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(無限大、極限、n乗根)

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22.を取り組んでみる。

$\begin{array}{l} y = {(\log n)}^{\frac{1}{n}} \\ \log y = \frac{1}{n} \log n \\ n \to \infty \Rightarrow \log y = 0 \\ \lim_{n \to \infty} {(\log n)}^{\frac{1}{n}} = 1 \end{array}$
$\begin{array}{l} y = {(\frac{\log n}{n})}^{\frac{1}{n}} \\ \log y = \frac{\log n}{n^{2}} \\ n \to \infty \Rightarrow \log y = 0 \\ \lim_{n \to \infty} {(\frac{\log n}{n})}^{\frac{1}{n}} = 1 \end{array}$
$\begin{array}{l} y = {(\frac{n}{e^{n}})}^{\frac{1}{n}} \\ \log y = \frac{1}{n} \log \frac{n}{e^{n}} \\ \log y = \frac{1}{n} (\log n - \log e^{n}) \\ \log y = \frac{\log n}{n} - \log e \\ n \to \infty \Rightarrow \log y = - \log e = \log e^{- 1} \\ \lim_{n \to \infty} {(\frac{n}{e^{n}})}^{\frac{1}{n}} = e^{- 1} \end{array}$
$\begin{array}{l} y = {(n \log n)}^{\frac{1}{n}} \\ \log y = \frac{1}{n} \log (n \log n) \\ \log y = \frac{1}{n} (\log n + \log (\log n)) \\ \log y = \frac{\log n}{n} + \frac{\log (\log n)}{n} \\ n \to \infty \Rightarrow \log y = 0 \\ \lim_{n \to \infty} {(n \log n)}^{\frac{1}{n}} = 1 \end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, log, E, exp, Limit, oo, plot

n = symbols('n', real=True)
fs = [log(n) ** (1 / n),
      (log(n) / n) ** (1 / n),
      (n / exp(n)) ** (1 / n),
      (n * log(n)) ** (1 / n)]

for i, f in enumerate(fs):
    print(f'({chr(ord("a") + i)})')
    l = Limit(f, n, oo)
    for t in [l, l.doit()]:
        pprint(t)
        print()
    print()

p = plot(1, 1 / E, *fs, (n, 1, 100), show=False, legend=True)
for i, color in enumerate(['red', 'red', 'green', 'blue', 'orange', 'brown']):
    p[i].line_color = color

p.save('sample22.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample22.py
(a)
    n ________
lim ╲╱ log(n) 
n─→∞          

1


(b)
        ________
       ╱ log(n) 
lim n ╱  ────── 
n─→∞╲╱     n    

1


(c)
       _______
    n ╱    -n 
lim ╲╱  n⋅ℯ   
n─→∞          

 -1
ℯ  


(d)
    n __________
lim ╲╱ n⋅log(n) 
n─→∞            

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="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<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="sample22.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 f1 = (n) => Math.log(n) ** (1 / n),
    f2 = (n) => (Math.log(n) / n) ** (1 / n),
    f3 = (n) => (n / Math.exp(n)) ** (1 / n),
    f4 = (n) => (n * Math.log(n)) ** (1 / n);

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 = [[x1, 1, x2, 1, 'red'],
                 [x1, 1 / Math.E, x2, 1 / Math.E, 'red']],
        fns = [[f1, 'green'],
               [f2, 'blue'],
               [f3, 'orange'],
               [f4, 'brown']],
        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19日火曜日

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(無限大、極限、n乗根)

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