数学 - Python - JavaScript - 基本理論(Basic Theory) - 微分(Differentiation) - 関数のグラフを描く(増減表、一次導関数、二次導関数、変曲点、下に凸、上に凸)

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

オイラーの贈物―人類の至宝eiπ=-1を学ぶ (吉田武(著)、東海大学出版会)の第Ⅰ部(基礎理論(Basic Theory))、3章(微分(Differentiation))、3.8(関数のグラフを描く)、問題5.を取り組んでみる。

$\begin{array}{l} f' (x) = - x \\ f'' (x) = - 1 \end{array}$

x 0

f''(x) - - -

f'(x) + 0 -

f(x) ↗ 1 ↘

備考上に凸 1 上に凸

\begin{array}{l} g' (x) = - x + \frac{1}{6} x^{3} \\ = \frac{1}{6} x (x^{2} - 6) \\ = \frac{1}{6} x (x + \sqrt{6}) (x - \sqrt{6}) \\ g'' (x) = - 1 + \frac{1}{2} x^{2} \\ = \frac{1}{2} (x^{2} - 2) \\ = \frac{1}{2} (x + \sqrt{2}) (x - \sqrt{2}) \end{array}

\begin{array}{l} g (- \sqrt{2}) = 1 - \frac{1}{2!} 2 + \frac{1}{4!} 4 \\ = 1 - 1 + \frac{1}{6} \\ = \frac{1}{6} \\ g (\sqrt{2}) = \frac{1}{6} \end{array}

x		-√6		-√2		0		√2		√6
f''(x)	-	0	+	+	+	0	-	-	-	0	+
f'(x)	+	+	+	0	-	-	-	0	+	+	+
f(x)	↗	↗	↗	$\frac{1}{6}$	↘	↘	↘	$\frac{1}{6}$	↗	↗	↗
備考	上に凸	変曲点	下に凸	下に凸	下に凸	変曲点	上に凸	上に凸	上に凸	変曲点	下に凸

コード(Emacs)

Python 3

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

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

print('5.')
x = symbols('x')
fs = [1 - 1 / factorial(2) * x ** 2,
      1 - 1 / factorial(2) * x ** 2 + 1 / factorial(4) * x ** 4]

for i, f in enumerate(fs, 1):
    print(f'[{i}]')
    pprint(f)
    for n in range(1, 3):
        d = Derivative(f, x, n)
        pprint(d)
        fn = d.doit()
        pprint(fn)
        pprint(solve(fn))
    try:
        p = plot(f, show=False, legend=True)
        p.save(f'sample5_{i}.svg')
    except Exception as err:
        print(type(err), err)
    print()

入出力結果(Terminal, IPython)

$ ./sample5.py
5.
[1]
   2    
  x     
- ── + 1
  2     
  ⎛   2    ⎞
d ⎜  x     ⎟
──⎜- ── + 1⎟
dx⎝  2     ⎠
-x
[0]
  2⎛   2    ⎞
 d ⎜  x     ⎟
───⎜- ── + 1⎟
  2⎝  2     ⎠
dx           
-1
[]

[2]
 4    2    
x    x     
── - ── + 1
24   2     
  ⎛ 4    2    ⎞
d ⎜x    x     ⎟
──⎜── - ── + 1⎟
dx⎝24   2     ⎠
 3    
x     
── - x
6     
[0, -√6, √6]
  2⎛ 4    2    ⎞
 d ⎜x    x     ⎟
───⎜── - ── + 1⎟
  2⎝24   2     ⎠
dx              
 2    
x     
── - 1
2     
[-√2, √2]

$

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="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="sample5.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 factorial = (n) => range(1, n + 1).reduce((prev, next) => prev * next, 1),
    g = (x) => 1 - 1 / factorial(2) * x ** 2 + 1 / factorial(4) * x ** 4,
    g1 = (x) => -x + 1 / 6 * x ** 3,
    h = (x0) => (x) => g1(x0) * (x - x0) + g(x0),
    g2 = (x) => -1 + 1 / 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),
        dx0 = parseFloat(input_dx0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [],
        fns = [[g, 'green'],
               [g1, 'blue'],
               [g2, 'brown']],
        fns1 = [],
        fns2 = [[h, '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]);
                }
            }
        });                 

    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 =
dx0 =

Kamimura's blog

2017年8月18日金曜日

数学 - Python - JavaScript - 基本理論(Basic Theory) - 微分(Differentiation) - 関数のグラフを描く(増減表、一次導関数、二次導関数、変曲点、下に凸、上に凸)

0 コメント:

コメントを投稿