[試題] 97下 周青松 微積分甲下 期末考
課程名稱︰微積分甲下
課程性質︰必修
課程教師︰周青松
開課學院：
開課系所︰
考試日期（年月日）︰98/6/12
考試時限（分鐘）：8：25～10：05
是否需發放獎勵金：YES
（如未明確表示，則不予發放）
試題 :
I.
2
A) Find f(t) given that { f'(t)=2costi-tsint j+2tk
{ f (0)=i+3k
B) Show that,for each integer n and all γ≠0, where γ=xi+yi+zk, and r=||γ||
n n-2
▽r =nr γ
II.
2 2
A) Find the directional derivative of the fuction f(x,y)=x +y at the point
(1,2) in the direction of the vector 2i-3j
2
B) Find the directional derivative of the fuction f(x,y,z)=2xz cosπy
at the point P(1,2,-1) toward the point Q(2,1,3)
III.
1 3 3
A) Use the chain rule to find the rate of change of f(x,y)= ─(x +y )
3
with respect to t along the ellipse γ(t) = acostｉ+bsintｊ
2
B) Use the chain rule to find the rate of change of f(x,y,z)=x y+zcosx
2 3
with respect to t along the ellipse γ(t) =ti+t j+t k
IV.
1/2 2 1/2 4
A) Evaluate ∫∫(x -y )dxdy with Ω the region enclosed by x=y and x=y
Ω
B) Evaluate the integral, taking Ω: 0≦x≦π/2 , 0≦y≦π/2
∫∫cos(x+y)dxdy
Ω
V.
A) Evaluate ∫∫∫xyz dxdydz with T the solid in the (看不懂老師寫啥)...
T 2
bounded by the parasolic cylinder z=4-x the plane z=0, the plane y=x, and
the plane y=0
B) Use the triple integral to find the volume enclosed by the ellipsed
2 2 2
x y z
— + — + — = 1
2 2 2
a b c
（每大題均20分）
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連續兩次都翻盤，X的！
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