## 2019年7月6日土曜日

### 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、差、商、極限

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

コード

Python 3

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

print('39.')

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

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 / factorial(2), 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('sample39.png')


C:\Users\...>py sample39.py
39.
⎛   2             ⎞
⎜  x              ⎟
⎜- ── + cos(x) - 1⎟
⎜  2              ⎟
lim ⎜─────────────────⎟
x─→0⁺⎜         2       ⎟
⎝        x        ⎠

-1

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

-1

C:\Users\...>