数学 - Python - JavaScript - 三角関数、グラフの描画、微分、導関数、二階微分、極値、変曲点、極限( @fmathsecond )ラジオ2さんのツイートより

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• 参考書籍
  • Pythonからはじめる数学入門
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問．f(x)=sin(x)/√(2+2cos(x))のグラフを描け．

— ラジオ2 (@fmathsecond) 2018年1月8日

ということで。極値点、変曲点、極限等を求めてグラフを描いてみた。

$\begin{array}{}f\left(x\right)=\frac{\sin x}{\sqrt{2+2\cos x}}=\frac{1}{\sqrt{2}}·\; <!--\; middle\; dot\; -->\frac{\sin x}{\sqrt{1+\cos x}}\\ f\text{'}\left(x\right)=\frac{1}{\sqrt{2}}·\; <!--\; middle\; dot\; -->\frac{\cos x\sqrt{1+\cos x}-\sin x·\; <!--\; middle\; dot\; -->\frac{1}{2}{\left(1+\cos x\right)}^{-\frac{1}{2}}·\; <!--\; middle\; dot\; -->\left(-\sin x\right)}{1+\cos x}\\ =\frac{1}{\sqrt{2}}\frac{\cos x\sqrt{1+\cos x}+{\sin }^{2}x·\; <!--\; middle\; dot\; -->\frac{1}{2\sqrt{1+\cos x}}}{1+\cos x}\\ =\frac{2\cos x\left(1+\cos x\right)+{\sin }^{2}x}{2\sqrt{2}{\left(1+\cos x\right)}^{\frac{3}{2}}}\\ =\frac{2\cos x+2{\cos }^{2}x+{\sin }^{2}x}{2\sqrt{2}{\left(1+\cos x\right)}^{\frac{3}{2}}}\\ =\frac{{\cos }^{2}x+2\cos x+1}{2\sqrt{2}{\left(1+\cos x\right)}^{\frac{3}{2}}}\\ =\frac{{\left(\cos x+1\right)}^2}{2\sqrt{2}{\left(1+\cos x\right)}^{\frac{3}{2}}}\\ =\frac{\sqrt{\cos x+1}}{2\sqrt{2}}\\ f\text{'}\text{'}\left(x\right)=\frac{{\left(\cos x+1\right)}^{-\frac{1}{2}}\left(-\sin x\right)}{4\sqrt{2}}\\ =\frac{-\sin x}{4\sqrt{2}\sqrt{\cos x+1}}\end{array}$

$\begin{array}{}x\ne \; <!--\; not\; equal\; to\; -->\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+2n\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\left(n\in \; <!--\; element\; of\; -->\text{ℤ <!-- double-struck capital Z -->}\right)\\ f\text{'}\left(x\right)>0\\ 2n\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\le \; <!--\; less-than\; or\; equal\; to\; -->x<\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+2n\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\\ f\text{'}\text{'}\left(x\right)\le \; <!--\; less-than\; or\; equal\; to\; -->0\\ -\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+2n\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}<x\le \; <!--\; less-than\; or\; equal\; to\; -->2n\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\\ f\text{'}\text{'}\left(x\right)\ge \; <!--\; greater-than\; or\; equal\; to\; -->0\\ f\left(0\right)=0\end{array}$

$\begin{array}{}\underset{x\to \; <!--\; rightwards\; arrow\; -->\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}-0}{\lim }f\left(x\right)\\ =\frac{1}{\sqrt{2}}\underset{x\to \; <!--\; rightwards\; arrow\; -->\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}-0}{\lim }\frac{\sqrt{{\sin }^{2}x}}{\sqrt{1+\cos x}}\\ =\frac{1}{\sqrt{2}}\underset{x\to \; <!--\; rightwards\; arrow\; -->\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}-0}{\lim }\sqrt{\frac{1-{\cos }^{2}x}{1+\cos x}}\\ =\frac{1}{\sqrt{2}}\underset{x\to \; <!--\; rightwards\; arrow\; -->\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}-0}{\lim }\sqrt{1-\cos x}\\ =\frac{1}{\sqrt{2}}·\; <!--\; middle\; dot\; -->\sqrt{2}\\ =1\end{array}$

$\begin{array}{}\underset{x\to \; <!--\; rightwards\; arrow\; -->\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+0}{\lim }f\left(x\right)\\ =\frac{1}{\sqrt{2}}\underset{x\to \; <!--\; rightwards\; arrow\; -->\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+0}{\lim }\frac{-\sqrt{{\sin }^{2}x}}{\sqrt{1+\cos x}}\\ =-1\end{array}$

開数 f のグラフの描画。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sin, cos, sqrt, Derivative, pi, Limit, plot

x = symbols('x')
f = sin(x) / sqrt(2 + 2 * cos(x))

for n in range(1, 3):
    Dn = Derivative(f, x, n)
    for t in [Dn, Dn.doit()]:
        pprint(t)
        print()
    print()

for dir in ['+', '-']:
    l = Limit(f, x, pi)
    try:
        for t in [l, l.doit()]:
            pprint(t)
            print()
        print()
    except Exception as err:
        print(type(err), err)

p = plot(f, -1, 1, show=False, legend=True)
for i, color in enumerate(['red', 'green', 'blue']):
    p[i].line_color = color

p.save('sample.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample.py
d ⎛     sin(x)     ⎞
──⎜────────────────⎟
dx⎜  ______________⎟
  ⎝╲╱ 2⋅cos(x) + 2 ⎠

                           2        
     cos(x)             sin (x)     
──────────────── + ─────────────────
  ______________                 3/2
╲╱ 2⋅cos(x) + 2    (2⋅cos(x) + 2)   


  2                  
 d ⎛     sin(x)     ⎞
───⎜────────────────⎟
  2⎜  ______________⎟
dx ⎝╲╱ 2⋅cos(x) + 2 ⎠

   ⎛                         2     ⎞       
   ⎜      6⋅cos(x)      3⋅sin (x)  ⎟       
√2⋅⎜-4 + ────────── + ─────────────⎟⋅sin(x)
   ⎜     cos(x) + 1               2⎟       
   ⎝                  (cos(x) + 1) ⎠       
───────────────────────────────────────────
                  ____________             
              8⋅╲╱ cos(x) + 1              


<class 'NotImplementedError'> 
<class 'NotImplementedError'> 
$

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="0.001" value="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">

<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="sample.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 f = (x) => Math.sin(x) / Math.sqrt(2 + 2 * Math.cos(x));

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 = [[x1, -1, x2, -1, 'red'],
                 [x1, 1, x2, 1, 'blue'],
                 [-Math.PI, y1, -Math.PI, y2, 'orange'],
                 [Math.PI, y1, Math.PI, y2, 'orange']],
        fns = [[f, 'green']],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, color]);
            }
        });

    fns1
        .forEach((o) => {
            let [f, color] = o;
            
            lines.push([x1, f(x1), x2, f(x2), 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 =