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)

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

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。

1. $\cosh' (t) = \frac{e^{t} - e^{- t}}{2}$
  
  $\sinh' (t) = \frac{e^{t} + e^{- t}}{2}$
  
  よって、
  
  $\begin{array}{l} \cosh' = \sinh \\ \sinh' = \cosh \end{array}$
2. $\begin{array}{l} \cosh^{2} t + \sinh^{2} t \\ = \frac{e^{2 t} + e^{- 2 t} + 2}{4} - \frac{e^{2 τ} + e^{- 2 t} - 2}{4} \\ = 1 \end{array}$
3. $y = \pm \sqrt{x^{2} - 1}$
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. $\begin{array}{l} t \geq 0 \\ x = \cosh t \\ x = \frac{e^{t} + e^{- t}}{2} \\ 2 x = e^{t} + e^{- t} \\ 2 x e^{t} = e^{2 t} + 1 \\ {(e^{t})}^{2} - 2 x e^{t} + 1 = 0 \\ e^{t} = \frac{x \pm \sqrt{x^{2} - 1}}{2} \\ \log e^{t} = \log \frac{x \pm \sqrt{x^{2} - 1}}{2} \\ t = \log \frac{x \pm \sqrt{x^{2} - 1}}{2} \end{array}$
  
  $\begin{array}{l} \frac{dt}{dx} = \frac{1}{\frac{x \pm \sqrt{x^{2} - 1}}{2}} \cdot \frac{1 \pm \frac{1}{2} {(x^{2} - 1)}^{- \frac{1}{\cdot 2^{2}}} x}{2} \\ = \frac{1 \pm {(x^{2} - 1)}^{- \frac{1}{2}} x}{x \pm \sqrt{x^{2} - 1}} \\ = \frac{\sqrt{x^{2} - 1} \pm x}{x \sqrt{x^{2} - 1} \pm (x^{2} - 1)} \end{array}$
  
  $\begin{array}{l} y = \frac{e^{t} - e^{- t}}{2} \\ z y = e^{t} - e^{- t} \\ 2 y e^{t} = e^{2 t} - 1 \\ {(e^{t})}^{2} + 2 y e^{t} - 1 = 0 \\ e^{t} = \frac{- y \pm \sqrt{y^{2} + 1}}{2} \\ e^{t} > 0 \\ e^{t} = \frac{- y + \sqrt{y^{2} + 1}}{2} \\ t = \log \frac{- y + \sqrt{y^{2} + 1}}{2} \end{array}$
  
  $\begin{array}{l} \frac{dt}{dy} = \frac{1}{\frac{- y + \sqrt{y^{2} + 1}}{2}} \cdot \frac{- 1 + \frac{1}{2} {(y^{2} + 1)}^{- \frac{1}{2}} \cdot 2 y}{2} \\ = \frac{- 1 + y {(y^{2} + 1)}^{- \frac{1}{2}}}{- y + \sqrt{y^{2} + 1}} \\ = \frac{\sqrt{y^{2} + 1} + y}{- y \sqrt{y^{2} + 1} + y^{2} + 1} \end{array}$

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();

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

Kamimura's blog

ほしい物リスト

2017年11月13日月曜日

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)

0 コメント:

コメントを投稿