2018年12月24日月曜日

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34.を取り組んでみる。


  1. 2辺の長さ を x cm、 y cm とする。

    周の長さは42 cm なので

    2 x + y = 42 x + y = 21

    円の直径は15 cm なので、

    x 2 + y 2 = 15 x 2 + y 2 = 1 5 2 x + y 2 - 2 x y = 1 5 2 2 1 2 - 2 x y = 1 5 2 2 x y = 2 1 2 - 1 5 2 2 x y = 21 + 15 21 - 15 2 x y = 36 · 6 x y = 108

    2次方程式の解の係数の関係により、 求める2辺の長さは、

    t 2 - 21 t + 108 = 0 t = 21 ± 441 - 432 2 = 21 ± 9 2 = 21 ± 3 2 = 9 , 12

コード(Emacs)

Python 3

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

print('34.')

x, y = symbols('x, y', positive=True)
eqs = (2 * (x + y) - 42, sqrt(x ** 2 + y ** 2) - 15)

pprint(solve(eqs, dict=True))

fs = solve(eqs[0], y) + solve(eqs[1], y)

pprint(fs)
p = plot(*fs, (x, -25, 25), ylim=(-25, 25), legend=True, show=False)
colors = ['red', 'green', 'green']

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

p.save('sample34.png')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample34.py
34.
[{x: 9, y: 12}, {x: 12, y: 9}]
⎡             ____________     ____________⎤
⎢            ╱    2           ╱    2       ⎥
⎣-x + 21, -╲╱  - x  + 225 , ╲╱  - x  + 225 ⎦
$

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="-20">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="20">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-20">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="20">

<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="sample34.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 + 21, 'red'],
           [x => -Math.sqrt(-(x ** 2) + 225), 'green'],
           [x => Math.sqrt(-(x ** 2) + 225), '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 = [[9, y1, 9, y2, 'blue'],
                 [12, y1, 12, y2, '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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