数学 - Python - 関数の変化をとらえる - 関数の極限と微分法 – いろいろな関数の導関数 - 逆三角関数とその微分(arcsin, arctan)

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〈4〉数列の極限，順列/順列・組合せ/確率/関数の極限と微分法(松坂和夫(著)、岩波書店)の第17章(関数の変化をとらえる - 関数の極限と微分法)、17.5(いろいろな関数の導関数)、逆三角関数とその微分、問55、56、57.を取り組んでみる。

1. $\arcsin \frac{1}{2} = \frac{π}{6}$
2. $\frac{1}{\sqrt{1 - \frac{1}{4}}} = \frac{2}{\sqrt{3}}$
3. $\arcsin (- \frac{1}{\sqrt{2}}) = - \frac{π}{4}$
4. $\frac{1}{\sqrt{1 - \frac{1}{2}}} = \sqrt{2}$
5. $\arcsin \frac{\sqrt{3}}{2} = \frac{π}{3}$
6. $\frac{1}{\sqrt{1 - \frac{3}{4}}} = 2$
7. $\arcsin (0) = 0$
8. $\frac{1}{\sqrt{1 - \frac{9}{25}}} = \frac{5}{4}$
1. $\arctan (- \sqrt{3}) = - \frac{π}{3}$
2. $\frac{1}{1 + 3} = \frac{1}{4}$
3. $\arctan 1 = \frac{π}{4}$
4. $\frac{1}{1 + 1} = \frac{1}{2}$
5. $\arctan (\tan \frac{5}{6} π) = \arctan (\tan (- \frac{π}{6})) = - \frac{π}{6}$
6. $\frac{1}{1 + 100} = \frac{1}{101}$
1. $\frac{1}{\sqrt{1 - \frac{x^{2}}{4}}} \cdot \frac{1}{2} = \frac{1}{\sqrt{4 - x^{2}}}$
2. $\frac{1}{1 + \frac{x^{2}}{9}} \cdot (- \frac{1}{3}) = - \frac{3}{9 + x^{2}}$
3. $\frac{1}{\sqrt{1 - \frac{4 x^{2} + 1 + 4 x}{5}}} \cdot \frac{2}{\sqrt{5}} = \frac{2}{\sqrt{- 4 x^{2} - 4 x + 4}} = \frac{1}{\sqrt{- x^{2} - x + 1}}$
4. $e^{\arctan x} \cdot \frac{1}{1 + x^{2}}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, asin, atan, sqrt, exp, Derivative

x = symbols('x')
print('57.')

funcs = [asin(x / 2),
         atan(- x / 3),
         asin((2 * x + 1) / sqrt(5)),
         exp(atan(x))]

for i, func in enumerate(funcs, 1):
    print('({})'.format(i))
    pprint(func)
    d = Derivative(func, x)
    f1 = d.doit()
    pprint(f1)
    pprint(factor(f1))
    pprint(expand(f1))
    pprint(f1.factor())

入出力結果(Terminal, IPython)

$ ./sample55.py
57.
(1)
    ⎛x⎞
asin⎜─⎟
    ⎝2⎠
        1        
─────────────────
       __________
      ╱    2     
     ╱    x      
2⋅  ╱   - ── + 1 
  ╲╱      4      
         1          
────────────────────
  __________________
╲╱ -(x - 2)⋅(x + 2) 
        1        
─────────────────
       __________
      ╱    2     
     ╱    x      
2⋅  ╱   - ── + 1 
  ╲╱      4      
         1          
────────────────────
  __________________
╲╱ -(x - 2)⋅(x + 2) 
(2)
     ⎛x⎞
-atan⎜─⎟
     ⎝3⎠
   -1     
──────────
  ⎛ 2    ⎞
  ⎜x     ⎟
3⋅⎜── + 1⎟
  ⎝9     ⎠
 -3   
──────
 2    
x  + 9
   -1     
──────────
  ⎛ 2    ⎞
  ⎜x     ⎟
3⋅⎜── + 1⎟
  ⎝9     ⎠
 -3   
──────
 2    
x  + 9
(3)
    ⎛√5⋅(2⋅x + 1)⎞
asin⎜────────────⎟
    ⎝     5      ⎠
           2⋅√5          
─────────────────────────
       __________________
      ╱            2     
     ╱    (2⋅x + 1)      
5⋅  ╱   - ────────── + 1 
  ╲╱          5          
          2          
─────────────────────
   __________________
  ╱      2           
╲╱  - 4⋅x  - 4⋅x + 4 
           2⋅√5          
─────────────────────────
       __________________
      ╱      2           
     ╱    4⋅x    4⋅x   4 
5⋅  ╱   - ──── - ─── + ─ 
  ╲╱       5      5    5 
          2          
─────────────────────
   __________________
  ╱      2           
╲╱  - 4⋅x  - 4⋅x + 4 
(4)
 atan(x)
ℯ       
 atan(x)
ℯ       
────────
  2     
 x  + 1 
 atan(x)
ℯ       
────────
  2     
 x  + 1 
 atan(x)
ℯ       
────────
  2     
 x  + 1 
 atan(x)
ℯ       
────────
  2     
 x  + 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="0.0001" 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">

<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="sample55.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.exp(Math.atan(x)),
    f1 = (x) => Math.exp(Math.atan(x)) / (1 + x ** 2),
    g = (x0) => (x) => f1(x0) * (x - x0) + f(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);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [];
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let lines = [];

    for (let x = x1; x <= x2; x += 100 * dx) {
        let g0 = g(x);

        lines.push([x1, g0(x1), x2, g0(x2), 'blue']);
    }

    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', 'green');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

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

Kamimura's blog

2017年6月9日金曜日

数学 - Python - 関数の変化をとらえる - 関数の極限と微分法 – いろいろな関数の導関数 - 逆三角関数とその微分(arcsin, arctan)

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