## 2019年7月5日金曜日

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

1. $\begin{array}{l}f\left(x\right)=\mathrm{cos}x-1-\frac{{x}^{2}}{2!}\\ f\text{'}\left(x\right)=-\mathrm{sin}x-x\\ {f}^{\left(2\right)}\left(x\right)=-\mathrm{sin}x-1\\ {f}^{\left(3\right)}\left(x\right)=-\mathrm{cos}x\\ {f}^{\left(4\right)}\left(x\right)=\mathrm{sin}x\\ {f}^{\left(5\right)}\left(x\right)=\mathrm{cos}x\\ {f}^{\left(6\right)}\left(x\right)=-\mathrm{sin}x\\ {f}^{\left(7\right)}\left(x\right)=-\mathrm{cos}x\\ f\left(x\right)=-\frac{1}{2!}{x}^{2}-\frac{1}{3!}{x}^{3}+\frac{1}{5!}{x}^{5}-\dots \\ \underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{cos}x-1-\frac{{x}^{2}}{2!}}{x}=0\end{array}$

コード

Python 3

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

print('38.')

x = symbols('x')
f = (cos(x) - 1 - x ** 2 / factorial(2)) / x

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

p = plot(cos(x) - 1 - x ** 2, x, 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('sample38.png')


C:\Users\...>py sample38.py
38.
⎛   2             ⎞
⎜  x              ⎟
⎜- ── + cos(x) - 1⎟
⎜  2              ⎟
lim ⎜─────────────────⎟
x─→0⁺⎝        x        ⎠

0

⎛   2             ⎞
⎜  x              ⎟
⎜- ── + cos(x) - 1⎟
⎜  2              ⎟
lim ⎜─────────────────⎟
x─→0⁻⎝        x        ⎠

0

C:\Users\...>