## 2017年12月27日水曜日

### 数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 指数関数と対数関数 - いくつかの応用(物質の分解率、時刻、残量)

1. 時刻 t における物質の量を

$f\left(t\right)=c{e}^{kt}$

とおく。

問題の仮定より、

$\begin{array}{}f\left(3\right)=\frac{90}{100}f\left(0\right)\\ c{e}^{3k}=\frac{9}{10}c\\ 3k=\mathrm{log}\frac{9}{10}\\ k=\frac{1}{3}\mathrm{log}\frac{9}{10}=\frac{1}{3}\left(\mathrm{log}9-\mathrm{log}10\right)\end{array}$

最初の量の半分が分解してしまう時刻。

$\begin{array}{}f\left(t\right)=\frac{1}{2}f\left(0\right)\\ {c}^{kt}=\frac{1}{2}C\\ kt=-\mathrm{log}2\\ t=\frac{-3\mathrm{log}2}{\mathrm{log}9-\mathrm{log}10}\\ =\frac{3\mathrm{log}2}{\mathrm{log}10-\mathrm{log}9}\end{array}$

コード(Emacs)

Python 3

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

t0 = 3 * log(2) / (log(10) - log(9))
C, t = symbols('C, t', real=True, nonnegative=True)
K = (log(9) - log(10)) / 3
f = C * exp(K * t)
eq = f - f.subs({t: 0}) / 2

t1 = solve(eq, t)[0]
for s in [f, eq, t1, t0.expand() == t1.expand()]:
pprint(s)
print()

C0 = 10
p = plot(f.subs({C: C0}), 90 * C0 / 100, C0, f.subs({C: C0, t: t0}),
(t, 0, t0), show=False, legend=True)
for i, color in enumerate(['red', 'green', 'blue', 'orange']):
p[i].line_color = color
p.save('sample8.svg')


$./sample8.py ⎛ log(10) log(9)⎞ t⋅⎜- ─────── + ──────⎟ ⎝ 3 3 ⎠ C⋅ℯ ⎛ log(10) log(9)⎞ t⋅⎜- ─────── + ──────⎟ ⎝ 3 3 ⎠ C C⋅ℯ - ─ 2 ⎛ 3 ⎞ ⎜ ─────────────────⎟ ⎜ -log(9) + log(10)⎟ log⎝2 ⎠ True$


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="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="20">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="20">
<br>
<label for="c0">C = </label>
<input id="c0" type="number" min="0" 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="sample8.js"></script>


JavaScript

let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
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_c0 = document.querySelector('#c0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
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),
c0 = parseFloat(input_c0.value);

if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}

let points = [],
f = (x) => c0 * Math.exp(Math.log(9 / 10) / 3 * x),
x0 = 3 * Math.log(2) / (Math.log(10) - Math.log(9)),
lines = [[x1, f(0), x2, f(0), 'red'],
[x1, 9 / 10 * f(0), x2, 9 / 10 * f(0), 'green'],
[x1, f(x0), x2, f(x0), 'blue']],
fns = [[f, 'orange']],
fns1 = [],
fns2 = [];

fns
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = f(x);

points.push([x, y, color]);
}
});

fns1
.forEach((o) => {
let [f, color] = o;

lines.push([x1, f(x1), x2, f(x2), color]);
});

fns2
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx0) {
let g = f(x);
lines.push([x1, g(x1), x2, g(x2), color]);
}
});

let xscale = d3.scaleLinear()
.domain([x1, x2])
let yscale = d3.scaleLinear()
.domain([y1, y2])

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();