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(累乗(べき乗)、三角関数(正弦、余弦、3倍角)、指数関数、対数関数)

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21.を取り組んでみる。

1. $\begin{array}{l} {[\frac{3}{4} x^{\frac{4}{3}}]}_{0}^{8} \\ = \frac{3}{4} \cdot 8^{\frac{4}{3}} \\ = \frac{3}{4} \cdot {(2^{3})}^{\frac{4}{3}} \\ = 12 \end{array}$
2. $\begin{array}{l} {[- \frac{1}{x}]}_{2}^{1} \\ = {[\frac{1}{x}]}_{1}^{2} \\ = \frac{1}{2} - 1 \\ = - \frac{1}{2} \end{array}$
3. $\begin{array}{l} {[- \cos θ]}_{0}^{\frac{π}{2}} \\ = - \cos \frac{π}{2} + \cos 0 \\ = 1 \end{array}$
4. $\begin{array}{l} {[\frac{1}{3} \sin 3 x]}_{- \frac{π}{6}}^{\frac{π}{3}} \\ = \frac{1}{3} (\sin (3 \cdot \frac{π}{3}) - \sin (3 \cdot (- \frac{π}{6}))) \\ = \frac{1}{3} (\sin π + \sin \frac{π}{2}) \\ = \frac{1}{3} (0 + 1) \\ = \frac{1}{3} \end{array}$
5. $\begin{array}{l} {[\frac{1}{2} e^{2 x}]}_{- 1}^{1} \\ = \frac{1}{2} (e^{2} - e^{- 2}) \end{array}$
6. $\begin{array}{l} \int_{e}^{1} \log x d x \\ = {[x \log x]}_{e}^{1} - \int_{e}^{1} x \frac{1}{x} d x \\ = (\log 1 - e \log e) - {[x]}_{e}^{1} \\ = - e - (1 - e) \\ = - 1 \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, plot, Rational, sin, pi, cos, exp, log, E

print('21.')
x = symbols('x')
fs = [(x ** Rational(1, 3), (0, 8)),
      (1 / x ** 2, (2, 1)),
      (sin(x), (0, pi / 2)),
      (cos(3 * x), (-pi / 6, pi / 3)),
      (exp(2 * x), (-1, 1)),
      (log(x), (E, 1))]


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'sample21_{i}.svg')
    except Exception as err:
        print(type(err), err)
    print()

入出力結果(Terminal, IPython)

$ ./sample21.py
21.
(1)
8         
⌠         
⎮ 3 ___   
⎮ ╲╱ x  dx
⌡         
0         
12

(2)
1      
⌠      
⎮ 1    
⎮ ── dx
⎮  2   
⎮ x    
⌡      
2      
-1/2

(3)
π          
─          
2          
⌠          
⎮ sin(x) dx
⌡          
0          
1

(4)
 π             
 ─             
 3             
 ⌠             
 ⎮  cos(3⋅x) dx
 ⌡             
-π             
───            
 6             
1/3

(5)
1         
⌠         
⎮   2⋅x   
⎮  ℯ    dx
⌡         
-1        
   -2    2
  ℯ     ℯ 
- ─── + ──
   2    2 

(6)
1          
⌠          
⎮ log(x) dx
⌡          
ℯ          
-1

$

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="sample21.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 f2 = (x) => 1 / x ** 2,
    f5 = (x) => Math.exp(2 * x),
    f6 = (x) => Math.log(x);

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 = [],
        fns = [[f2, 'red'],
               [f5, 'green'],
               [f6, 'blue']],
        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26日土曜日

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(累乗(べき乗)、三角関数(正弦、余弦、3倍角)、指数関数、対数関数)

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