数学 - Python - JavaScript - 曲線の性質、最大・最小 - 微分法の応用 – 曲線の凹凸、曲線をえがくこと - 凸関数の定義のいいかえ(一般化、対数関数)

数学読本〈5〉

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
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第18章(曲線の性質、最大・最小 - 微分法の応用)、18.3(曲線の凹凸、曲線をえがくこと)、凸関数の定義のいいかえ、問41、42、43.を取り組んでみる。

41. 
    

    帰納法により証明。

    $\begin{array}{l}{t}_n=1\\ 1\le i\le n-1\\ t_i=0\\ f\left(t_n{x}_n\right)=f\left(x_n\right)=t_{n}f\left(x_n\right)\\ \\ t_n<1\\ {\displaystyle \sum _{i=1}^{n-1}{t}_i}+t_n=1\\ {\displaystyle \sum _{i=1}^{n-1}{t}_i}=1-t_n\\ {\displaystyle \sum _{i=1}^{n-1}\frac{t_i}{1-t_n}}=1\\ \frac{t_i}{1-t_n}\ge 0\\ \\ f\left({\displaystyle \sum _{i=1}^n{t}_{i}x}\right)\\ =f\left({\displaystyle \sum _{i=1}^{n-1}{t}_{i}x}+t_{n}x\right)\\ =f\left(\left(1-t_n\right){\displaystyle \sum _{i=1}^{n-1}\frac{t_i}{1-t_n}x}+t_{n}x\right)\\ \le \left(1-t_n\right)f\left({\displaystyle \sum _{i=1}^{n-1}\frac{t_i}{1-t_n}x}\right)+t_{n}f\left(x\right)\\ \le \left(1-t_n\right)\left({\displaystyle \sum _{i=1}^{n-1}\left(\frac{t_i}{1-t_n}f\left(x\right)\right)}\right)+t_{n}f\left(x\right)\\ ={\displaystyle \sum _{i=1}^{n-1}\left(t_{i}f\left(x\right)\right)}+t_{n}f\left(x\right)\\ ={\displaystyle \sum _{i=1}^n\left(t_{i}f\left(x\right)\right)}\end{array}$
42. 
    $\begin{array}{l}f\left(\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}\right)\\ =f\left({\displaystyle \sum _{i=1}^n\frac{1}{n}{x}_i}\right)\\ {\displaystyle \sum _{i=1}^n\frac{1}{n}}=1\\ \frac{1}{n}\ge 0\end{array}$

    前問、問41を利用。

    $\begin{array}{l}f\left(\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}\right)\\ =f\left({\displaystyle \sum _{i=1}^n\frac{1}{n}{x}_i}\right)\\ {\displaystyle \sum _{i=1}^n\frac{1}{n}}=1\\ \frac{1}{n}\ge 0\\ f\left({\displaystyle \sum _{i=1}^n\frac{1}{n}{x}_i}\right)\le {\displaystyle \sum _{i=1}^n\frac{1}{n}f\left(x_i\right)}\\ =\frac{1}{n}{\displaystyle \sum _{i=1}^{n}f\left(x_i\right)}\end{array}$

    fがIに置いて凹のとき、前問の問41は不等号の向きが反対になるので、本問も不等号の向きが反対になる。

    $f\left({\displaystyle \sum _{i=1}^n\frac{1}{n}{x}_i}\right)\ge \frac{1}{n}{\displaystyle \sum _{i=1}^{n}f\left(x_i\right)}$
43. 
    

    前問、問42の凹の場合を利用。

    $\begin{array}{l}\log \frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}\ge \frac{1}{n}{\displaystyle \sum _{i=1}^n\log x_i}\\ =\frac{1}{n}\log {\displaystyle \prod _{i=1}^n{x}_i}\\ =\log {\left({\displaystyle \prod _{i=1}^n{x}_i}\right)}^{\frac{1}{n}}\\ \\ \frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}\ge {\left({\displaystyle \prod _{i=1}^n{x}_i}\right)}^{\frac{1}{n}}\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, log, plot

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

p = plot(log(x + 1) - log(x), (x, 0.1, 50), show=False, legend=True)
p.save('sample41.svg')

入出力結果(Terminal, IPython)

$ ./sample41.py
$

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="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" 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="sample41.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) => Math.log(x),
    f1 = (x) => 1 / x,
    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, 'green']],
        fns1 = [],
        fns2 = [[g, 'orange']];

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

            if (Math.abs(y) < Infinity) {
                points.push([x, y, color]);
            }
        }
    });
    fns1.forEach((o) => {
        let [fn, color] = o;
        
        lines.push([x1, fn(x1), x2, fn(x2), color]);
    });
    fns2.forEach((o) => {
        let [fn, color] = o;

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