## 2019年6月18日火曜日

### 数学 - Python - 解析学 - 級数 - テイラーの公式 - 三角関数(正弦)、累乗(べき乗、平方)、極限

1. $\begin{array}{l}\mathrm{sin}x=x-\frac{1}{3!}{x}^{3}+\dots \\ {\mathrm{sin}}^{2}x={x}^{2}-\frac{1}{3!}{x}^{4}+\dots \\ \mathrm{sin}\left({x}^{2}\right)={x}^{2}-\frac{1}{3!}{x}^{6}+\dots \\ \underset{x\to 0}{\mathrm{lim}}\frac{{\mathrm{sin}}^{2}x}{\mathrm{sin}\left({x}^{2}\right)}=1\end{array}$

コード

Python 3

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

print('21.')

x = symbols('x', real=True)
f = sin(x) ** 2 / sin(x ** 2)

for dir in ['+', '-']:
l = Limit(f, x, 0, dir=dir)
for o in [l, l.doit()]:
pprint(o)
print()

p = plot(sin(x), sin(x) ** 2, sin(x ** 2), f,
(x, -5, 5),
ylim=(-5, 5),
legend=True,
show=False)
colors = ['red', 'green', 'blue', 'brown', 'orange',
'purple', 'pink', 'gray', 'skyblue', 'yellow']

for o, color in zip(p, colors):
o.line_color = color

p.show()
p.save('sample21.png')


C:\Users\...>py sample21.py
21.
⎛   2   ⎞
⎜sin (x)⎟
lim ⎜───────⎟
x─→0⁺⎜   ⎛ 2⎞⎟
⎝sin⎝x ⎠⎠

1

⎛   2   ⎞
⎜sin (x)⎟
lim ⎜───────⎟
x─→0⁻⎜   ⎛ 2⎞⎟
⎝sin⎝x ⎠⎠

1

C:\Users\...>