2017年8月17日木曜日

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)、置換積分法、問13.を取り組んでみる。


    1. 3x=t 1= dt dx x=3t ( 3t ) t ( 1 ) dt dx dx = ( 3t ) t dt = ( 3 t 1 2 t 3 2 )dt = 2 5 t 5 2 2 t 3 2 = 2 5 ( 3x ) 5 2 2 ( 3x ) 3 2

    2. 3x =t 1 2 ( 3x ) 1 2 = dt dx 1 2t = dt dx 3x= t 2 x=3 t 2 ( 3 t 2 )t( 2t )dt =2 ( t 2 3 ) t 2 dt =2 ( t 4 3 t 2 )dt =2( 1 5 t 5 t 3 ) =2( 1 5 ( 3x ) 5 2 ( 3x ) 3 2 )

    3. logx=t 1 x = dt dx t 2 dt = 1 3 t 3 = ( logx ) 3 3

    4. x 2 +1=t 2x= dt dx 1 2 2x x 2 +1 dx = 1 2 1 t dx = 1 2 2 t 1 2 = x 2 +1

    5. x 2 +1=t 2x= dt dx 1 2 2x ( x 2 +1 ) n dx = 1 2 t n |dt = 1 2 1 n+1 t n+1 = ( x 2 +1 ) 1n 2( 1n )

コード(Emacs)

Python 3

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

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

print('13.')
x = symbols('x')
n = symbols('n', positive=True, integer=True)
fs = [x * sqrt(3 - x),
      x * sqrt(3 - x),
      log(x) ** 2 / x,
      x / sqrt(x ** 2 + 1),
      x / (x ** 2 + 1) ** n]


for i, f in enumerate(fs, 1):
    print(f'({i})')
    I = Integral(f, x)
    for o in [I, I.doit()]:
        pprint(o.factor())
        print()
    try:
        p = plot(f, show=False, legend=True)
        p.save(f'sample13_{i}.svg')
    except Exception as err:
        print(type(err), err)
    print()

入出力結果(Terminal, IPython)

$ ./sample13.py
13.
(1)
⌠                
⎮     ________   
⎮ x⋅╲╱ -x + 3  dx
⌡                

⎧     2   _______           _______          _______             
⎪2⋅ⅈ⋅x ⋅╲╱ x - 3    2⋅ⅈ⋅x⋅╲╱ x - 3    12⋅ⅈ⋅╲╱ x - 3       │x│    
⎪──────────────── - ─────────────── - ──────────────  for ─── > 1
⎪       5                  5                5              3     
⎨                                                                
⎪    2   ________         ________        ________               
⎪ 2⋅x ⋅╲╱ -x + 3    2⋅x⋅╲╱ -x + 3    12⋅╲╱ -x + 3                
⎪ ─────────────── - ────────────── - ─────────────     otherwise 
⎩        5                5                5                     


(2)
⌠                
⎮     ________   
⎮ x⋅╲╱ -x + 3  dx
⌡                

⎧     2   _______           _______          _______             
⎪2⋅ⅈ⋅x ⋅╲╱ x - 3    2⋅ⅈ⋅x⋅╲╱ x - 3    12⋅ⅈ⋅╲╱ x - 3       │x│    
⎪──────────────── - ─────────────── - ──────────────  for ─── > 1
⎪       5                  5                5              3     
⎨                                                                
⎪    2   ________         ________        ________               
⎪ 2⋅x ⋅╲╱ -x + 3    2⋅x⋅╲╱ -x + 3    12⋅╲╱ -x + 3                
⎪ ─────────────── - ────────────── - ─────────────     otherwise 
⎩        5                5                5                     


(3)
⌠           
⎮    2      
⎮ log (x)   
⎮ ─────── dx
⎮    x      
⌡           

   3   
log (x)
───────
   3   


(4)
⌠               
⎮      x        
⎮ ─────────── dx
⎮    ________   
⎮   ╱  2        
⎮ ╲╱  x  + 1    
⌡               

   ________
  ╱  2     
╲╱  x  + 1 


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

⎧                           ⎛ 2    ⎞                                   
⎪                        log⎝x  + 1⎠                                   
⎪                        ───────────                          for n = 1
⎪                             2                                        
⎪                                                                      
⎨                2                                                     
⎪               x                             1                        
⎪- ─────────────────────────── - ───────────────────────────  otherwise
⎪              n             n               n             n           
⎪      ⎛ 2    ⎞      ⎛ 2    ⎞        ⎛ 2    ⎞      ⎛ 2    ⎞            
⎩  2⋅n⋅⎝x  + 1⎠  - 2⋅⎝x  + 1⎠    2⋅n⋅⎝x  + 1⎠  - 2⋅⎝x  + 1⎠            

<class 'ValueError'> The same variable should be used in all univariate expressions being plotted.

$

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="n0">n = </label>
<input id="n0" type="number" min="2" 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="sample13.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_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 f3 = (x) => Math.log(x) ** 2 / 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),
        n0 = parseInt(input_n0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [],
        f5 = (x) => x / (x ** 2 + 1) ** n0,
        fns = [[f3, 'green'],
               [f5, '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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