2017年8月13日日曜日

学習環境

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)、19.2(不定積分の計算)、x^αの不定積分、問8.を取り組んでみる。


    1. x2 x +1 x dx = 1dx 2 x 1 2 dx + 1 x dx =x4 x 1 2 +logx

    2. x 2 +2x+1 x 2 dx = 1dx + 2 x dx + x 2 dx =x2log| x | x 1

    3. x 2 dx 2 x dx + 3dx 4xdx = x 1 2log| x |+3x2 x 2

    4. x x +3x+3 x +1 x dx = x 1 2 dx + 3dx + 3 x 1 2 dx + 1 x dx = 2 3 x 3 2 +3x+6 x 1 2 +logx

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, sqrt, Rational, plot

print('8.')
x = symbols('x')
fs = [(sqrt(x) - 1) ** 2 / x,
      (x + 1) ** 2 / x ** 2,
      (1 - 2 * x + 3 * x ** 2 - 4 * x ** 3) / x ** 2,
      (sqrt(x) + 1) ** 3 / x]


for i, f in enumerate(fs, 1):
    print(f'({i})')
    I = Integral(f, x)
    for o in [I, I.doit()]:
        pprint(o)
        print()
    p = plot(f, show=False, legend=True)
    p.save(f'sample8_{i}.svg')
    print()

入出力結果(Terminal, IPython)

$ ./sample8.py
8.
(1)
⌠             
⎮         2   
⎮ (√x - 1)    
⎮ ───────── dx
⎮     x       
⌡             

-4⋅√x + x + log(x)


(2)
⌠            
⎮        2   
⎮ (x + 1)    
⎮ ──────── dx
⎮     2      
⎮    x       
⌡            

               1
x + 2⋅log(x) - ─
               x


(3)
⌠                           
⎮      3      2             
⎮ - 4⋅x  + 3⋅x  - 2⋅x + 1   
⎮ ─────────────────────── dx
⎮             2             
⎮            x              
⌡                           

     2                    1
- 2⋅x  + 3⋅x - 2⋅log(x) - ─
                          x


(4)
⌠             
⎮         3   
⎮ (√x + 1)    
⎮ ───────── dx
⎮     x       
⌡             

   3/2                      
2⋅x                         
────── + 6⋅√x + 3⋅x + log(x)
  3                         


$

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="sample8.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 f1 = (x) => (Math.sqrt(x) - 1) ** 2 / x,
    f2 = (x) => (x + 1) ** 2 / x ** 2,
    f3 = (x) => (1 - 2 * x + 3 * x ** 2 - 4 * x ** 3) / x ** 2,
    f4 = (x) => (Math.sqrt(x) + 1) ** 3 / 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 = [],
        lines = [],
        fns = [[f1, 'red'],
               [f2, 'green'],
               [f3, 'blue'],
               [f4, 'orange']],
        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();







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