数学 - Python - JavaScript - 曲線の性質、最大・最小 - 微分法の応用 – 関数の近似、テイラーの定理 - 1次の近似式(指数関数、対数関数、累乗根、三角関数(正弦、余弦、正接))

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)、18.5(関数の近似、テイラーの定理)、1次の近似式、問53、54.を取り組んでみる。

53. 1. 
       $\begin{array}{l}{10.02}^3\\ ={\left(10+0.02\right)}^3\\ ={10}^3{\left(1+\frac{2}{1000}\right)}^3\\ \simeq 1000·\left(1+\frac{3·2}{1000}\right)\\ =1006\end{array}$
    2. 
       $\begin{array}{l}{\left(9.012\right)}^{\frac{1}{2}}\\ ={\left(9\left(1+\frac{12}{9000}\right)\right)}^{\frac{1}{2}}\\ =3{\left(1+\frac{1}{750}\right)}^{\frac{1}{2}}\\ \simeq 3\left(1+\frac{1}{2·750}\right)\\ =3+\frac{1}{2·250}\\ =3.002\end{array}$
    3. 
       $\begin{array}{l}{1.984}^{-1}\\ {\left(2-0.016\right)}^{-1}\\ =2^{-1}{\left(1-0.008\right)}^{-1}\\ \simeq \frac{1+0.008}{2}\\ =0.504\end{array}$
    4. 
       $\begin{array}{l}{7.9904}^{-\frac{1}{3}}\\ ={\left(8-0.0096\right)}^{-\frac{1}{3}}\\ =8^{-\frac{1}{3}}{\left(1-0.0012\right)}^{-\frac{1}{3}}\\ \simeq \frac{1+0.0004}{2}\\ =0.5002\end{array}$
54. 1. 
       $\begin{array}{l}f\left(x\right)=e^x\\ f\text{'}\left(x\right)=e^x\\ f\left(0\right)=1\\ f\text{'}\left(0\right)=1\\ e^x\fallingdotseq 1+x\end{array}$
    2. 
       $\begin{array}{l}f\left(x\right)=\tan x\\ f\text{'}\left(x\right)=\frac{1}{{\cos }^{2}x}\\ f\left(0\right)=0\\ f\text{'}\left(0\right)=1\\ \tan x\fallingdotseq 0+1x=x\end{array}$
    3. 
       $\begin{array}{l}f\left(h\right)=\sin \left(a+h\right)\\ f\text{'}\left(h\right)=\cos \left(a+h\right)\\ f\left(0\right)=\sin a\\ f\text{'}\left(0\right)=\cos a\\ \sin \left(a+h\right)\fallingdotseq \sin a+h\cos a\end{array}$
    4. 
       $\begin{array}{l}f\left(h\right)=\log \left(a+h\right)\\ f\text{'}\left(h\right)=\frac{1}{a+h}\\ f\left(0\right)=\log a\\ f\text{'}\left(0\right)=\frac{1}{a}\\ \log \left(a+h\right)\fallingdotseq \log a+\frac{h}{a}\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sin, cos, log, tan, exp, plot

print('54.')

x = symbols('x')
fs = [(exp(x), 1 + x),
      (tan(x), x),
      (sin(2 + x), sin(2) + x * cos(2)),
      (log(2 + x), log(2) + x / 2)]

for i, (f, g) in enumerate(fs, 1):
    print(f'({i})')
    p = plot(f, g, (x, -0.1, 0.1), show=False, legend=True)
    for j, color in enumerate(['red', 'green']):
        p[j].line_color = color
    p.save(f'sample54_{i}.svg')

入出力結果(Terminal, IPython)

$ ./sample53.py
54.
(1)
(2)
(3)
(4)
$

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">

<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="sample53.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 f1 = (x) => Math.exp(x),
    f2 = (x) => 1 + x,
    g1 = (x) => Math.tan(x),
    g2 = (x) => x,
    h1 = (x) => Math.sin(2 + x),
    h2 = (x) => Math.sin(2) + x * Math.cos(2),
    l1 = (x) => Math.log(2 + x),
    l2 = (x) => Math.log(2) + x / 2;

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 = [[f1, 'red'], [f2, 'green'],
               [g1, 'blue'], [g2, 'orange'],
               [h1, 'purple'], [h2, 'brown'],
               [l1, 'skyblue'], [l2, 'yellow']],
        fns1 = [],
        fns2 = [];

    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 =