2017年8月9日水曜日

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1-11、12、13、14、15、16.を取り組んでみる。

      • f'( x )= 2( 3x+1 )( 2x3 )3 ( 3x+1 ) 2 = 11 ( 3x+1 ) 2 f''( x )= 11·2( 3x+1 )3 ( 3x+1 ) 4 = 66 ( 3x+1 ) 3 3x+1=0 x= 1 3
      • 凹の区間。
      • x> 1 3
      • 凸の区間。
      • x< 1 3
      • f'( x )=4 x 3 +4 =4( x 3 +1 ) f''( x )=4·3 x 2 0
      • 凹の区間。
      • ϕ
      • 凸の区間。
      • f'( x )=5 x 4 +1 f''( x )=20 x 3 +1 20 x 3 +1=0 x 3 = 1 20 x= 20 1 3
      • 凹の区間。
      • x< 20 1 3
      • 凸の区間。
      • x> 20 1 3
      • f'( x )=6 x 5 +6 =6( x 5 +1 ) f''( x )=6·5 x 4 0
      • 凹の区間。
      • ϕ
      • 凸の区間。
      • f'( x )=7 x 6 +1 f''( x )=42 x 5
      • 凹の区間。
      • x<0
      • 凸の区間。
      • x>0
      • f'( x )=8 x 7 +1 f''( x )=56 x 6 0
      • 凹の区間。
      • ϕ
      • 凸の区間。

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, solve, Derivative, Limit, plot, sin, cos, tan

x = symbols('x')
fs = [(2 * x - 3) / (3 * x + 1),
      x ** 4 + 4 * x,
      x ** 3 + x,
      x ** 6 + 6 * x,
      x ** 7 + x,
      x ** 8 + x]


for i, f in enumerate(fs, 11):
    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')
    print()

入出力結果(Terminal, IPython)

$ ./sample1_11.py
11.
  2         
 d ⎛2⋅x - 3⎞
───⎜───────⎟
  2⎝3⋅x + 1⎠
dx          
  ⎛3⋅(2⋅x - 3)    ⎞
6⋅⎜─────────── - 2⎟
  ⎝  3⋅x + 1      ⎠
───────────────────
              2    
     (3⋅x + 1)     
[]

12.
  2          
 d ⎛ 4      ⎞
───⎝x  + 4⋅x⎠
  2          
dx           
    2
12⋅x 
[0]

13.
  2        
 d ⎛ 3    ⎞
───⎝x  + x⎠
  2        
dx         
6⋅x
[0]

14.
  2          
 d ⎛ 6      ⎞
───⎝x  + 6⋅x⎠
  2          
dx           
    4
30⋅x 
[0]

15.
  2        
 d ⎛ 7    ⎞
───⎝x  + x⎠
  2        
dx         
    5
42⋅x 
[0]

16.
  2        
 d ⎛ 8    ⎞
───⎝x  + x⎠
  2        
dx         
    6
56⋅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="0.0001" 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="dx0">dx0 = </label>
<input id="dx0" type="number" 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="sample1_11.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) => (2 * x - 3) / (3 * x + 1),
    f1 = (x) => 11 / (3 * x + 1) ** 2,
    f2 = (x) => -66 / (3 * x + 1) ** 3,
    g = (x0) => (x) => f1(x0) * (x - x0) + f(x0);
        
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;
    }    

    let points = [],
        lines = [],
        fns = [[f, 'red'],
               [f1, 'green'],
               [f2, 'blue']],
        fns1 = [],
        fns2 = [[g, 'orange']];

    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();








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