2017年7月23日日曜日

学習環境

解析入門〈2〉(松坂 和夫(著)、岩波書店)の第8章(積分の計算)、8.2(定積分の計算)、問題1-1、2、3.を取り組んでみる。


    1. axa 1sint1 aasinta x=asint 0xa 0asinta 0sint1 0t π 2 dx=acostdt a 2 x 2 =a 1 sin 2 t =acost 0 π 2 a( cost )a( cost ) dt = a 2 0 π 2 cos 2 t dt = a 2 0 π 2 cos2t+cos0 2 dt = a 2 0 π 2 cos2t+1 2 dt = a 2 2 [ 1 2 sin2t+t ] 0 π 2 = a 2 2 ( ( 1 2 sinπ+ π 2 )( 1 2 sin0+0 ) ) = a 2 2 ( π 2 ) = π a 2 4

    2. [ arcsin x a ] 0 a =arcsin1arcsin0 = π 2

    3. 0 1 1 ( x+ 1 2 ) 2 + 3 4 dx 0 1 1 ( x+ 1 2 ) 2 + ( 3 2 ) 2 dx t=x+ 1 2 x=0,t= 1 2 x=1,t= 3 2 x=t 1 2 dx=dt 1 2 3 2 1 t 2 + ( 3 2 ) 2 dx = 1 3 2 [ arctan x 3 2 ] 1 2 3 2 = 2 3 ( arctan 3 2 3 2 arctan 1 2 3 2 ) = 2 3 ( arctan 3 arctan 1 3 ) = 2 3 ( π 3 π 6 ) = 2π 6 3 = π 3 3

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, sqrt, Pow

print('1.')
x = symbols('x', real=True)
a = symbols('a', positive=True)
fs = [(sqrt(Pow(a, 2) - Pow(x, 2)), (0, a)),
      (1 / sqrt(Pow(a, 2) - Pow(x, 2)), (0, a)),
      (1 / (1 + x + x ** 2), (0, 1))]

for i, (f, (x1, x2)) in enumerate(fs, 1):
    print(f'({i})')
    I = Integral(f, (x, x1, x2))
    pprint(I)
    I = I.doit()
    pprint(I)
    print('factor:')
    pprint(I.factor())
    print('expand:')
    pprint(I.expand())

入出力結果(Terminal, IPython)

$ ./sample1.py
1.
(1)
a                
⌠                
⎮    _________   
⎮   ╱  2    2    
⎮ ╲╱  a  - x   dx
⌡                
0                
   2
π⋅a 
────
 4  
factor:
   2
π⋅a 
────
 4  
expand:
   2
π⋅a 
────
 4  
(2)
a                
⌠                
⎮      1         
⎮ ──────────── dx
⎮    _________   
⎮   ╱  2    2    
⎮ ╲╱  a  - x     
⌡                
0                
π
─
2
factor:
π
─
2
expand:
π
─
2
(3)
1              
⌠              
⎮     1        
⎮ ────────── dx
⎮  2           
⎮ x  + x + 1   
⌡              
0              
√3⋅π
────
 9  
factor:
√3⋅π
────
 9  
expand:
√3⋅π
────
 9  
$

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="-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="a0">a = </label>
<input id="a0" type="number" min="0" 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,
    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_a0 = document.querySelector('#a0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
             input_a0],
    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 f3 = (x) => 1 / (1 + 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),
        a0 = parseFloat(input_a0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        lines = [[a0, y1, a0, y2, 'red'],
                 [1, y1, 1, y2, 'blue']],
        f1 = (x) => Math.sqrt(a0 ** 2 - x ** 2),
        f2 = (x) => 1 / Math.sqrt(a0 ** 2 - x ** 2),
        fns = [[f1, 'green'],
               [f2, 'brown'],
               [f3, 'orange']];

    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]);
                }
            }
        });
    
    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('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.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.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
    p(fns.join('\n'));
};

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








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