课程名称︰微积分甲下
课程性质︰理工学院必修
课程教师︰官彦良
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰108.04.11
考试时限（分钟）：30
试题 :
-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.