数学 – Python - JavaScript - 数学はここから始まる - 数 – 高次方程式 – 高次方程式の解法(直方体の体積(容積))

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)、高次方程式の解法の問29.を取り組んでみる。

$\begin{array}{l} x (12 - 2 x) (16 - 2 x) = 128 \\ x (6 - x) (8 - x) = 32 \\ x^{3} - 14 x^{2} + 48 x - 32 = 0 \\ (x - 4) (x^{2} - 10 x + 8) = 0 \\ x^{2} - 10 x + 8 = 0 \\ x = 5 \pm \sqrt{25 - 8} \\ = 5 \pm \sqrt{17} \\ x = 4, 5 \pm \sqrt{17} \end{array}$

x の範囲について。

$\begin{array}{l} 12 - 2 x > 0 \\ x < 6 \\ 0 < x < 6 \end{array}$

よって、求める x の値は、

$x = 4, 5 - \sqrt{17}$

コード(Emacs)

Python 3

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

print('29.')

x = symbols('x', real=True)
f = x * (12 - 2 * x) * (16 - 2 * x)
eq = f - 128
xs = solve(eq, x)

pprint(xs)
print()

for x0 in xs:
    if 0 < x0 < 6:
        pprint(x0)
    print()

p = plot(f, 128, (x, -5, 10), ylim=(-10, 150), show=False, legend=True)
colors = ['red', 'green']

for i, color in enumerate(colors):
    p[i].line_color = color

p.save('sample29.png')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample29.py
29.
[4, -√17 + 5, √17 + 5]

4

-√17 + 5


$

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="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-15">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="15">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-150">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="150">

<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="sample29.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 * (12 - 2 * x) * (16 - 2 * x), 'red']];
    
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 = [[x1, 128, x2, 128, 'green'],
                 [4, y1, 4, y2, 'blue'],
                 [5 - Math.sqrt(17), y1, 5 - Math.sqrt(17), y2, 'brown'],
                 [5 + Math.sqrt(17), y1, 5 + Math.sqrt(17), y2, 'orange']];
    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();

r = dx =
x1 = x2 =
y1 = y2 =

Kamimura's blog

2018年12月14日金曜日

数学 – Python - JavaScript - 数学はここから始まる - 数 – 高次方程式 – 高次方程式の解法(直方体の体積(容積))

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