課程名稱︰微積分甲下
課程性質︰理工學院必修
課程教師︰官彥良
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰108.04.11
考試時限（分鐘）：30
試題 :
-x^2-y^2
1. (30%) Find all critical points of the function f(x,y)=xye^ and clas-
ify them.
2. Evaluate the following integrals.
1 1
(a) (20%)∫∫√(y)sinydydx
0 x^2
√2 √(2-y^2) 1 1
(b) (20%)∫ ∫ (x+y)/(x^2+y^2)dxdy + ∫∫ (x+y)/(x^2+y^2)dxdy
1 0 0 1-y
√2 √(2-x^2)
+∫ ∫ (x+y)/(x^2+y^2)dydx.
1 0
3. (30%) Find the points on the intersections of the plane x+y+2z=2 and the
paraboloid z = x^2 + y^2 that are nearest to and farthest from the origin.