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 - 指数関数・対数関数の微分

数学読本〈4〉

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
  • 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

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)、指数関数・対数関数の微分、問47、48、49、50、51、52、53.を取り組んでみる。

47. 
    $\frac{1}{-x}·\left(-1\right)=\frac{1}{x}$
48. 1. 
       $3e^{3x}$
    2. 
       $2x{e}^{x^2}$
    3. 
       $\frac{4}{4x}=\frac{1}{x}$
    4. 
       $2^x\log 2$
    5. 
       $2x{e}^{-x}-x^2{e}^{-x}$
    6. 
       $\frac{2x}{\left(x^2+1\right)\log 10}$
    7. 
       $e^x\sin x+e^x\cos x$
    8. 
       $\frac{-\frac{1}{x}}{{\left(\log x\right)}^2}$
    9. 
       $\frac{1-\log x}{x^2}$
    10. 
        $e^x\log x+e^x\frac{1}{x}$
    11. 
        $e^x\cos e^x$
    12. 
        $3e^{\sin 3x}\cos 3x$
    13. 
        $\frac{-\sin x}{\cos x}=-\tan x$
    14. 
        $\frac{1}{x}\sin \left(\log x\right)$
49. 1. 
       $\begin{array}{l}y\text{'}=e^x\\ y-1=x\end{array}$
    2. 
       $\begin{array}{l}y\text{'}=\frac{1}{x}\\ y-0=x-1\end{array}$
    3. 
       $\begin{array}{l}y\text{'}=2e^{2x}\\ y-e^2=2e^2\left(x-1\right)\end{array}$
    4. 
       $\begin{array}{l}y\text{'}=\frac{1}{x}\\ y-1=\frac{1}{e}\left(x-e\right)\end{array}$
    5. 
       $\begin{array}{l}y\text{'}=e^x+x{e}^x\\ y=x\end{array}$
    6. 
       $\begin{array}{l}y\text{'}=\frac{1-\log x}{x^2}\\ y=x-1\end{array}$
50. 
    $\begin{array}{l}y\text{'}=e^x\\ y-e^a=e^a\left(x-a\right)\\ 0-e^a=e^a\left(0-a\right)\\ e^a=a{e}^a\\ a=1\\ \\ y-e=e\left(x-1\right)\\ y=ex\end{array}$
51. 
    $\begin{array}{l}y\text{'}=\frac{1}{x}\\ y-\log a=\frac{1}{a}\left(x-a\right)\\ 0-\log a=\frac{1}{a}\left(0-a\right)\\ -\log a=-1\\ \log a=1\\ a=e\\ y-\log e=\frac{1}{e}\left(x-e\right)\\ y-1=\frac{x}{e}-1\\ y=\frac{x}{e}\end{array}$
52. 
    $\begin{array}{l}f\left(x\right)=a^x\\ f\text{'}\left(x\right)=a^x\log a\\ \underset{h\to 0}{\lim }\frac{a^{0+h}-a^0}{h}\\ =\underset{h\to 0}{\lim }\frac{a^{0+h}-a^0}{h}\\ =f\text{'}\left(0\right)\\ =\log a\end{array}$
53. 
    $\begin{array}{l}h=\frac{1}{n}\\ n\to \infty \Rightarrow h\to 0\\ \underset{n\to \infty }{\lim }{\left(1+\frac{1}{n}\right)}^n\\ =\underset{h\to 0}{\lim }{\left(1+h\right)}^h\\ =e\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, log, Derivative, Limit, S

print('47.')
x = symbols('x', negative=True)

f = log(-x)
d = Derivative(f, x, 1)
f1 = d.doit()
pprint(d)
pprint(f1)

print('52.')
h = symbols('h')
a = symbols('a', positive=True)
f = (a ** h - 1) / h
l = Limit(f, h, 0)
pprint(l)
pprint(l.doit())

print('53.')
n = symbols('n', positive=True, integer=True)
l = Limit((1 + 1 / n) ** n, n, S.Infinity)
pprint(l)
pprint(l.doit())

入出力結果(Terminal, IPython)

$ ./sample47.py
47.
d          
──(log(-x))
dx         
1
─
x
52.
      h    
     a  - 1
 lim ──────
h─→0⁺  h   
log(a)
53.
           n
    ⎛    1⎞ 
lim ⎜1 + ─⎟ 
n─→∞⎝    n⎠ 
ℯ
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="1.5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="3">
<br>
<label for="n0">n = </label>
<input id="n0" type="number" min="1" step="1" value="100">

<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="sample47.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_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_y1, input_y2,
             input_n0],
    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 = (n) => (1 + 1 / n) ** n;

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value),
        n0 = parseInt(input_n0.value);
        
    
    if (r === 0 || y1 > y2) {
        return;
    }

    let points = [];
    
    for (let x = 1; x <= n0; x += 1) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let lines = [[0, Math.E, n0, Math.E]];

    let xscale = d3.scaleLinear()
        .domain([0, n0])
        .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([[0, 0, n0, 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, i) => i <= 1 ? 'black' : 'blue');
    
    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 =
y1 = y2 =
n =