数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 指数関数と対数関数 - 指数関数(双曲線関数、cosh(双曲線余弦関数)、sinh(双曲線正弦関数)、導関数、パラメーター表示、逆関数、グラフの描画)

解析入門 原書第3版
原著

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)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 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

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8章(指数関数と対数関数)、2(指数関数)、練習問題32.を取り組んでみる。

32. 1. 
       
       $\cosh \text{'}\left(t\right)=\frac{e^t-e^{-t}}{2}$
       $\sinh \text{'}\left(t\right)=\frac{e^t+e^{-t}}{2}$

       よって、

       $\begin{array}{}\cosh \text{'}=\sinh \\ \sinh \text{'}=\cosh \end{array}$
    2. 
       
       $\begin{array}{}{\cosh }^{2}t+{\sinh }^{2}t\\ =\frac{e^{2t}+e^{-2t}+2}{4}-\frac{e^{2\mathrm{\tau \; <!--\; greek\; small\; letter\; tau\; -->}}+e^{-2t}-2}{4}\\ =1\end{array}$
    3. 
       
       $y=\pm \; <!--\; plus-minus\; sign\; -->\sqrt{x^2-1}$
    4. 
       
       $\cosh t>0$

       なので、 曲線の x が正の部分が問題の方程式によりパラメーター表示される。

    5. 
       
       $\begin{array}{}t\ge \; <!--\; greater-than\; or\; equal\; to\; -->0\\ x=\cosh t\\ x=\frac{e^t+e^{-t}}{2}\\ 2x=e^t+e^{-t}\\ 2x{e}^t=e^{2t}+1\\ {\left(e^t\right)}^2-2x{e}^t+1=0\\ e^t=\frac{x\pm \; <!--\; plus-minus\; sign\; -->\sqrt{x^2-1}}{2}\\ \log e^t=\log \frac{x\pm \; <!--\; plus-minus\; sign\; -->\sqrt{x^2-1}}{2}\\ t=\log \frac{x\pm \; <!--\; plus-minus\; sign\; -->\sqrt{x^2-1}}{2}\end{array}$
       $\begin{array}{}\frac{\mathrm{dt}}{\mathrm{dx}}=\frac{1}{\frac{x\pm \; <!--\; plus-minus\; sign\; -->\sqrt{x^2-1}}{2}}·\; <!--\; middle\; dot\; -->\frac{1\pm \; <!--\; plus-minus\; sign\; -->\frac{1}{2}{\left(x^2-1\right)}^{-\frac{1}{·\; <!--\; middle\; dot\; -->2^2}}x}{2}\\ =\frac{1\pm \; <!--\; plus-minus\; sign\; -->{\left(x^2-1\right)}^{-\frac{1}{2}}x}{x\pm \; <!--\; plus-minus\; sign\; -->\sqrt{x^2-1}}\\ =\frac{\sqrt{x^2-1}\pm \; <!--\; plus-minus\; sign\; -->x}{x\sqrt{x^2-1}\pm \; <!--\; plus-minus\; sign\; -->\left(x^2-1\right)}\end{array}$
       $\begin{array}{}y=\frac{e^t-e^{-t}}{2}\\ zy=e^t-e^{-t}\\ 2y{e}^t=e^{2t}-1\\ {\left(e^t\right)}^2+2y{e}^t-1=0\\ e^t=\frac{-y\pm \; <!--\; plus-minus\; sign\; -->\sqrt{y^2+1}}{2}\\ e^t>0\\ e^t=\frac{-y+\sqrt{y^2+1}}{2}\\ t=\log \frac{-y+\sqrt{y^2+1}}{2}\end{array}$
       $\begin{array}{}\frac{\mathrm{dt}}{\mathrm{dy}}=\frac{1}{\frac{-y+\sqrt{y^2+1}}{2}}·\; <!--\; middle\; dot\; -->\frac{-1+\frac{1}{2}{\left(y^2+1\right)}^{-\frac{1}{2}}·\; <!--\; middle\; dot\; -->2y}{2}\\ =\frac{-1+y{\left(y^2+1\right)}^{-\frac{1}{2}}}{-y+\sqrt{y^2+1}}\\ =\frac{\sqrt{y^2+1}+y}{-y\sqrt{y^2+1}+y^2+1}\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, cosh, sinh, Derivative, plot, sqrt, solve
import random

print('32.')
print('(a)')
t = symbols('t')
f = cosh(t)
g = sinh(t)

for h in [f, g]:
    D = Derivative(h, t, 1)
    for s in [D, D.doit()]:
        pprint(s)
        print()
    print()

print('(b)')
for _ in range(10):
    print((f**2 - g ** 2).subs({t: random.random() * 10}))

print('(c)')
x, y = symbols('x, y')
eq = x ** 2 - y ** 2 - 1
ys = solve(eq, y)

p = plot(*ys, show=False, legend=True)
for i, color in enumerate(['red', 'green']):
    p[i].line_color = color

p.save('sample32_c.svg')

print('(d)')
ts1 = solve(f - y, t)
for t0 in ts1:
    D = Derivative(t0, y, 1)
    for s in [D, D.doit()]:
        pprint(s)
        print()
    print()

p = plot(f, *[t0.subs({y: t}) for t0 in ts1],
         ylim=(-10, 10), show=False, legend=True)
for i, color in enumerate(['red', 'green', 'blue']):
    p[i].line_color = color
p.save('sample32_d1.svg')

ts2 = solve(g - y, t)
for t0 in ts2:
    D = Derivative(t0, y, 1)
    for s in [D, D.doit()]:
        pprint(s)
        print()
    print()

p = plot(g, *[t0.subs({y: t}) for t0 in ts2],
         ylim=(-10, 10), show=False, legend=True)
for i, color in enumerate(['red', 'green', 'blue']):
    p[i].line_color = color

p.save('sample32_d2.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample32.py
32.
(a)
d          
──(cosh(t))
dt         

sinh(t)


d          
──(sinh(t))
dt         

cosh(t)


(b)
1.00000000000182
1.00000000000000
0.999999999999986
1.00000000000000
0.999999999999886
1.00000000000000
1.00000000000000
1.00000000000000
0.999999999999773
1.00000000000000
(c)
(d)
  ⎛   ⎛       ________⎞⎞
d ⎜   ⎜      ╱  2     ⎟⎟
──⎝log⎝y - ╲╱  y  - 1 ⎠⎠
dy                      

       y         
- ─────────── + 1
     ________    
    ╱  2         
  ╲╱  y  - 1     
─────────────────
        ________ 
       ╱  2      
 y - ╲╱  y  - 1  


  ⎛   ⎛       ________⎞⎞
d ⎜   ⎜      ╱  2     ⎟⎟
──⎝log⎝y + ╲╱  y  - 1 ⎠⎠
dy                      

     y         
─────────── + 1
   ________    
  ╱  2         
╲╱  y  - 1     
───────────────
       ________
      ╱  2     
y + ╲╱  y  - 1 


  ⎛   ⎛       ________⎞⎞
d ⎜   ⎜      ╱  2     ⎟⎟
──⎝log⎝y - ╲╱  y  + 1 ⎠⎠
dy                      

       y         
- ─────────── + 1
     ________    
    ╱  2         
  ╲╱  y  + 1     
─────────────────
        ________ 
       ╱  2      
 y - ╲╱  y  + 1  


  ⎛   ⎛       ________⎞⎞
d ⎜   ⎜      ╱  2     ⎟⎟
──⎝log⎝y + ╲╱  y  + 1 ⎠⎠
dy                      

     y         
─────────── + 1
   ________    
  ╱  2         
╲╱  y  + 1     
───────────────
       ________
      ╱  2     
y + ╲╱  y  + 1 


$

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="-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="sample312js"></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.cosh(x),
    g = (x) => Math.log((x + Math.sqrt(x ** 2 - 1)) / 2),
    h = (x) => Math.log((x - Math.sqrt(x ** 2 - 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);

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

    let points = [],
        lines = [],
        fns = [[(x) => x, 'red'],
               [f, 'green'],
               [g, 'blue'],
               [h, 'orange']],
        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();

(x) => x,red
(x) => Math.cosh(x),green
(x) => Math.log((x + Math.sqrt(x ** 2 - 1)) / 2),blue
(x) => Math.log((x - Math.sqrt(x ** 2 - 1)) / 2),orange

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