## 2017年10月20日金曜日

### 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 - 対数関数(接線の方程式、微分、導関数)

1. $\begin{array}{l}\frac{d}{dx}\mathrm{log}\left(x+1\right)=\frac{1}{x+1}\\ \\ f\left(x\right)=\frac{1}{3+1}\left(x-3\right)+\mathrm{log}\left(3+1\right)\\ =\frac{1}{4}\left(x-3\right)+\mathrm{log}4\\ =\frac{1}{4}x-\frac{3}{4}+2\mathrm{log}2\end{array}$

2. $\begin{array}{l}\frac{d}{dx}\mathrm{log}\left(2x-5\right)\\ =\frac{1}{2x-5}·2\\ =\frac{2}{2x-5}\\ \\ f\left(x\right)=\frac{2}{2·4-5}\left(x-4\right)+\mathrm{log}\left(2·4-5\right)\\ =\frac{2}{3}\left(x-4\right)+\mathrm{log}3\\ =\frac{2}{3}x-\frac{8}{3}+\mathrm{log}3\end{array}$

コード(Emacs)

Python 3

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

print('4.')
x = symbols('x')
fs = [(log(x + 1), 3),
(log(2 * x - 5), 4)]

for i, (f, x0) in enumerate(fs, 3):
print(f'{i}.')
D = Derivative(f, x, 1)
f1 = D.doit()
g = f1.subs({x: x0}) * (x - x0) + f.subs({x: x0})
for t in [D, f1, g]:
pprint(t)
print()
p = plot(f, g, show=False, legend=True)
p.save(f'sample{i}.svg')
print()


$./sample4.py 4. 3. d ──(log(x + 1)) dx 1 ───── x + 1 x 3 ─ - ─ + log(4) 4 4 4. d ──(log(2⋅x - 5)) dx 2 ─────── 2⋅x - 5 2⋅x 8 ─── - ─ + log(3) 3 3$


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="-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">
<br>
<label for="x0">x0 = </label>
<input id="x0" type="number" value="3">
<label for="a0">a0 = </label>
<input id="a0" type="number" value="1">
<label for="b0">b0 = </label>
<input id="b0" type="number" value="2">

<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,
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_x0 = document.querySelector('#x0'),
input_a0 = document.querySelector('#a0'),
input_b0 = document.querySelector('#b0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_x0, input_a0, input_b0],
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),
x0 = parseFloat(input_x0.value),
a0 = parseFloat(input_a0.value),
b0 = parseFloat(input_b0.value);

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

let points = [],
lines = [[x0, y1, x0, y2, 'red']],
f = (x) => Math.log(a0 * x + b0),
f1 = (x) => 1 / (a0 * x + b0) * a0,
g = (x) => f1(x0) * (x - x0) + f(x0),
fns = [[f, 'green']],
fns1 = [[g, 'blue']],
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();