東北大学 工学研究科 土木工学専攻 2023年2月実施 基礎科目 [1] 微分積分

Author

Miyake

Description

Kai

1.

\[ \begin{aligned} \lim_{x \to 0} \left( \frac{a^x + b^x}{2} \right)^{\frac{1}{x}} &= \lim_{x \to 0} \left( 1 + \frac{\log (ab)}{2} x + O(x^2) \right)^{\frac{1}{x}} \\ &= e^{\frac{1}{2} \log (ab)} \\ &= \sqrt{ab} \end{aligned} \]

2.

(1)

\[ \begin{aligned} \mathrm{d} x &= \mathrm{d} r \cos \theta - r \mathrm{d} \theta \sin \theta \\ \mathrm{d} y &= \mathrm{d} r \sin \theta + r \mathrm{d} \theta \cos \theta \end{aligned} \]

(2)

\[ \begin{aligned} x \mathrm{d} y - y \mathrm{d} x &= r^2 \mathrm{d} \theta \end{aligned} \]

3.

4.