## 2017年11月3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 - 指数関数(指数関数、対数関数、累乗、接線の方程式)

1. $\begin{array}{l}y\text{'}=\frac{1}{x}\\ g\left(x\right)=\frac{1}{e}\left(x-e\right)+\mathrm{log}e\\ =\frac{1}{e}x-1+1\\ =\frac{x}{e}\end{array}$

2. $\begin{array}{l}y\text{'}=\mathrm{log}x+x·\frac{1}{x}=\mathrm{log}x+1\\ g\left(x\right)=\left(\mathrm{log}e+1\right)\left(x-e\right)+e\mathrm{log}e\\ =2x-2e+e\\ =2x-e\end{array}$

3. $\begin{array}{l}g\left(x\right)=\left(\mathrm{log}2+1\right)\left(x-2\right)+2\mathrm{log}2\\ =\left(\mathrm{log}2+1\right)x-2\mathrm{log}2-2+2\mathrm{log}2\\ =\left(\mathrm{log}2+1\right)x-2\end{array}$

4. $\begin{array}{l}y\text{'}=\frac{1}{{x}^{3}}·3{x}^{2}=\frac{3}{x}\\ g\left(x\right)=\frac{3}{e}\left(x-e\right)+\mathrm{log}{e}^{3}\\ =\frac{3}{e}x-3+3\\ =\frac{3}{e}x\end{array}$

5. $\begin{array}{l}y\text{'}=\frac{-\frac{1}{x}}{{\left(\mathrm{log}x\right)}^{2}}\\ =-\frac{1}{x{\left(\mathrm{log}x\right)}^{2}}\\ g\left(x\right)=-\frac{1}{e{\left(\mathrm{log}e\right)}^{2}}\left(x-e\right)+\frac{1}{\mathrm{log}e}\\ =-\frac{1}{e}\left(x-e\right)+1\\ =-\frac{1}{e}x+2\end{array}$

6. $\begin{array}{l}g\left(x\right)=-\frac{1}{2{\left(\mathrm{log}2\right)}^{2}}\left(x-2\right)+\frac{1}{\mathrm{log}2}\\ =-\frac{1}{2{\left(\mathrm{log}2\right)}^{2}}x+\frac{1}{{\left(\mathrm{log}2\right)}^{2}}+\frac{1}{\mathrm{log}2}\end{array}$

7. $\begin{array}{l}y\text{'}={e}^{2x}2=2{e}^{2x}\\ g\left(x\right)=2{e}^{2}\left(x-1\right)+{e}^{2}\\ =2{e}^{2}x-{e}^{2}\end{array}$

8. $\begin{array}{l}y\text{'}={e}^{x}+x{e}^{x}={e}^{x}\left(1+x\right)\\ g\left(x\right)={e}^{2}\left(1+2\right)\left(x-2\right)+2{e}^{2}\\ =3{e}^{2}x-6{e}^{2}+2{e}^{2}\\ =3{e}^{2}x-4{e}^{2}\end{array}$

9. $\begin{array}{l}g\left(x\right)={e}^{5}\left(1+5\right)\left(x-5\right)+5{e}^{5}\\ =6{e}^{5}-30{e}^{5}+5{e}^{5}\\ =6{e}^{5}-25{e}^{5}\end{array}$

10. $\begin{array}{l}y\text{'}={e}^{-x}+x{e}^{-x}\left(-1\right)\\ ={e}^{-x}\left(1-x\right)\\ g\left(x\right)={e}^{-0}\left(1-0\right)\left(x-0\right)+0\\ =x\end{array}$

11. $\begin{array}{l}y\text{'}={e}^{-x}\left(-1\right)=-{e}^{-x}\\ g\left(x\right)=-{e}^{0}\left(x-0\right)+{e}^{0}\\ =-x+1\end{array}$

12. $\begin{array}{l}y\text{'}=2x{e}^{-x}+{x}^{2}{e}^{-x}\left(-1\right)\\ =2x{e}^{-x}-{x}^{2}{e}^{-x}\\ ={e}^{-x}x\left(2-x\right)\\ g\left(x\right)={e}^{-1}\left(2-1\right)\left(x-1\right)+{e}^{-1}\\ ={e}^{-1}x-{e}^{-1}+{e}^{-1}\\ ={e}^{-1}x\end{array}$

コード(Emacs)

Python 3

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

x = symbols('x')
fs = [(log(x), E),
(x * log(x), E),
(x * log(x), 2),
(log(x ** 3), E),
(1 / log(x), E),
(1 / log(x), 2),
(exp(2 * x), 1),
(x * exp(x), 2),
(x * exp(x), 5),
(x * exp(-x), 0),
(exp(-x), 0),
(x ** 2 * exp(-x), 1)]

for i, (f, x0) in enumerate(fs, 9):
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.expand()]:
pprint(t)
print()
print()

p = plot(f, g, (x, x0 - 0.5, x0 + 0.5), show=False, legend=True)
for j, color in enumerate(['green', 'blue']):
p[j].line_color = color
p.save(f'sample{i}.svg')


$./sample9.py 9. d ──(log(x)) dx 1 ─ x -1 x⋅ℯ 10. d ──(x⋅log(x)) dx log(x) + 1 2⋅x - ℯ 11. d ──(x⋅log(x)) dx log(x) + 1 x⋅log(2) + x - 2 12. d ⎛ ⎛ 3⎞⎞ ──⎝log⎝x ⎠⎠ dx 3 ─ x -1 3⋅x⋅ℯ 13. d ⎛ 1 ⎞ ──⎜──────⎟ dx⎝log(x)⎠ -1 ───────── 2 x⋅log (x) -1 - x⋅ℯ + 2 14. d ⎛ 1 ⎞ ──⎜──────⎟ dx⎝log(x)⎠ -1 ───────── 2 x⋅log (x) x 1 1 - ───────── + ────── + ─────── 2 log(2) 2 2⋅log (2) log (2) 15. d ⎛ 2⋅x⎞ ──⎝ℯ ⎠ dx 2⋅x 2⋅ℯ 2 2 2⋅x⋅ℯ - ℯ 16. d ⎛ x⎞ ──⎝x⋅ℯ ⎠ dx x x x⋅ℯ + ℯ 2 2 3⋅x⋅ℯ - 4⋅ℯ 17. d ⎛ x⎞ ──⎝x⋅ℯ ⎠ dx x x x⋅ℯ + ℯ 5 5 6⋅x⋅ℯ - 25⋅ℯ 18. d ⎛ -x⎞ ──⎝x⋅ℯ ⎠ dx -x -x - x⋅ℯ + ℯ x 19. d ⎛ -x⎞ ──⎝ℯ ⎠ dx -x -ℯ -x + 1 20. d ⎛ 2 -x⎞ ──⎝x ⋅ℯ ⎠ dx 2 -x -x - x ⋅ℯ + 2⋅x⋅ℯ -1 x⋅ℯ$


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="sample9.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'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
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 f = (x) => Math.log(x),
f1 = (x) => 1 / x,
g = (x) => f1(Math.E) * (x - Math.E) + f(Math.E);

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 = [[Math.E, y1, Math.E, y2, 'red']],
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();