## 2019年7月9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)、指数関数、直線、商、極限

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

コード

Python 3

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

print('42.')

x = symbols('x')
num = sin(x) + exp(x) - 1
den = x
f = num / den
for d in ['+', '-']:
l = Limit(f, x, 0, dir=d)
for o in [l, l.doit()]:
pprint(o)
print()

p = plot(num, den, 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('sample42.png')


C:\Users\...>py sample42.py
42.
⎛ x             ⎞
⎜ℯ  + sin(x) - 1⎟
lim ⎜───────────────⎟
x─→0⁺⎝       x       ⎠

2

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

2

C:\Users\...>