2018年12月27日木曜日

学習環境

数学読本〈1〉数・式の計算/方程式/不等式 (松坂 和夫(著)、岩波書店)の第3章(数学の威力を発揮する - 方程式)、3.3(高次方程式)、連立2次方程式の問36.を取り組んでみる。



    1. x 2 - 5 x y + 6 y 2 = 0 x - 3 y x - 2 y = 0 x - 3 y = 0 y = 1 3 x x 2 + 1 3 x 2 + 2 · 1 9 x 2 = 56 9 x 2 + 3 x 2 + 2 x 2 = 504 14 x 2 = 504 x 2 = 36 x = ± 6 , y = ± 2 x - 2 y = 0 y = 1 2 x x 2 + 1 2 x 2 + 1 2 x 2 = 56 2 x 2 = 56 x 2 = 28 x = ± 2 7 , y = ± 7

      (複号同順)


    2. 10 x 2 - 4 y 2 = - 6 - 4 x 2 + x y + 2 y 2 = 6 6 x 2 + x y - 2 y 2 = 0 2 x - y 3 x + 2 y = 0 2 x - y = 0 y = 2 x 10 x 2 - 16 x 2 = - 6 x 2 = 1 x = ± 1 , y = ± 2 3 x + 2 y = 0 y = - 3 2 x 10 x 2 - 4 · 9 4 x 2 = - 6 x 2 = - 6 , x = ± 6 i , y = 3 6 2 i

      (複号同順)


      ここから

      y = - 6 x x 2 + 36 x 2 = 13 x 4 - 13 x 2 + 36 = 0 x 2 - 9 x 2 - 4 = 0 x = ± 3 , y = 2 x = ± 2 , y = 3

      (複号同順)


      ここから

      y = - 15 x x 2 - 225 x 2 = 16 x 4 - 16 x 2 - 225 = 0 x 2 - 25 x 2 + 9 = 0 x = ± 5 , y = 3 x = ± 3 i , y = ± 5 i

      (複号同順)


    3. 3 x y = 18 x y = 6 x + y = 5

      2次方程式の解と係数の関係により、

      t 2 - 5 t + 6 = 0 t - 3 t - 2 = 0 t = 2 , 3 x = 2 , y = 3 x = 3 , y = 2

    4. 60 x y + 48 x - 36 y - 24 = 0 60 x y + 45 x - 35 y - 20 = 0 3 x - y - 4 = 0 y = 3 x - 4 5 x 3 x - 4 + 4 x - 3 3 x - 4 - 2 = 0 15 x 2 - 20 x + 4 x - 9 x + 12 - 2 = 0 15 x 2 - 25 x + 10 = 0 3 x 2 - 5 x + 2 = 0 x - 1 3 x - 2 = 0 x = 1 , y = - 1 x = 2 3 , y = - 2

    5. y = 2 - x x 3 + 2 - x 3 = - 4 x 3 + 8 - 12 x + 6 x 2 - x 3 + 4 = 0 6 x 2 - 12 x + 12 = 0 x 2 - 2 x + 2 = 0 x = 1 ± 1 - 2 = 1 ± i y = 1 i

      (複号同順)


    6. 5 x 2 - y 2 = 65 x 2 + y 2 x 2 - y 2 = - 65 x 2 - y 2 x 2 + y 2 + 5 = 0 x 2 + y 2 = 0 y 2 = - x 2 x 2 + x 2 = 13 x 2 = 13 2 x = ± 13 2 y 2 = - 13 2 y = ± 13 2 i x = ± 13 2 , y = ± 13 2 i x 2 + y 2 + 5 = 0 y 2 = - x 2 - 5 x 2 + x 2 + 5 = 13 x 2 = 4 x = ± 2 y 2 = - 4 - 5 = - 9 y = ± 3 i

      (符号の順番は任意)

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Rational, solve, plot

print('36.')

x, y = symbols('x, y')

ts = [(x ** 2 + x * y + 2 * y ** 2 - 56, x ** 2 - 5 * x * y + 6 * y ** 2),
      (5 * x ** 2 - 2 * y ** 2 + 3, -4 * x ** 2 + x * y + 2 * y ** 2 - 6),
      (x ** 2 + y ** 2 - 13, x * y - 6),
      (x ** 2 - y ** 2 - 16, x * y + 15),
      (2 * y + x + y - 11, 2 * x * y - x - y - 7),
      (5 * x * y + 4 * x - 3 * y - 2, 12 * x * y + 9 * x - 7 * y - 4),
      (x + y - 2, x ** 3 + y ** 3 + 4),
      (x ** 2 - y ** 2 - 13, x ** 4 - y ** 4 + 65)]

for i, t in enumerate(ts, 1):
    print(f'({i})')
    pprint(solve(t, dict=True))


fss = [solve(eq, y) for eq in ts[0]]
fs, gs = fss

p = plot(*fs, *gs, legend=True, show=False)
colors = ['red', 'red', 'green', 'green']
for i, color in enumerate(colors):
    p[i].line_color = color
p.save('sample36.png')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample36.py
36.
(1)
[{x: -6, y: -2}, {x: 6, y: 2}, {x: -2⋅√7, y: -√7}, {x: 2⋅√7, y: √7}]
(2)
⎡                              ⎧             3⋅√6⋅ⅈ⎫  ⎧            -3⋅√6⋅ⅈ ⎫⎤
⎢{x: -1, y: -2}, {x: 1, y: 2}, ⎨x: -√6⋅ⅈ, y: ──────⎬, ⎨x: √6⋅ⅈ, y: ────────⎬⎥
⎣                              ⎩               2   ⎭  ⎩               2    ⎭⎦
(3)
[{x: -3, y: -2}, {x: -2, y: -3}, {x: 2, y: 3}, {x: 3, y: 2}]
(4)
[{x: -5, y: 3}, {x: 5, y: -3}, {x: -3⋅ⅈ, y: -5⋅ⅈ}, {x: 3⋅ⅈ, y: 5⋅ⅈ}]
(5)
[{x: 2, y: 3}, {x: 8, y: 1}]
(6)
[{x: 2/3, y: -2}, {x: 1, y: -1}]
(7)
[{x: 1 - ⅈ, y: 1 + ⅈ}, {x: 1 + ⅈ, y: 1 - ⅈ}]
(8)
[{x: -2, y: -3⋅ⅈ}, {x: -2, y: 3⋅ⅈ}, {x: 2, y: -3⋅ⅈ}, {x: 2, y: 3⋅ⅈ}]
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.01">
<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="sample36.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';

let fns = [[x => -x / 4 - Math.sqrt(-7 * x ** 2 + 448) / 4, 'red'],
           [x => -x / 4 + Math.sqrt(-7 * x ** 2 + 448) / 4, 'red'],
           [x => x / 3, 'green'],
           [x => x / 2, 'green']];
    
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 = [],
        x0 = 10,
        y0 = 12,
        lines = [[-2 * Math.sqrt(7), y1, -2 * Math.sqrt(7), y2, 'blue'],
                 [x1, -Math.sqrt(7), x2, -Math.sqrt(7), 'blue'],
                 [2 * Math.sqrt(7), y1, 2 * Math.sqrt(7), y2, 'brown'],
                 [x1, Math.sqrt(7), x2, Math.sqrt(7), 'brown']];
        
    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                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('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].forEach((fs) => p(fs.join('\n')));
};

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







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