2018年10月27日土曜日

数学 – Python - JavaScript - 数学はここから始まる - 数 – 方程式とその解法 – 2次方程式の解法(一般化、解の公式)

1. $\begin{array}{}x=\frac{-5±\sqrt{25+96}}{8}\\ =\frac{-5±\sqrt{121}}{8}\\ =\frac{-5±11}{8}\\ =-2,\frac{3}{4}\end{array}$

2. $\begin{array}{}x=\frac{2±\sqrt{4+16}}{2}\\ =\frac{2\left(1±\sqrt{1+4}\right)}{2}\\ =1±\sqrt{5}\end{array}$

3. $\begin{array}{}x=\frac{-8±\sqrt{{8}^{2}+4·16}}{2}\\ =\frac{2\left(-4±2\sqrt{4+16}\right)}{2}\\ =-4±2\sqrt{5}\end{array}$

4. $\begin{array}{}x=\frac{11±\sqrt{121-76}}{2}\\ =\frac{11±\sqrt{45}}{2}\\ =\frac{11±3\sqrt{5}}{2}\end{array}$

5. $\begin{array}{}x=\frac{-17±\sqrt{1{7}^{2}-24·12}}{12}\\ =\frac{-17±\sqrt{289-288}}{12}\\ =-\frac{3}{2},-\frac{4}{3}\end{array}$

6. $\begin{array}{}x=\frac{4±\sqrt{{4}^{2}-4·3·\left(-2\right)}}{2·3}\\ =\frac{2\left(2±\sqrt{4+6}\right)}{2-3}\\ =\frac{2±\sqrt{10}}{3}\end{array}$

コード(Emacs)

Python 3

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

print('4.')

x = symbols('x', real=True)

eqs = [(4, 5, -6),
(1, -2, -4),
(1, 8, -16),
(1, -11, 19),
(6, 17, 12),
(3, -4, -2)]

for i, (a, b, c) in enumerate(eqs, 1):
print(f'({i})')
pprint(solve(a * x ** 2 + b * x + c, dict=True))
print()

p = plot(*[a * x ** 2 + b * x + c for a, b, c in eqs], show=False, legend=True)
colors = ['red', 'green', 'blue', 'orange', 'brown', 'purple']
for i, color in enumerate(colors):
p[i].line_color = color
p.save('sample4.svg')


$./sample4.py 4. (1) [{x: -2}, {x: 3/4}] (2) [{x: 1 + √5}, {x: -√5 + 1}] (3) [{x: -4 + 4⋅√2}, {x: -4⋅√2 - 4}] (4) ⎡⎧ 3⋅√5 11⎫ ⎧ 3⋅√5 11⎫⎤ ⎢⎨x: - ──── + ──⎬, ⎨x: ──── + ──⎬⎥ ⎣⎩ 2 2 ⎭ ⎩ 2 2 ⎭⎦ (5) [{x: -3/2}, {x: -4/3}] (6) ⎡⎧ 2 √10⎫ ⎧ √10 2⎫⎤ ⎢⎨x: ─ + ───⎬, ⎨x: - ─── + ─⎬⎥ ⎣⎩ 3 3 ⎭ ⎩ 3 3⎭⎦$


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="0.001" 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">

<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'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
p = (x) => pre0.textContent += x + '\n';

let fns = [[x => 4 * x ** 2 + 5 * x - 6, '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 = [[-2, y1, -2, y2, 'green'],
[3 / 4, y1, 3 / 4, y2, 'blue']];

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();