2017年9月19日火曜日

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4.を取り組んでみる。


    • S=| α β ( a x 2 +bx+c )dx | =| α β a( xα )( xβ )dx | =| a 6 ( βα ) 3 | = | a | 6 ( βα ) 3

    • α= b D 2a β= b+ D 2a βα= D a S= | a | 6 D 3 2 a 3 = D 3 2 6 a 2

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, solve

print('4.')
a, b, c, x = symbols('a b c x')
f = a * x ** 2 + b * x + c
s = solve(f, x)
pprint(s)
print()

α, β = s
I = abs(Integral(f, (x, α, β)))
for g in [I, I.doit(), abs(a / 6 * (β - α) ** 3)]:
    pprint(g.factor())
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample4.py
4.
⎡        _____________   ⎛       _____________⎞ ⎤
⎢       ╱           2    ⎜      ╱           2 ⎟ ⎥
⎢-b + ╲╱  -4⋅a⋅c + b    -⎝b + ╲╱  -4⋅a⋅c + b  ⎠ ⎥
⎢─────────────────────, ────────────────────────⎥
⎣         2⋅a                     2⋅a           ⎦

│                                           _____________                     
│           _____________                  ╱           2                      
│          ╱           2            b    ╲╱  -4⋅a⋅c + b                       
│   b    ╲╱  -4⋅a⋅c + b          - ─── - ────────────────              b    ╲╱
│- ─── - ────────────────          2⋅a         2⋅a                  - ─── - ──
│  2⋅a         2⋅a                          ⌠                         2⋅a     
│           ⌠                               ⎮                2                
│           ⎮             c dx +            ⎮             a⋅x  dx +           
│           ⌡                               ⌡                                 
│           _____________                   _____________                     
│          ╱           2                   ╱           2                      
│   b    ╲╱  -4⋅a⋅c + b             b    ╲╱  -4⋅a⋅c + b                b    ╲╱
│- ─── + ────────────────        - ─── + ────────────────           - ─── + ──
│  2⋅a         2⋅a                 2⋅a         2⋅a                    2⋅a     

                     │
 _____________       │
╱           2        │
  -4⋅a⋅c + b         │
──────────────       │
    2⋅a              │
 ⌠                   │
 ⎮             b⋅x dx│
 ⌡                   │
 _____________       │
╱           2        │
  -4⋅a⋅c + b         │
──────────────       │
    2⋅a              │

│         _____________         _____________│
│        ╱           2     2   ╱           2 │
│  2⋅c⋅╲╱  -4⋅a⋅c + b     b ⋅╲╱  -4⋅a⋅c + b  │
│- ──────────────────── + ───────────────────│
│          3⋅a                       2       │
│                                 6⋅a        │

│       _____________         _____________│
│      ╱           2     2   ╱           2 │
│4⋅c⋅╲╱  -4⋅a⋅c + b     b ⋅╲╱  -4⋅a⋅c + b  │
│──────────────────── - ───────────────────│
│         a                       2        │
│                                a         │
────────────────────────────────────────────
                     6                      

$

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" value="1">
<label for="b0">b = </label>
<input id="b0" type="number" value="3">
<label for="c0">c = </label>
<input id="c0" type="number" 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="sample4.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'),
    input_b0 = document.querySelector('#b0'),    
    input_c0 = document.querySelector('#c0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_b0, input_c0],
    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 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),
        b0 = parseFloat(input_b0.value),        
        c0 = parseFloat(input_c0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        f = (x) => a0 * x ** 2 + b0 * x + c0,
        lines = [],
        fns = [[f, 'green']],
        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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