数学 - Python - JavaScript - 曲線の性質、最大・最小 - 微分法の応用 – 平均値の定理 - 平均値の定理(ロルの定理の応用、水平化)

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, 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)、18.1(平均値の定理)、平均値の定理、問1.を取り組んでみる。

1. 1. 
      $\begin{array}{l}f\text{'}\left(x\right)=2x\\ \frac{b^2-a^2}{b-a}=2c\\ c=\frac{a+b}{2}=1\end{array}$
   2. 
      $\begin{array}{l}f\text{'}\left(x\right)=3x^2\\ \frac{b^3-a^3}{b-a}=3c^2\\ c^2=\frac{a^2+ab+b^2}{3}=\frac{4-6+9}{3}=\frac{7}{3}\\ c=\pm \sqrt{\frac{7}{3}}\end{array}$
   3. 
      $\begin{array}{l}f\text{'}\left(x\right)=\frac{1}{x}\\ \frac{\log b-\log a}{b-a}=\frac{1}{c}\\ c=\frac{b-a}{\log \frac{b}{a}}=e-1\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, log, E, Derivative, solve

print('1.')
x, c = symbols('x c')
fs = [(x ** 2, 0, 2),
      (x ** 3, -2, 3),
      (log(x), 1, E)]

for i, (f, a, b) in enumerate(fs, 1):
    print('({0})'.format(i))
    pprint(f)
    f1 = Derivative(f, x, 1).doit()
    pprint(f1)
    pprint(solve((f.subs({x: b}) - f.subs({x: a})) /
                 (b - a) - f1.subs({x: c}), c, dict=True))

入出力結果(Terminal, IPython)

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

from sympy import pprint, symbols, log, E, Derivative, solve

print('1.')
x, c = symbols('x c')
fs = [(x ** 2, 0, 2),
      (x ** 3, -2, 3),
      (log(x), 1, E)]

for i, (f, a, b) in enumerate(fs, 1):
    print('({0})'.format(i))
    pprint(f)
    f1 = Derivative(f, x, 1).doit()
    pprint(f1)
    pprint(solve((f.subs({x: b}) - f.subs({x: a})) /
                 (b - a) - f1.subs({x: c}), c, dict=True))

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="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<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="c0">c = </label>
<input id="c0" type="number" step="0.1" value="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.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_c0 = document.querySelector('#c0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_c0],
    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,
    f1 = (x) => 2 * x,
    g = (x) => x ** 3,
    g1 = (x) => 3 * x ** 2,
    h = (x) => Math.log(x),
    h1 = (x) => 1 / x,
    l = (f, f1, 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),
        c0 = parseFloat(input_c0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        fns = [[f, 'red'], [g, 'green'], [h, 'blue']],
        lines = [],
        fns1 = [[f1, 0, 2], [g1, -2, 3], [h1, 1, Math.E]];

    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, i) => {
        let [fn1, a, b] = o,
            [fn, color] = fns[i],
            l0 = l(fn, fn1, c0);

        lines.push([x1, l0(x1), x2, l0(x2), color]);
        lines.push([a, fn(a), b, fn(b), 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);
    p(fns);
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

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