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） ------------------------------------------------------------------------------ 連續兩次都翻盤，X的！ --

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