2017年8月6日日曜日

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6章(曲線をえがくこと)、3(凸関数)、練習問題1-1、2、3.を取り組んでみる。

      • f'( x )= 2x( x3 )( x 2 +2 ) ( x3 ) 2 = x 2 6x2 ( x3 ) 2 f''( x )= ( 2x6 ) ( x3 ) 2 ( x 2 6x2 )2( x3 ) ( x3 ) 4 = 2 ( x3 ) 3 2( x3 )( x 2 6x2 ) ( x3 ) 4 = 2( ( x3 ) 2 ( x 2 6x2 ) ) ( x3 ) 3 = 2( x 2 6x+9 x 2 +6x+2 ) ( x3 ) 3 = 2·11 ( x3 ) 3
      • 凹の区間は x < 3。
      • 凸の区間は x > 3。
      • f'( x )= ( x 2 +1 )( x3 )2x ( x 2 +1 ) 2 = x 2 +12 x 2 +6x ( x 2 +1 ) 2 = x 2 +6x+1 ( x 2 +1 ) 2 f''( x )= ( 2x+6 ) ( x 2 +1 ) 2 ( x 2 +6x+1 )2( x 2 +1 )2x ( x 2 +1 ) 4 = 2( x3 )( x 2 +1 )( x 2 +6x+1 )4x ( x 2 +1 ) 3 = 2( x 3 +3 x 2 x+3+2 x 3 12 x 2 2x ) ( x 2 +1 ) 3 = 2( x 3 9 x 2 3x+3 ) ( x 2 +1 ) 3
      • 凹の区間。
      • 凸の区間。
      • f'( x )= x 2 +1( x+1 )2x ( x 2 +1 ) 2 = x 2 +12 x 2 2x ( x 2 +1 ) 2 = x 2 2x+1 ( x 2 +1 ) 2 f''( x )= ( 2x2 ) ( x 2 +1 ) 2 ( x 2 2x+1 )2( x 2 +1 )2x ( x 2 +1 ) 4 = 2( x+1 )( x 2 +1 )+( x 2 +2x1 )4x ( x 2 +1 ) 3 = 2 x 3 2 x 2 2x2+4 x 3 +8 x 2 4x ( x 2 +1 ) 3 = 2 x 3 +6 x 2 6x2 ( x 2 +1 ) 3 = 2( x 3 +3 x 2 3x1 ) ( x 2 +1 ) 3 = 2( x1 )( x 2 +4x+1 ) ( x 2 +1 ) 3 x 2 +4x+1=0 x=2± 41 =2± 3
      • 凹の区間。
      • x<2 3 ,2+ 3 <x<1
      • 凸の区間。
      • 2 3 <x<2+ 3 ,1<x

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, solve, Derivative, Limit, plot

x = symbols('x')
fs = [(x ** 2 + 2) / (x - 3),
      (x - 3) / (x ** 2 + 1),
      (x + 1) / (x ** 2 + 1)]

for i, f in enumerate(fs, 1):
    print(f'{i}.')
    d = Derivative(f, x, 2)
    pprint(d)
    f2 = d.doit()
    pprint(f2)
    pprint(solve(f2))
    p = plot(f, show=False, legend=True)
    p.save(f'sample1_{i}.svg')

入出力結果(Terminal, IPython)

$ ./sample1.py
1.
  2⎛ 2    ⎞
 d ⎜x  + 2⎟
───⎜──────⎟
  2⎝x - 3 ⎠
dx         
  ⎛                2     ⎞
  ⎜   2⋅x         x  + 2 ⎟
2⋅⎜- ───── + 1 + ────────⎟
  ⎜  x - 3              2⎟
  ⎝              (x - 3) ⎠
──────────────────────────
          x - 3           
[]
2.
  2        
 d ⎛x - 3 ⎞
───⎜──────⎟
  2⎜ 2    ⎟
dx ⎝x  + 1⎠
  ⎛   2                  ⎞
  ⎜4⋅x ⋅(x - 3)          ⎟
2⋅⎜──────────── - 3⋅x + 3⎟
  ⎜    2                 ⎟
  ⎝   x  + 1             ⎠
──────────────────────────
                2         
        ⎛ 2    ⎞          
        ⎝x  + 1⎠          
⎡    ⎛  1   √3⋅ⅈ⎞ 3 ___________               10                              
⎢3 + ⎜- ─ - ────⎟⋅╲╱ 30 + 10⋅ⅈ  + ──────────────────────────, 3 + ────────────
⎢    ⎝  2    2  ⎠                 ⎛  1   √3⋅ⅈ⎞ 3 ___________      ⎛  1   √3⋅ⅈ⎞
⎢                                 ⎜- ─ - ────⎟⋅╲╱ 30 + 10⋅ⅈ       ⎜- ─ + ────⎟
⎣                                 ⎝  2    2  ⎠                    ⎝  2    2  ⎠

10               ⎛  1   √3⋅ⅈ⎞ 3 ___________            10        3 ___________
────────────── + ⎜- ─ + ────⎟⋅╲╱ 30 + 10⋅ⅈ , 3 + ───────────── + ╲╱ 30 + 10⋅ⅈ 
 3 ___________   ⎝  2    2  ⎠                    3 ___________                
⋅╲╱ 30 + 10⋅ⅈ                                    ╲╱ 30 + 10⋅ⅈ                 
                                                                              

⎤
⎥
⎥
⎥
⎦
3.
  2        
 d ⎛x + 1 ⎞
───⎜──────⎟
  2⎜ 2    ⎟
dx ⎝x  + 1⎠
  ⎛   2                  ⎞
  ⎜4⋅x ⋅(x + 1)          ⎟
2⋅⎜──────────── - 3⋅x - 1⎟
  ⎜    2                 ⎟
  ⎝   x  + 1             ⎠
──────────────────────────
                2         
        ⎛ 2    ⎞          
        ⎝x  + 1⎠          
[1, -2 - √3, -2 + √3]
$

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="-20">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="20">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-20">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="20">

<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="sample1.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 f = (x) => (x ** 2 + 2) / (x - 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 = [[f, 'green']],
        fns1 = [],
        fns2 = [];

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

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







0 コメント:

コメントを投稿

Comments on Google+: