2017年9月24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20章(面積、体積、長さ - 積分法の応用)、20.1(面積)、面積の公式、問9.を取り組んでみる。


  1. y= 4x =2 x y= 1 4 x 2 2 x = 1 4 x 2 8 x = x 2 64x= x 4 x 4 64x=0 x( x 3 64 )=0 x=0,4 0 4 ( 2 x 1 4 x 2 )dx = [ 4 3 x 3 2 1 12 x 3 ] 0 4 = 4 3 · 4 3 2 1 12 · 4 3 = 4 3 · ( 2 2 ) 3 2 16 3 = 32 3 16 3 = 16 3 y=3x 2 x =3x 4x=9+ x 2 6x x 2 10x+9=0 ( x1 )( x9 )=0 x=1,9 y0 3x0 x3 x=1 1 4 x 2 =3x x 2 +4x12=0 ( x+6 )( x2 )=0 x=6,2 y0 3x0 x3 x=2 0 1 ( 2 x 1 4 x 2 )dx + 1 2 ( ( 3x ) 1 4 x 2 )dx = 0 1 2 x dx + 0 1 ( 1 4 x 2 )dx + 1 2 ( 3x )dx ++ 1 2 ( 1 4 x 2 )dx = [ 4 3 x 3 2 ] 0 1 + [ 3x 1 2 x 2 ] 1 2 + 0 2 ( 1 4 x 2 )dx = 4 3 +( ( 62 )( 3 1 2 ) ) 1 4 [ 1 3 x 3 ] 0 2 = 4 3 +4 5 2 1 4 · 8 3 = 16+48308 12 = 26 12 = 13 6 13 6 :( 16 3 13 6 ) = 13 6 : 19 6 =13:19

コード(Emacs)

Python 3

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

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

print('9.')
x = symbols('x')
f = 2 * sqrt(x)
g = Rational(1, 4) * x ** 2
h = 3 - x
pprint(solve(f - g))
pprint(solve(f - h))
pprint(solve(g - h))

S1 = Integral(f - g, (x, 0, 1)) + Integral(h - g, (x, 1, 2))
S2 = Integral(f - h, (x, 1, 2)) + Integral(f - g, (x, 2, 4))

for S in [S1, S2]:
    pprint(S)
    print()
    pprint(S.doit())
    print()

pprint((S1.doit() / S2.doit()).expand())

入出力結果(Terminal, Jupyter(IPython))

$ ./sample9.py
9.
[0, 4]
[1]
[-6, 2]
2                     1               
⌠                     ⌠               
⎮ ⎛   2        ⎞      ⎮ ⎛        2⎞   
⎮ ⎜  x         ⎟      ⎮ ⎜       x ⎟   
⎮ ⎜- ── - x + 3⎟ dx + ⎮ ⎜2⋅√x - ──⎟ dx
⎮ ⎝  4         ⎠      ⎮ ⎝       4 ⎠   
⌡                     ⌡               
1                     0               

13/6

                      4               
                      ⌠               
2                     ⎮ ⎛        2⎞   
⌠                     ⎮ ⎜       x ⎟   
⎮ (2⋅√x + x - 3) dx + ⎮ ⎜2⋅√x - ──⎟ dx
⌡                     ⎮ ⎝       4 ⎠   
1                     ⌡               
                      2               

19/6

13
──
19
$

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="sample9.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) => 2 * Math.sqrt(x),
    g = (x) => 1 / 4 * x ** 2,
    h = (x) => 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 = [[f, 'red'],
               [g, 'green']],
        fns1 = [[h, 'blue']],
        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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