数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 曲線をえがくこと - 凸関数(有理式)

学習環境

Surface 3 (4G LTE)、Surface 3 タイプカバー、Surface ペン(端末)
Windows 10 Pro (OS)
数式入力ソフト(TeX, MathML): MathType
MathML対応ブラウザ: Firefox、Safari
MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
参考書籍

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4.を取り組んでみる。

1. $\begin{array}{l} f' (x) = 1 - \frac{1}{x^{2}} \\ f'' (x) = \frac{2 x}{x^{4}} = \frac{2}{x^{3}} \end{array}$
  凹の区間。
  $x < 0$
  凸の区間。
  $0 > x$
2. $\begin{array}{l} f' (x) = \frac{x^{2} + 1 - x 2 x}{{(x^{2} + 1)}^{2}} \\ = \frac{- x^{2} + 1}{{(x^{2} + 1)}^{2}} \\ f'' (x) = \frac{- 2 x {(x^{2} + 1)}^{2} - (- x^{2} + 1) 2 (x^{2} + 1) 2 x}{{(x^{2} + 1)}^{4}} \\ = \frac{- 2 x (x^{2} + 1) + 4 x (x^{2} - 1)}{{(x^{2} + 1)}^{3}} \\ = \frac{2 x (- x^{2} - 1 + 2 x^{2} - 2)}{{(x^{2} + 1)}^{3}} \\ = \frac{2 x (x^{2} - 3)}{{(x^{2} + 1)}^{3}} \end{array}$
  凹の区間。
  $x < - \sqrt{3}, 0 < x < \sqrt{3}$
  凸の区間。
  $- \sqrt{3} < x < 0, \sqrt{3} < x$
3. $\begin{array}{l} f' (x) = \frac{x^{2} - 1 - x 2 x}{{(x^{2} - 1)}^{2}} \\ = \frac{- x^{2} - 1}{{(x^{2} - 1)}^{2}} \\ f'' (x) = \frac{- 2 x {(x^{2} - 1)}^{2} + (x^{2} + 1) 2 (x^{2} - 1) 2 x}{{(x^{2} - 1)}^{4}} \\ = \frac{- 2 x (x^{2} - 1) + 4 x (x^{2} + 1)}{{(x^{2} - 1)}^{3}} \\ = \frac{2 x (- x^{2} + 1 + 2 x^{2} + 2)}{{(x^{2} - 1)}^{3}} \\ = \frac{2 x (x^{2} + 3)}{{(x^{2} - 1)}^{3}} \end{array}$
  凹の区間。
  $x < - 1, 0 < x < 1$
  凸の区間。
  $- 1 < x < 0, 1 < x$

コード(Emacs)

Python 3

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

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

print('3.')

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


for i, f in enumerate(fs):
    c = chr(ord('a') + i)
    print(c)
    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'sample3_{c}.svg')
    print()

入出力結果(Terminal, IPython)

$ ./sample3.py
3.
a
  2       
 d ⎛    1⎞
───⎜x + ─⎟
  2⎝    x⎠
dx        
2 
──
 3
x 
[]

b
  2        
 d ⎛  x   ⎞
───⎜──────⎟
  2⎜ 2    ⎟
dx ⎝x  + 1⎠
    ⎛    2     ⎞
    ⎜ 4⋅x      ⎟
2⋅x⋅⎜────── - 3⎟
    ⎜ 2        ⎟
    ⎝x  + 1    ⎠
────────────────
           2    
   ⎛ 2    ⎞     
   ⎝x  + 1⎠     
[0, -√3, √3]

c
  2        
 d ⎛  x   ⎞
───⎜──────⎟
  2⎜ 2    ⎟
dx ⎝x  - 1⎠
    ⎛    2     ⎞
    ⎜ 4⋅x      ⎟
2⋅x⋅⎜────── - 3⎟
    ⎜ 2        ⎟
    ⎝x  - 1    ⎠
────────────────
           2    
   ⎛ 2    ⎞     
   ⎝x  - 1⎠     
[0, -√3⋅ⅈ, √3⋅ⅈ]

$

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">

<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="sample3.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 fa = (x) => x + 1 / x,
    fc = (x) => x / (x ** 2 - 1);
        
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 = [[-1, y1, -1, y2, 'red'],
                 [0, y1, 0, y2, 'red'],
                 [1, y1, 1, y2, 'red']],
        fns = [[fa, 'green'],
               [fc, 'orange']],
        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();

r = dx =
x1 = x2 =
y1 = y2 =

Kamimura's blog

2017年8月11日金曜日

数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 曲線をえがくこと - 凸関数(有理式)

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