数学 - Python - JavaScript - 解析学 - 積分 - 積分法 - 上方和および下方和(対数関数、幾何学的定義、面積)

解析入門 原書第3版
原著

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
  • 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

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9章(積分法)、5(上方和および下方和)、練習問題6.を取り組んでみる。

6. 
   

   関数 f を

   $f\left(x\right)=\frac{1}{x}$

   とする。

   区間

   $\left[1,2\right]$

   を考える。

   各小区間の長さが

   $\frac{1}{n}$

   である分割の上方和、下方和を考える。

   上方和。

   $\begin{array}{}\frac{1}{n}\left(\frac{1}{1+\frac{0}{n}}+\frac{1}{1+\frac{1}{n}}+\dots \; <!--\; horizontal\; ellipsis\; -->+\frac{1}{1+\frac{n-1}{n}}\right)\\ =\frac{1}{n}\underset{k=0}{\overset{n-1}{\sum \; <!--\; n-ary\; summation\; -->}}\frac{n}{n+k}\\ =\underset{k=0}{\overset{n-1}{\sum \; <!--\; n-ary\; summation\; -->}}\frac{1}{n+k}\end{array}$

   下方和。

   $\underset{k=1}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}\frac{1}{n+k}$

   また、

   $\log 2$

   は x 軸、 f、直線

   $x=1,x=2$

   で囲まれた面積なので、

   $\underset{k=1}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}\frac{1}{n+k}\le \; <!--\; less-than\; or\; equal\; to\; -->\log 2\le \; <!--\; less-than\; or\; equal\; to\; -->\underset{k=0}{\overset{n-1}{\sum \; <!--\; n-ary\; summation\; -->}}\frac{1}{n+k}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, log, summation

i, n = symbols('i, n', integer=True)
x = symbols('x')
f = 1 / x
u = 1 / n * summation(f.subs({x: 1 + i / n}), (i, 0, n - 1))
l = 1 / n * summation(f.subs({x: 1 + i / n}), (i, 1, n))

for t in [u, l]:
    pprint(t.factor())
    print()

pprint(float(log(2)))
print()
for n0 in range(1, 100, 10):
    print(f'n = {n0}')
    for s in [u, l]:
        print(float(s.subs({n: n0})))
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample6.py
n - 1      
 ____      
 ╲         
  ╲     1  
   ╲  ─────
   ╱  i + n
  ╱        
 ╱         
 ‾‾‾‾      
i = 0      

  n        
 ____      
 ╲         
  ╲     1  
   ╲  ─────
   ╱  i + n
  ╱        
 ╱         
 ‾‾‾‾      
i = 1      

0.6931471805599453

n = 1
1.0
0.5

n = 11
0.7163904507944756
0.6709359053399301

n = 21
0.7051936256951331
0.6813841018856093

n = 31
0.7012767246542976
0.6851476923962331

n = 41
0.6992819190214862
0.6870867970702668

n = 51
0.698073169409205
0.6882692478405776

n = 61
0.6972623372115745
0.6890656159000991

n = 71
0.6966807053467657
0.689638451825639

n = 81
0.696243126118449
0.6900702866122761

n = 91
0.6959019805909358
0.6904074750964303

$

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="n0">n = </label>
<input id="n0" type="number" min="1" step="1" value="2">

<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_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, 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 = (x) => 1 / x,
    u = (n) => range(0, n).map((i) => 1 / (n + i)).reduce((x, y) => x + y),
    l = (n) => range(1, n + 1).map((i) => 1 / (n + i)).reduce((x, y) => x + y);

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),
        n0 = parseInt(input_n0.value, 10);

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

    let points = [],
        lines = [],
        fns = [[f, 'green']],
        fns1 = [],
        fns2 = [];

    for (let i = 1; i <= 2; i += 1 / n0) {
        lines.push([i, y1, i, y2, 'red']);
    }
    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')));
    p(`log 2 = ${Math.log(2)}`);
    [u, l].forEach((s) => p(s(n0)));
};

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

r = dx =
x1 = x2 =
y1 = y2 =
n =