(d)

5. Find the total differential for  z=2xsin y-3x2y2 . (10 分)

(a)

(b)

(c)

(a)請求出函數在 x=0 與 x=1 之間之弧長(arc length) (6 分)，

，(b)並求出此函數在此區間繞 x 軸旋轉所得到之體積。(6 分)

8. 請利用泰勒級數(Taylor Series) 的方法在 c=π 處展開 f (x)= x2 cos x，並寫出前 3 個非 0 項。 (7 分)

9. 有一極坐標函數為r = f (θ ) = 1- cosθ 。請求出此函數所圍出來之面積(7 分)。

一、英譯中：請將下面這段英文譯成正確、流暢的中文（30 分） The global issues that we face today can seem overwhelming—too much for any person, any one organization, or any one government to tackle. As we size up the global challenges we face—from advancing gender equality to eliminating poverty—it’s crucial to recognize that these issues do not exist in silos. For instance, progress we have made on issues related to health and security are exacerbated by climate change. Only by looking at these challenges holistically, can we have a better chance at making progress on a range of issues. Facing different problems in human existence, what can humans accomplish collectively? That’s where the Sustainable Development Goals come in, a framework designed by the United Nations for peace and prosperity for both people and the planet.

1. Find the value of the limit .（10%）