数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 平均値の定理 - 増加・減少関数(方程式、双曲線と点との距離の最小点)

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

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5章(平均値の定理)、3(増加・減少関数)、補充問題8、9、10.を取り組んでみる。

$\begin{array}{l} f (y) = {(y - 1)}^{2} + {(\frac{y^{2}}{4} - 2)}^{2} \\ f' (y) = 2 (y - 1) + 2 (\frac{y^{2}}{4} - 2) \frac{1}{2} y \\ = \frac{y^{3}}{4} - 2 \\ = \frac{1}{4} (y^{3} - 8) \\ f' (2) = 0 \\ (1, 2) \end{array}$
$\begin{array}{l} y^{2} = x^{2} - 1 \\ y = \pm \sqrt{x^{2} - 1} \\ x \leq - 1, 1 \leq x \\ f (x) = x^{2} + {(\sqrt{x^{2} - 1} - 1)}^{2} \\ = x^{2} + x^{2} - 1 + 1 - 2 \sqrt{x^{2} - 1} \\ = 2 (x^{2} - \sqrt{x^{2} - 1}) \\ f' (x) = 2 (2 x - \frac{1}{2} {(x^{2} - 1)}^{- \frac{1}{2}} 2 x) \\ = 2 x (2 - {(x^{2} - 1)}^{- \frac{1}{2}}) \\ 2 - {(x^{2} - 1)}^{- \frac{1}{2}} = 0 \\ 2 {(x^{2} - 1)}^{\frac{1}{2}} - 1 = 0 \\ 4 (x^{2} - 1) - 1 = 0 \\ 4 x^{2} - 5 = 0 \\ x = \pm \frac{\sqrt{5}}{2} \\ (\pm \frac{\sqrt{5}}{2}, \frac{1}{2}) \end{array}$
$\begin{array}{l} y = x (x^{2} - 3) \\ = x (x + \sqrt{3}) (x - \sqrt{3}) \\ f (x) = {(x - 11)}^{2} + {((x^{3} - 3 x) - 1)}^{2} \\ f' (x) = 2 (x - 11) + 2 (x^{3} - 3 x - 1) (3 x^{2} - 3) \\ = 2 x - 22 + 6 (x^{3} - 3 x - 1) (x^{2} - 1) \\ = 2 (x - 11 + 3 (x^{5} - 4 x^{3} - x^{2} + 3 x + 1)) \\ = 2 (3 x^{5} - 12 x^{3} - 3 x^{2} + 10 x - 8) \\ = 2 (x - 2) (3 x^{4} + 6 x^{3} - 3 x + 4) \\ f' (2) = 0 \\ (2, 2) \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Derivative, solve, plot, sqrt

x = symbols('x')

fs = [(x - 1) ** 2 + (x ** 2 / 4 - 2) ** 2,
      x ** 2 - sqrt(x ** 2 - 1),
      (x - 11) ** 2 + (x ** 3 - 3 * x - 1) ** 2]

for i, f in enumerate(fs, 8):
    print('{0}.'.format(i))
    d = Derivative(f, x, 1)
    f1 = d.doit()
    pprint(d)
    pprint(f1)
    pprint(solve(f1, x))

p = plot(sqrt(4 * x), -sqrt(4 * x), show=False, legend=True)
p.save('sample8.svg')
p = plot(sqrt(x ** 2 - 1), -sqrt(x ** 2 - 1), show=False, legend=True)
p.save('sample9.svg')
p = plot(x ** 3 - 3 * x, show=False, legend=True)
p.save('sample10.svg')

入出力結果(Terminal, IPython)

$ ./sample8.py
8.
  ⎛                   2⎞
  ⎜           ⎛ 2    ⎞ ⎟
d ⎜       2   ⎜x     ⎟ ⎟
──⎜(x - 1)  + ⎜── - 2⎟ ⎟
dx⎝           ⎝4     ⎠ ⎠
  ⎛ 2    ⎞          
  ⎜x     ⎟          
x⋅⎜── - 2⎟ + 2⋅x - 2
  ⎝4     ⎠          
[2, -1 - √3⋅ⅈ, -1 + √3⋅ⅈ]
9.
  ⎛        ________⎞
d ⎜ 2     ╱  2     ⎟
──⎝x  - ╲╱  x  - 1 ⎠
dx                  
           x     
2⋅x - ───────────
         ________
        ╱  2     
      ╲╱  x  - 1 
⎡   -√5   √5⎤
⎢0, ────, ──⎥
⎣    2    2 ⎦
10.
  ⎛                          2⎞
d ⎜        2   ⎛ 3          ⎞ ⎟
──⎝(x - 11)  + ⎝x  - 3⋅x - 1⎠ ⎠
dx                             
      ⎛   2    ⎞ ⎛ 3          ⎞     
2⋅x + ⎝6⋅x  - 6⎠⋅⎝x  - 3⋅x - 1⎠ - 22
⎡             ___________            ___________            ___________       
⎢     1      ╱ 3   √39⋅ⅈ     1      ╱ 3   √39⋅ⅈ     1      ╱ 3   √39⋅ⅈ     1  
⎢2, - ─ -   ╱  ─ - ───── , - ─ +   ╱  ─ - ───── , - ─ -   ╱  ─ + ───── , - ─ +
⎣     2   ╲╱   4     6       2   ╲╱   4     6       2   ╲╱   4     6       2  

     ___________⎤
    ╱ 3   √39⋅ⅈ ⎥
   ╱  ─ + ───── ⎥
 ╲╱   4     6   ⎦
$

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="-2">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="14">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-2">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="14">
<br>
<label for="x0">x = </label>
<input id="x0" type="number" value="2">

<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="sample8.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_x0 = document.querySelector('#x0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_x0],
    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 ** 3 - 3 * x,
    g = (x) => Math.sqrt((x - 11) ** 2 + (x ** 3 - 3 * x - 1) ** 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),
        x0 = parseFloat(input_x0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    
    
    let points = [[11, 1, 'red']],
        lines = [[x0, f(x0), 11, 1, 'blue'], [x0, y1, x0, y2, 'brown']],
        fns = [[f, 'green'], [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]);
                }
            }
        });                 
    
    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.join('\n'));
};

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

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

Kamimura's blog

2017年6月26日月曜日

数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 平均値の定理 - 増加・減少関数(方程式、双曲線と点との距離の最小点)

0 コメント:

コメントを投稿