x = symbols('x')
f = exp(-1 / x ** 2)

for n in range(2, 5):
    D = Derivative(f, x, n)
    fn = D.doit()
    for t in [D, fn]:
        pprint(t)
        print()
    for dir in ['+', '-']:
        l = Limit(fn, x, 0, dir=dir)
        for t in [l, l.doit()]:
            pprint(t)
            print()
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample14.py
   ⎛ -1 ⎞
   ⎜ ───⎟
  2⎜   2⎟
 d ⎜  x ⎟
───⎝ℯ   ⎠
  2      
dx       

             -1 
             ───
               2
  ⎛     2 ⎞   x 
2⋅⎜-3 + ──⎟⋅ℯ   
  ⎜      2⎟     
  ⎝     x ⎠     
────────────────
        4       
       x        

     ⎛             -1 ⎞
     ⎜             ───⎟
     ⎜               2⎟
     ⎜  ⎛     2 ⎞   x ⎟
     ⎜2⋅⎜-3 + ──⎟⋅ℯ   ⎟
     ⎜  ⎜      2⎟     ⎟
     ⎜  ⎝     x ⎠     ⎟
 lim ⎜────────────────⎟
x─→0⁺⎜        4       ⎟
     ⎝       x        ⎠

0

     ⎛             -1 ⎞
     ⎜             ───⎟
     ⎜               2⎟
     ⎜  ⎛     2 ⎞   x ⎟
     ⎜2⋅⎜-3 + ──⎟⋅ℯ   ⎟
     ⎜  ⎜      2⎟     ⎟
     ⎜  ⎝     x ⎠     ⎟
 lim ⎜────────────────⎟
x─→0⁻⎜        4       ⎟
     ⎝       x        ⎠

0


   ⎛ -1 ⎞
   ⎜ ───⎟
  3⎜   2⎟
 d ⎜  x ⎟
───⎝ℯ   ⎠
  3      
dx       

                 -1 
                 ───
                   2
  ⎛    9    2 ⎞   x 
4⋅⎜6 - ── + ──⎟⋅ℯ   
  ⎜     2    4⎟     
  ⎝    x    x ⎠     
────────────────────
          5         
         x          

     ⎛                 -1 ⎞
     ⎜                 ───⎟
     ⎜                   2⎟
     ⎜  ⎛    9    2 ⎞   x ⎟
     ⎜4⋅⎜6 - ── + ──⎟⋅ℯ   ⎟
     ⎜  ⎜     2    4⎟     ⎟
     ⎜  ⎝    x    x ⎠     ⎟
 lim ⎜────────────────────⎟
x─→0⁺⎜          5         ⎟
     ⎝         x          ⎠

0

     ⎛                 -1 ⎞
     ⎜                 ───⎟
     ⎜                   2⎟
     ⎜  ⎛    9    2 ⎞   x ⎟
     ⎜4⋅⎜6 - ── + ──⎟⋅ℯ   ⎟
     ⎜  ⎜     2    4⎟     ⎟
     ⎜  ⎝    x    x ⎠     ⎟
 lim ⎜────────────────────⎟
x─→0⁻⎜          5         ⎟
     ⎝         x          ⎠

0


   ⎛ -1 ⎞
   ⎜ ───⎟
  4⎜   2⎟
 d ⎜  x ⎟
───⎝ℯ   ⎠
  4      
dx       

                        -1 
                        ───
                          2
  ⎛      75   36   4 ⎞   x 
4⋅⎜-30 + ── - ── + ──⎟⋅ℯ   
  ⎜       2    4    6⎟     
  ⎝      x    x    x ⎠     
───────────────────────────
              6            
             x             

     ⎛                        -1 ⎞
     ⎜                        ───⎟
     ⎜                          2⎟
     ⎜  ⎛      75   36   4 ⎞   x ⎟
     ⎜4⋅⎜-30 + ── - ── + ──⎟⋅ℯ   ⎟
     ⎜  ⎜       2    4    6⎟     ⎟
     ⎜  ⎝      x    x    x ⎠     ⎟
 lim ⎜───────────────────────────⎟
x─→0⁺⎜              6            ⎟
     ⎝             x             ⎠

0

     ⎛                        -1 ⎞
     ⎜                        ───⎟
     ⎜                          2⎟
     ⎜  ⎛      75   36   4 ⎞   x ⎟
     ⎜4⋅⎜-30 + ── - ── + ──⎟⋅ℯ   ⎟
     ⎜  ⎜       2    4    6⎟     ⎟
     ⎜  ⎝      x    x    x ⎠     ⎟
 lim ⎜───────────────────────────⎟
x─→0⁻⎜              6            ⎟
     ⎝             x             ⎠

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="0.001" value="0.0001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-1.5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="1.5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-1.5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1.5">

<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="sample14.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 f1 = (x) => 2 * Math.exp(-1 / x ** 2) / x ** 3,
    f2 = (x) => 2 * Math.exp(-1 / x ** 2) * (2 - 3 * x ** 2) / x ** 6,
    f3 = (x) => 4 * (6 - 9 / x ** 2 + 2 / x ** 4) * Math.exp(-1 / x ** 2) / x ** 5;

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 = [[f1, 'red'],
               [f2, 'green'],
               [f3, 'blue']],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, color]);
            }
        });

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

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







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