## 2019年5月1日水曜日

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

1. $\begin{array}{l}\mathrm{tan}{x}^{2}\\ =\frac{\mathrm{sin}{x}^{2}}{\mathrm{cos}{x}^{2}}\\ =\frac{{x}^{2}-\frac{1}{3!}{\left({x}^{2}\right)}^{3}+\frac{1}{5!}{\left({x}^{2}\right)}^{5}-\dots }{1-\frac{1}{2!}{\left({x}^{2}\right)}^{2}+\frac{1}{4!}{\left({x}^{2}\right)}^{4}-\dots }\\ {\mathrm{sin}}^{2}x={\left(x-\frac{1}{3!}{x}^{3}+\frac{1}{5!}{x}^{5}-\dots \right)}^{2}\\ \underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{tan}{x}^{2}}{{\mathrm{sin}}^{2}x}=1\end{array}$

コード

Python 3

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

print('15.')

x = symbols('x')
f = tan(x ** 2)
g = sin(x) ** 2
h = f / g
l = Limit(h, x, 0)

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

p = plot(f, g, h,
1,
(x, -5, 5),
ylim=(-5, 5),
show=False,
legend=True)

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('sample15.png')


C:\Users\...>py sample15.py
15.
⎛   ⎛ 2⎞⎞
⎜tan⎝x ⎠⎟
lim ⎜───────⎟
x─→0⁺⎜   2   ⎟
⎝sin (x)⎠

1

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

1

C:\Users\...>