課程名稱：微積分乙下
課程性質︰共同必修
課程教師︰張樹城
開課學院：管院
開課系所︰會計系/財金系/工管系企管組/國企系/地理系/經濟系
考試日期（年月日）︰2011/06/23
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(20%) Compute the followings:
(a)
∫∫e^(x^2+y^2)dxdy, where R={(x,y):x^2+y^2≦1, y≧0}
R
(b)
1 1/2
∫∫ e^(-x^2) dxdy
0 y/2
(c)
∫∫∫dxdydz, where D={(x,y,z):0≦x≦1, 0≦y≦1-x, 0≦z≦1-(x+y)}
D
(d)
∫∫∫dxdydx, where D={(x,y,z):x^2+y^2+z^2≦1}
D
2.(15%) Define
f(x,y)=x^2+3xy+y^2
(a)Find all critical points of f(x,y) in the interior of unit circle:x^2+y^2<1
(b)By using the method of Lagrange multiplier to find all extremal points of
f(x,y) on the circle:x^2+y^2=1
(c)Find all extremal points on the disk:x^2+y^2≦1
3.(15%)
(a)Find dy/dx for x^2+siny-2y=0
(b)Find dz/dx and dz/dy for z=(e^x)sin(y+z)
4.(15%) Define
xy(x^2-y^2)
f(x,y)= ------------- ;(x,y)≠(0,0)
x^2+y^2
= 0 ;(x,y)=(0,0)
Show that
(a) fx(0,0)=fy(0,0)=0
(b) fxy(0,0)≠fyx(0,0)
5.(10%)Show that
2xy+zx^100+zy^10
f(x,y,z)=------------------ ;(x,y,z)≠(0,0,0)
x^2+y^2+z^2
=0 ;(x,y,z)=(0,0,0)
is discontinuous at (0,0,0)
6.(10%)
(a)Define
F(x,y,z)=yi+xj+4k
Find a real-valued function f(x,y,z) such that ▽f=F
(b)Evaluate the following line integral
(2,3,-1)
∫ ydx+xdy+4dz
(1,1,1)
7.(15%)Define
-y x
M(x,y)=--------- and N(x,y)=---------
x^2+y^2 x^2+y^2
(a)Compute the line integral
∫ Mdx+Ndy , where Cr is the (counterclockwise) circle of radius r.
Cr
(b)Compute the line integral
∫ Mdx+Ndy, where C is any (counterclockwise) simple closed curve in which
C
enclosed region containing the point (0,0)
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