試題如附件。

第 2 題：

\(\displaystyle \lim_{n\to\infty} \frac{1}{n}\left(\sin\frac{\pi}{2n}+\sin\frac{2\pi}{2n}+\sin\frac{3\pi}{2n}+\cdots+\sin\frac{n\pi}{2n}\right)\)

　\(\displaystyle = \lim_{n\to\infty} \sum_{k=1}^n\sin\frac{k\pi}{2n}\cdot\frac{1}{n}\)

　\(\displaystyle = \frac{2}{\pi}\lim_{n\to\infty} \sum_{k=1}^n\sin\frac{k\pi}{2n}\cdot\frac{\displaystyle\frac{\pi}{2}-0}{n}\)

　\(\displaystyle = \frac{2}{\pi}\int_0^{\pi/2} \sin(x) dx\)

　\(\displaystyle = \frac{2}{\pi}\left(-\cos x\right)\Bigg|_0^{\pi/2}\)

　\(\displaystyle = \frac{2}{\pi}\)

註：下圖是 \(n=10\) 時，

qq.png (25.61 KB)

2013-4-26 22:22

第 1 題：

令 \(\vec{v}=(3,1), \vec{u}=(1,3)\)

則 \(\vec{OP}=\sin\alpha\,\vec{v}+\cos\beta\,\vec{u}\)

因為 \(\displaystyle 0\leq\sin\alpha\leq\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}\leq\cos\beta\leq1\)

所以，所求＝\(\displaystyle \left(\frac{\sqrt{3}}{2}-0\right)\left(1-\frac{\sqrt{3}}{2}\right)\big|\Bigg|\begin{array}{cc}3&1\\ 1&3\end{array}\Bigg|\big|=4\sqrt{3}-6\)