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 - 第2次導関数の符号と凹凸(変曲点)

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

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)、第2次導関数の符号と凹凸、問33、34、35.を取り組んでみる。

1. $\begin{array}{l} y' = - 3 x^{2} + 6 x \\ y'' = - 6 x + 6 = - 6 (x - 1) \\ (1, - 1 + 3) = (1, 2) \end{array}$
2. $\begin{array}{l} y' = 2 (x^{2} - 1) 2 x = 4 (x^{3} - x) \\ y'' = 4 (3 x^{2} - 1) \\ (\pm \frac{1}{\sqrt{3}}, {(\frac{1}{3} - 1)}^{2}) = (\pm \frac{1}{\sqrt{3}}, \frac{4}{9}) \end{array}$
3. $\begin{array}{l} y' = \frac{1}{\cos^{2} x} \\ y'' = \frac{2 \cos x \sin x}{\cos^{4} x} = \frac{\sin 2 x}{\cos^{4} x} \\ - π < 2 x < π \\ (0, 0) \end{array}$
$\begin{array}{l} a \neq 0 \\ f (x) = a x^{3} + b x^{2} + c x + d \\ f' (x) = 3 a x^{2} + 2 b x + c \\ f'' (x) = 6 a x + 2 b = 2 (3 a x + b) \\ (- \frac{b}{3 a}, f (- \frac{b}{3 a})) \end{array}$
$\begin{array}{l} g (x) = f (x) - (f' (a) (x - a) + f (a)) \\ g' (x) = f' (x) - f' (a) \\ g' (a) = 0 \\ x < a, a < x \\ g' (x) > 0 \\ g (a) = 0 \\ x < a \\ g (x) < 0 \\ x > a \\ g (x) > 0 \end{array}$

コード(Emacs)

Python 3

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

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

print('33.')
x = symbols('x')

fs = [-x ** 3 + 3 * x ** 2,
      (x ** 2 - 1) ** 2,
      tan(x)]

for i, f in enumerate(fs, 1):
    print('({0})'.format(i))
    d = Derivative(f, x, 2)
    pprint(d)
    f2 = d.doit()
    pprint(f2)
    s = solve(f2, x)
    pprint(s)
    for x0 in s:
        pprint(x0)
        pprint(f.subs({x: x0}))
        print()
    print()
    p = plot(f, show=False, legend=True)
    p.save('sample33_{0}.svg'.format(i))

print('34.')
a = symbols('a', nonzero=True)
b, c, d = symbols('b c d')
f = a * x ** 3 + b * x ** 2 + c * x + d
d = Derivative(f, x, 2)
pprint(d)
f2 = d.doit()
pprint(f2)
s = solve(f2, x)
pprint(s)
for x0 in s:
    pprint(x)
    pprint(f.subs({x: x0}))

入出力結果(Terminal, IPython)

$ ./sample33.py
33.
(1)
  2             
 d ⎛   3      2⎞
───⎝- x  + 3⋅x ⎠
  2             
dx              
6⋅(-x + 1)
[1]
1
2


(2)
  2⎛        2⎞
 d ⎜⎛ 2    ⎞ ⎟
───⎝⎝x  - 1⎠ ⎠
  2           
dx            
  ⎛   2    ⎞
4⋅⎝3⋅x  - 1⎠
⎡-√3   √3⎤
⎢────, ──⎥
⎣ 3    3 ⎦
-√3 
────
 3  
4/9

√3
──
3 
4/9


(3)
  2        
 d         
───(tan(x))
  2        
dx         
  ⎛   2       ⎞       
2⋅⎝tan (x) + 1⎠⋅tan(x)
[0, -∞⋅ⅈ, ∞⋅ⅈ]
0
0

-∞⋅ⅈ
-ⅈ

∞⋅ⅈ
ⅈ


34.
  2                       
 ∂ ⎛   3      2          ⎞
───⎝a⋅x  + b⋅x  + c⋅x + d⎠
  2                       
∂x                        
2⋅(3⋅a⋅x + b)
⎡-b ⎤
⎢───⎥
⎣3⋅a⎦
x
              3
    b⋅c    2⋅b 
d - ─── + ─────
    3⋅a       2
          27⋅a
$

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="sample33.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 f1 = (x) => 1 / Math.cos(x) ** 2,
    f2 = (x) => Math.sin(2 * x) / Math.cos(x) ** 4,
    g = (x0) => (x) => f2(x0) * (x - x0) + f1(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 = [[f1, 'blue']],
        fns1 = [],
        fns2 = [[g, 'green']];

    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 =

Kamimura's blog

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2017年7月3日月曜日

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 - 第2次導関数の符号と凹凸(変曲点)

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