2018年10月12日金曜日

学習環境

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3部(積分)、第11章(積分の計算)、補充問題(いろいろな問題)7.を取り組んでみる。


  1. 問題.のヒントより、

    0 1 1 - x 2 n dx = 0 1 1 - x 1 + x n dx 0 1 1 - x n dx

    が成り立つ。


    また、

    1 - x n dx = x 1 - x n - x n - 1 1 - x n - 1 dx = x 1 - x n - n - 1 + 1 - x 1 - x n - 1 dx = x 1 - x n + n 1 - x n - 1 dx - n 1 - x n dx n + 1 1 - x n dx = x 1 - x n + n 1 - x n - 1 dx 1 - x n dx = 1 n + 1 x 1 - x n + n 1 - x n - 1 dx

    よって、

    0 1 1 - x n dx = 1 n + 1 x 1 - x n 0 1 + n n + 1 0 1 1 - x n - 1 dx = n n + 1 0 1 1 - x n - 1 dx

    また、

    0 1 1 - x 0 dx = 1 0 1 1 - x 1 dx = x - 1 2 x 2 0 1 = 1 2 0 1 1 - x 2 dx = 0 1 1 - 2 x + x 2 dx = x - x 2 + 1 3 x 3 0 1 = 1 3

    となるので、

    0 1 1 - x n dx = 1 n + 1

    と推測。

    このことと、上記のことより、

    0 1 1 - x n dx = n n + 1 · 1 n - 1 + 1 = 1 n + 1

    となるので帰納法によりすべての非負整数に対して成り立つ。

    ゆえに、最初の不等式から

    0 1 1 - x 2 n dx 1 n + 1

    が成り立つ。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, plot

print('7.')

x = symbols('x')
n = symbols('n', integer=True, nonegative=True)
f = (1 - x ** 2) ** n
g = (1 - x) ** n

I = Integral(f, (x, 0, 1))
for t in [I, I.doit()]:
    pprint(t)
    print()

d = {n: 5}
p = plot(f.subs(d), g.subs(d), (1 / (n + 1)).subs(d),
         (x, 0, 1), legend=True, show=False)
colors = ['red', 'green', 'blue']
for i, color in enumerate(colors):
    p[i].line_color = color
p.save('sample7.svg')

for n0 in range(10):
    a = float(I.subs({n: n0}).doit())
    b = 1 / (n0 + 1)
    print(f'n = {n0}: {a:<10.5} {b:<10.5} {a >= b}')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample7.py
7.
1               
⌠               
⎮           n   
⎮ ⎛   2    ⎞    
⎮ ⎝- x  + 1⎠  dx
⌡               
0               

 ┌─  ⎛1/2, -n │  ⎞
 ├─  ⎜        │ 1⎟
2╵ 1 ⎝  3/2   │  ⎠

n = 0: 1.0        1.0        True
n = 1: 0.66667    0.5        True
n = 2: 0.53333    0.33333    True
n = 3: 0.45714    0.25       True
n = 4: 0.40635    0.2        True
n = 5: 0.36941    0.16667    True
n = 6: 0.34099    0.14286    True
n = 7: 0.31826    0.125      True
n = 8: 0.29954    0.11111    True
n = 9: 0.28377    0.1        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">

<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="sample7.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_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let f = (x) => (1 - x ** 2) ** 2,
    g = (x) => (1 - x) ** 2,
    fns = [[f, 'red'],
           [g, 'green']];

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 = [],
        n0 = 2,
        yn = 1 / (2 + 1),
        lines = [[x1, yn, x2, yn, 'blue'],
                 [0, y1, 0, y2, 'brown'],
                 [1, y1, 1, y2, 'orange']];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, 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].forEach((fs) => p(fs.join('\n')));
};

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







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