数学 - Python - JavaScript - 細分による加法 - 積分法 – 定積分の性質と計算 - 簡単な例(整級数)

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

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂和夫(著)、岩波書店)の第19章(細分による加法 - 積分法)、19.3(定積分の性質と計算)、簡単な例、問20.を取り組んでみる。

1. $\begin{array}{l} {[3 x]}_{- 2}^{5} \\ = 3 (5 - (- 2)) \\ = 21 \end{array}$
2. $\begin{array}{l} 2 \int_{0}^{1} 2 x^{2} d x + 2 \int_{0}^{1} 5 d x \\ = 2 {[\frac{2}{3} x^{3} + 5 x]}_{0}^{1} \\ = 2 (\frac{2}{3} + 5) \\ = \frac{34}{3} \end{array}$
3. $\begin{array}{l} \int_{0}^{3} (t^{2} - 6 t + 9) d x \\ = {[\frac{1}{3} t^{3} - 3 t^{2} + 9 t]}_{0}^{3} \\ = 9 - 27 + 27 \\ = 9 \end{array}$
4. $\begin{array}{l} {[\frac{1}{4} y^{4} - 2 y^{2}]}_{3}^{- 1} \\ = (\frac{1}{4} - 2) - (\frac{81}{4} - 18) \\ = - \frac{80}{4} + 16 \\ = - 20 + 16 \\ = - 4 \end{array}$
5. $\begin{array}{l} \int_{2}^{4} {(x - 4)}^{3} d x \\ = \int_{2}^{4} (x^{3} + 3 (- 4) x^{2} + 3 \cdot 16 x - 64) d x \\ = \int_{2}^{4} (x^{3} - 12 x^{2} + 48 x - 64) d x \\ = {[\frac{1}{4} x^{4} - 4 x^{3} + 24 x^{2} - 64 x]}_{2}^{4} \\ = \frac{1}{4} (256 - 16) - 4 (64 - 8) + 24 (16 - 4) - 64 (4 - 2) \\ = \frac{240}{4} - 4 \cdot 56 + 24 \cdot 12 - 64 \cdot 2 \\ = 60 - 224 + 288 - 128 \\ = - 4 \end{array}$
6. $\begin{array}{l} \int_{1}^{0} (8 x^{3} + 3 (- 1) 4 x^{2} + 3 {(- 1)}^{2} 2 x - 1) d x \\ = \int_{1}^{0} (8 x^{3} - 12 x^{2} + 6 x - 1) d x \\ = {[2 x^{4} - 4 x^{3} + 3 x^{2} - x]}_{1}^{0} \\ = - (2 - 4 + 3 - 1) \\ = 0 \end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, Integral, plot

print('20.')
x = symbols('x')
fs = [(3, (-2, 5)),
      (2 * x ** 2 + 3 * x + 5, (-1, 1)),
      ((x - 3) ** 2, (0, 3)),
      (x ** 3 - 4 * x, (3, -1)),
      ((4 - x) ** 3, (4, 2)),
      ((2 * x - 1) ** 3, (1, 0))]

for i, (f, (x1, x2)) in enumerate(fs, 1):
    print(f'({i})')
    I = Integral(f, (x, x1, x2))
    for o in [I, I.doit()]:
        pprint(o)
    try:
        p = plot(f, show=False, legend=True)
        p.save(f'sample20_{i}.svg')
    except Exception as err:
        print(type(err), err)
    print()

入出力結果(Terminal, IPython)

$ ./sample20.py
20.
(1)
5      
⌠      
⎮  3 dx
⌡      
-2     
21
<class 'TypeError'> __init__() missing 1 required positional argument: 'var_start_end'

(2)
1                     
⌠                     
⎮  ⎛   2          ⎞   
⎮  ⎝2⋅x  + 3⋅x + 5⎠ dx
⌡                     
-1                    
34/3

(3)
3            
⌠            
⎮        2   
⎮ (x - 3)  dx
⌡            
0            
9

(4)
-1              
⌠               
⎮  ⎛ 3      ⎞   
⎮  ⎝x  - 4⋅x⎠ dx
⌡               
3               
-4

(5)
2             
⌠             
⎮         3   
⎮ (-x + 4)  dx
⌡             
4             
-4

(6)
0              
⌠              
⎮          3   
⎮ (2⋅x - 1)  dx
⌡              
1              
0

$

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="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<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="sample20.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 f = (x) => 3;

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 = [],
        x3 = -2,
        x4 = 5,
        lines = [[x3, y1, x3, y2, 'red'],
                 [x4, y1, x4, y2, 'red']],
        fns = [[f, 'green']],
        fns1 = [],
        fns2 = [];

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

            if (Math.abs(y) < Infinity) {
                points.push([x, y, color]);
            }
        }
    });
    fns1.forEach((o) => {
        let [fn, color] = o;
        
        lines.push([x1, fn(x1), x2, fn(x2), color]);
    });
    fns2.forEach((o) => {
        let [fn, color] = o;

        for (let x = x1; x <= x2; x += dx0) {
            let g = fn(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8月25日金曜日

数学 - Python - JavaScript - 細分による加法 - 積分法 – 定積分の性質と計算 - 簡単な例(整級数)

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