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(累乗、第n次導関数、帰納法)

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, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
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6.を取り組んでみる。

1. n = 1のとき。
  $\begin{array}{l} \frac{d}{d x} (x e^{x}) = e^{x} + x e^{x} \\ = (x + 1) e^{x} \end{array}$
  よって、n = 1の場合は成り立つ。
  $\begin{array}{l} \frac{d^{n}}{{(d x)}^{n}} (x e^{x}) \\ = \frac{d}{d x} (\frac{d^{n - 1}}{{(d x)}^{n - 1}} (x e^{x})) \\ = \frac{d}{d x} ((x + n - 1) e^{x}) \\ = e^{x} + (x + n - 1) e^{x} \\ = (x + n) e^{x} \end{array}$
  ゆえに、帰納法により任意の整数 n ≥ 1に対して成り立つ。(証明終)
2. n = 1のとき。
  $\begin{array}{l} \frac{d}{d x} (x e^{- x}) \\ = e^{- x} + x e^{- x} (- 1) \\ = (- 1) (x - 1) e^{- x} \end{array}$
  よって、n = 1 の場合は成り立つ。
  $\begin{array}{l} \frac{d^{n}}{{(d x)}^{n}} (x e^{- x}) \\ = \frac{d}{d x} (\frac{d^{n - 1}}{{(d x)}^{n - 1}} (x e^{- x})) \\ = \frac{d}{d x} ({(- 1)}^{n - 1} (x - (n - 1)) e^{- x}) \\ = {(- 1)}^{n - 1} (e^{- x} + (x - (n - 1)) e^{- x} (- 1)) \\ = {(- 1)}^{n} (- 1 + x - n + 1) e^{- x} \\ = {(- 1)}^{n} (x - n) e^{- x} \end{array}$
  よって帰納法より、任意の整数 n ≥ 1 に対し成り立つ。(証明終)
3. n = 0のとき。
  $\begin{array}{l} \frac{d^{0 + 1}}{{(d x)}^{0 + 1}} (x^{0} \log x) \\ = \frac{d}{d x} \log x \\ = \frac{1}{x} \\ = \frac{0!}{x!} \end{array}$
  第1次導関数について成り立つ。
  $\begin{array}{l} \frac{d^{n + 1}}{{(d x)}^{n + 1}} (x^{n} \log x) \\ = \frac{d^{n}}{{(d x)}^{n}} (\frac{d}{d x} (x^{n} \log x)) \\ = \frac{d^{n}}{{(d x)}^{n}} (n x^{n - 1} \log x + x^{n} \frac{1}{x}) \\ = \frac{d^{n}}{{(d x)}^{n}} (n x^{n - 1} \log x + x^{n - 1}) \\ = (n \frac{d^{n}}{{(d x)}^{n}} (x^{n - 1} \log x) + \frac{d^{n}}{{(d x)}^{n}} x^{n - 1}) \\ = n \cdot \frac{(n - 1)!}{x} + 0 \\ = \frac{n!}{x} \end{array}$
  よって帰納法より、任意の整数n ≥ 0 に対して成り立つ。(証明終)

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Derivative, exp, log, factorial, plot

print('6.')
x = symbols('x')
n = symbols('n', integer=True)
fs = [(x * exp(x), (x + n) * exp(x)),
      (x * exp(-x), (-1) ** n * (x - n) * exp(-x)),
      (x ** (n - 1) * log(x), factorial(n - 1) / x)]

for i, (f, fn) in enumerate(fs):
    print(f'({chr(ord("a") + i)})')
    for t in [f, fn]:
        pprint(t)
        print()
    for n0 in range(1, 11):
        print(Derivative(f.subs({n: n0}), x,
                         n0).doit().factor() == fn.subs({n: n0}))
    print()

p = plot(*map(lambda x: x[0].subs({n: 1}), fs), show=False, legend=True)
for i, color in enumerate(['red', 'green', 'blue']):
    p[i].line_color = color

p.save('sample6.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample6.py
6.
(a)
   x
x⋅ℯ 

         x
(n + x)⋅ℯ 

True
True
True
True
True
True
True
True
True
True

(b)
   -x
x⋅ℯ  

    n           -x
(-1) ⋅(-n + x)⋅ℯ  

True
True
True
True
True
True
True
True
True
True

(c)
 n - 1       
x     ⋅log(x)

(n - 1)!
────────
   x    

True
True
True
True
True
True
True
True
True
True

$

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="-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">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" step="0.1" value="0.1">

<label for="n0">n0 = </label>
<input id="n0" type="number" min="0" step="1" min="1" value="1">

<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="sample6.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_dx0 = document.querySelector('#dx0'),
    input_n0 = document.querySelector('#n0'),        
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_dx0, 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 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),
        dx0 = parseFloat(input_dx0.value),
        n0 = parseInt(input_n0.value, 10);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [],
        f = (x) => (x + (n0 - 1)) * Math.exp(x),
        f1 = (x) => (x + n0) * Math.exp(x),
        g = (x0) => (x) => f1(x0) * (x - x0) + f(x0),
        fns = [[f, 'red']],
        fns1 = [],
        fns2 = [[g, 'green']];

    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 =
dx0 = n0 =

Kamimura's blog

2017年10月31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 - 指数関数(累乗、第n次導関数、帰納法)

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