## 2018年1月14日日曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - 積分法 - 上方和および下方和(累乗(べき乗、2乘)、関数、区間、分割)

1. 上方和。

$\begin{array}{}\frac{1}{2}\left({\left(\frac{3}{2}\right)}^{2}+{2}^{2}\right)\\ =\frac{1}{2}\left(\frac{9}{4}+4\right)\\ =\frac{25}{8}\end{array}$

下方和。

$\begin{array}{}\frac{1}{2}\left({1}^{2}+{\left(\frac{3}{2}\right)}^{2}\right)\\ =\frac{1}{2}\left(1+\frac{9}{4}\right)\\ =\frac{13}{8}\end{array}$

2. 上方和。

$\begin{array}{}\frac{1}{3}\left({\left(\frac{4}{3}\right)}^{2}+{\left(\frac{5}{3}\right)}^{2}+{2}^{2}\right)\\ =\frac{1}{3}·\frac{16+25+36}{9}\\ =\frac{1}{3}\frac{77}{9}\\ =\frac{77}{27}\end{array}$

下方和。

$\begin{array}{}\frac{1}{3}\left({\left(\frac{3}{3}\right)}^{2}+{\left(\frac{4}{3}\right)}^{2}+{\left(\frac{5}{3}\right)}^{2}\right)\\ =\frac{1}{3}·\frac{9+16+25}{9}\\ =\frac{50}{27}\end{array}$

3. 上方和。

$\begin{array}{}\frac{1}{4}\left({\left(\frac{5}{4}\right)}^{2}+{\left(\frac{6}{4}\right)}^{2}+{\left(\frac{7}{4}\right)}^{2}+{\left(\frac{8}{4}\right)}^{2}\right)\\ =\frac{1}{4}·\frac{1}{16}\frac{1}{6}·\left(8\left(8+1\right)\left(16+1\right)-4\left(4+1\right)\left(8+1\right)\right)\\ =\frac{1}{16·6}\left(2·9·17-5·9\right)\\ =\frac{1}{16}·\frac{1}{2}\left(2·3·17-15\right)\\ =\frac{102-15}{32}\\ =\frac{87}{32}\end{array}$

下方和。

$\begin{array}{}\frac{1}{4}\left({\left(\frac{4}{4}\right)}^{2}+{\left(\frac{5}{4}\right)}^{2}+{\left(\frac{6}{4}\right)}^{2}+{\left(\frac{7}{4}\right)}^{2}\right)\\ =\frac{1}{4}·\frac{1}{16}·\frac{1}{6}\left(7·8·15-3·4·7\right)\\ =\frac{1}{16}·\frac{1}{6}\left(7·2·15-3·7\right)\\ =\frac{1}{16}·\frac{7}{2}\left(2·5-7\right)\\ =\frac{7}{32}·3\\ =\frac{63}{32}\end{array}$

4. 上方和。

$\begin{array}{}\frac{1}{n}\left({\left(1+\frac{1}{n}\right)}^{2}+{\left(1+\frac{2}{n}\right)}^{2}+\dots +{\left(1+\frac{n}{n}\right)}^{2}\right)\\ =\frac{1}{n}\left({\left(\frac{n+1}{n}\right)}^{2}+{\left(\frac{n+2}{n}\right)}^{2}+\dots +{\left(\frac{n+n}{n}\right)}^{2}\right)\\ =\frac{1}{{n}^{3}}·\frac{1}{6}\left(\left(2n\right)\left(2n+1\right)\left(4n+1\right)-n\left(n+1\right)\left(2n+1\right)\right)\\ =\frac{1}{6{n}^{2}}·\left(2n+1\right)\left(2\left(4n+1\right)-\left(n+1\right)\right)\\ =\frac{\left(2n+1\right)\left(7n+1\right)}{6{n}^{2}}\end{array}$

下方和。

$\begin{array}{}\frac{1}{n}\left({1}^{2}+{\left(1+\frac{1}{n}\right)}^{2}+\dots +{\left(1+\frac{n-1}{n}\right)}^{2}\right)2\\ =\frac{1}{n}\left(\frac{{n}^{2}}{{n}^{2}}+\frac{{\left(n+1\right)}^{2}}{{n}^{2}}+\dots +\frac{\left(2n-1\right)}{{n}^{2}}\right)\\ =\frac{1}{{n}^{3}}\frac{1}{6}\left(\left(2n-1\right)\left(2n\right)\left(4n-2+1\right)-\left(n-1\right)n\left(2n-2+1\right)\right)\\ =\frac{1}{6{n}^{3}}\left(\left(2n-1\right)\left(2n\right)\left(4n-1\right)-\left(n-1\right)n\left(2n-1\right)\right)\\ =\frac{1}{6{n}^{2}}\left(2n-1\right)\left(8n-2-n+1\right)\\ =\frac{\left(2n-1\right)\left(7n-1\right)}{6{n}^{2}}\end{array}$

コード(Emacs)

Python 3

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

i, n = symbols('i, n', integer=True)
x = symbols('x')
f = x ** 2
u = 1 / n * summation(f.subs({x: 1 + i / n}), (i, 1, n))
l = 1 / n * summation(f.subs({x: 1 + i / n}), (i, 0, n - 1))

for n0 in [2, 3, 4]:
d = {n: n0}
for g in [u, l]:
pprint(g.subs(d))
print()
print()

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


$./sample1.py 25/8 13/8 77 ── 27 50 ── 27 87 ── 32 63 ── 32 (2⋅n + 1)⋅(7⋅n + 1) ─────────────────── 2 6⋅n (2⋅n - 1)⋅(7⋅n - 1) ─────────────────── 2 6⋅n$


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="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<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="sample1.js"></script>


JavaScript

let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
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) => x ** 2;

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])
let yscale = d3.scaleLinear()
.domain([y1, y2])

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();