## 2019年6月30日日曜日

### 数学 - Python - 解析学 - 級数 - テイラーの公式 - 連続関数、n+1次導関数絶対値、定数、n次テイラー多項式、差、商、極限

1. $\begin{array}{l}f\left(x\right)-{P}_{n}\left(x\right)\\ =\frac{{f}^{\left(n+1\right)}\left(0\right)}{\left(n+1\right)!}{x}^{\left(n+1\right)}+..\\ \underset{x\to 0}{\mathrm{lim}}\frac{f\left(x\right)-{P}_{n}\left(x\right)}{{x}^{2}}=0\end{array}$

2. $\underset{x\to 0}{\mathrm{lim}}\frac{f\left(x\right)-{P}_{n}\left(x\right)}{{x}^{n}}=0$

3. $\underset{x\to 0}{\mathrm{lim}}\frac{f\left(x\right)-{P}_{n}\left(x\right)}{{x}^{n-1}}=0$

コード

Python 3

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

print('33.')

x = symbols('x')
f = sin(x)
n = 4
p = sum([Derivative(f, x, k).subs({x: 0}).doit() / factorial(k) * x ** k
for k in range(n)])
ns = [2, n, n - 1]
fs = [(f - p) / x ** n0 for n0 in ns]
for i, g in enumerate(fs):
print(f'({chr(ord("a") + i)})')
for d in ['+', '-']:
l = Limit(g, x, 0, dir=d)
for o in [l, l.doit()]:
pprint(o)
print()

p = plot(f, p, *fs,
(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('sample33.png')


C:\Users\...>py sample33.py
33.
(a)
⎛ 3             ⎞
⎜x              ⎟
⎜── - x + sin(x)⎟
⎜6              ⎟
lim ⎜───────────────⎟
x─→0⁺⎜        2      ⎟
⎝       x       ⎠

0

⎛ 3             ⎞
⎜x              ⎟
⎜── - x + sin(x)⎟
⎜6              ⎟
lim ⎜───────────────⎟
x─→0⁻⎜        2      ⎟
⎝       x       ⎠

0

(b)
⎛ 3             ⎞
⎜x              ⎟
⎜── - x + sin(x)⎟
⎜6              ⎟
lim ⎜───────────────⎟
x─→0⁺⎜        4      ⎟
⎝       x       ⎠

0

⎛ 3             ⎞
⎜x              ⎟
⎜── - x + sin(x)⎟
⎜6              ⎟
lim ⎜───────────────⎟
x─→0⁻⎜        4      ⎟
⎝       x       ⎠

0

(c)
⎛ 3             ⎞
⎜x              ⎟
⎜── - x + sin(x)⎟
⎜6              ⎟
lim ⎜───────────────⎟
x─→0⁺⎜        3      ⎟
⎝       x       ⎠

0

⎛ 3             ⎞
⎜x              ⎟
⎜── - x + sin(x)⎟
⎜6              ⎟
lim ⎜───────────────⎟
x─→0⁻⎜        3      ⎟
⎝       x       ⎠

0

C:\Users\...>