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 - 大きさの程度(対数、累乗、積、曲線、極値点、変曲点、導関数、二回微分、極限)

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)、4(大きさの程度)、練習問題18.を取り組んでみる。

1. $\begin{array}{l} f' (x) = \log x + x \cdot \frac{1}{x} = \log x + 1 \\ f'' (x) = \frac{1}{x} \\ f' (x) = 0 \\ \log x = - 1 \\ x = \frac{1}{e} \\ x < \frac{1}{e} \\ f' (x) < 0 \\ x > \frac{1}{e} \\ f' (x) > 0 \\ \lim_{x \to + 0} f (x) = 0 \\ \lim_{x \to \infty} f (x) = \infty \end{array}$
  
  曲線の描画。
2. $\begin{array}{l} f' (x) = 2 x \log x + x^{2} \frac{1}{x} \\ = 2 x \log x + x \\ = x (2 \log x + 1) \\ f'' (x) = 2 \log x + 2 x \cdot \frac{1}{x} + 1 \\ = 2 \log x + 3 \\ f' (x) = 0 \\ 2 \log x + 1 = 0 \\ \log x = - \frac{1}{2} \\ x = e^{- \frac{1}{2}} \\ f'' (x) = 0 \\ 2 \log x + 3 = 0 \\ \log x = - \frac{3}{2} \\ x = e^{- \frac{3}{2}} \\ \lim_{x \to + 0} f (x) = 0 \\ \lim_{x \to \infty} f (x) = \infty \end{array}$
  
  曲線の描画。
3. $\begin{array}{l} f' (x) = {(\log x)}^{2} + x_{2} (\log x) \cdot \frac{1}{x} \\ = (\log x) (\log x + 2) \\ f'' (x) = \frac{1}{x} (\log x + 2) + (\log x) \frac{1}{x} \\ = \frac{2}{x} (\log x + 1) \\ f' (x) = 0 \\ \log x = 0 \\ x = 1 \\ \log x = - 2 \\ x = e^{- 2} \\ f'' (x) = 0 \\ \log x = - 1 \\ x = e^{- 1} \\ \lim_{x \to + 0} f (x) = 0 \\ \lim_{x \to \infty} f (x) = \infty \end{array}$
  
  曲線の描画。
4. $\begin{array}{l} f' (x) = \frac{\log x - x \cdot \frac{1}{x}}{{(\log x)}^{2}} \\ = \frac{\log x - 1}{{(\log x)}^{2}} \\ f'' (x) = \frac{\frac{1}{x} \cdot {(\log x)}^{2} - (\log x - 1) \cdot 2 (\log x) \cdot \frac{1}{x}}{{(\log x)}^{4}} \\ = \frac{\log x - 2 \log x + 2}{x {(\log x)}^{3}} \\ = \frac{2 - 16 g x}{x {(\log x)}^{3}} \\ f' (x) = 0 \\ \log x = 1 \\ x = e \\ f'' (x) = 0 \\ \log x = 2 \\ x = e^{2} \\ \lim_{x \to + 1} f (x) = \infty \\ \lim_{x \to \infty} f (x) = \infty \end{array}$
  
  曲線の描画。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, log, Limit, solve, Derivative, oo

x = symbols('x')
fs = [(x * log(x), 0),
      (x ** 2 * log(x), 0),
      (x * (log(x)) ** 2, 0),
      (x / log(x), 1)]

for i, (f, x0) in enumerate(fs):
    print(f'({chr(ord("a") + i)})')
    for n in range(1, 3):
        Dn = Derivative(f, x, n)
        fn = Dn.doit()
        for t in [Dn, fn, solve(fn)]:
            pprint(t)
            print()
        print()
    for x1 in [x0, oo]:
        l = Limit(f, x, x1)
        for t in [l, l.doit()]:
            pprint(t)
            print()
        print()
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample18.py
(a)
d           
──(x⋅log(x))
dx          

log(x) + 1

⎡ -1⎤
⎣ℯ  ⎦


  2          
 d           
───(x⋅log(x))
  2          
dx           

1
─
x

[]


 lim (x⋅log(x))
x─→0⁺          

0


lim (x⋅log(x))
x─→∞          

∞



(b)
d ⎛ 2       ⎞
──⎝x ⋅log(x)⎠
dx           

2⋅x⋅log(x) + x

⎡ -1/2⎤
⎣ℯ    ⎦


  2           
 d ⎛ 2       ⎞
───⎝x ⋅log(x)⎠
  2           
dx            

2⋅log(x) + 3

⎡ -3/2⎤
⎣ℯ    ⎦


     ⎛ 2       ⎞
 lim ⎝x ⋅log(x)⎠
x─→0⁺           

0


    ⎛ 2       ⎞
lim ⎝x ⋅log(x)⎠
x─→∞           

∞



(c)
d ⎛     2   ⎞
──⎝x⋅log (x)⎠
dx           

   2              
log (x) + 2⋅log(x)

⎡    -2⎤
⎣1, ℯ  ⎦


  2           
 d ⎛     2   ⎞
───⎝x⋅log (x)⎠
  2           
dx            

2⋅(log(x) + 1)
──────────────
      x       

⎡ -1⎤
⎣ℯ  ⎦


     ⎛     2   ⎞
 lim ⎝x⋅log (x)⎠
x─→0⁺           

0


    ⎛     2   ⎞
lim ⎝x⋅log (x)⎠
x─→∞           

∞



(d)
d ⎛  x   ⎞
──⎜──────⎟
dx⎝log(x)⎠

  1         1   
────── - ───────
log(x)      2   
         log (x)

[ℯ]


  2        
 d ⎛  x   ⎞
───⎜──────⎟
  2⎝log(x)⎠
dx         

       2   
-1 + ──────
     log(x)
───────────
      2    
 x⋅log (x) 

⎡ 2⎤
⎣ℯ ⎦


     ⎛  x   ⎞
 lim ⎜──────⎟
x─→1⁺⎝log(x)⎠

∞


    ⎛  x   ⎞
lim ⎜──────⎟
x─→∞⎝log(x)⎠

∞



$

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="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-1">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-1">
<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="sample18.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 fa = (x) => x * Math.log(x),
    fb = (x) => x ** 2 * Math.log(x),
    fc = (x) => x * Math.log(x) ** 2,
    fd = (x) => x / Math.log(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 = [[1 / Math.E, y1, 1 / Math.E, y2, 'red'],
                 [x1, -1 / Math.E, x2, -1 / Math.E, 'red'],
                 [1 / Math.sqrt(Math.E), y1, 1 / Math.sqrt(Math.E), y2, 'green'],
                 [x1, -1 / (2 * Math.E), x2, -1 / (2 * Math.E), 'green'],
                 [Math.exp(-2), y1, Math.exp(-2), y2, 'blue'],
                 [x1, 4 * Math.exp(-2), x2, 4 * Math.exp(-2), 'blue'],
                 [Math.E, y1, Math.E, y2, 'orange'],
                 [x1, Math.E, x2, Math.E, 'orange']],
                 
        fns = [[fa, 'red'],
               [fb, 'green'],
               [fc, 'blue'],
               [fd, '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

2017年12月15日金曜日

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 - 大きさの程度(対数、累乗、積、曲線、極値点、変曲点、導関数、二回微分、極限)

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