[轉錄][試題] 97下 周青松 微積分甲下 期末考

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※ [本文轉錄自 NTU-Exam 看板 #1ACSH6oI ]

作者: dryadb23152 (【宅online】能量全開!!) 看板: NTU-Exam
標題: [試題] 97下 周青松 微積分甲下 期末考
時間: Fri Jun 12 10:58:13 2009

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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