※ [本文轉錄自 NTU-Exam 看板 #1AfnpocO ] 作者: iwantowaylaw (我要成為太鼓達人) 看板: NTU-Exam 標題: [試題] 時間: Wed Sep 9 11:11:45 2009 課程名稱︰微積分甲下暑修 課程性質︰ 課程教師︰周青松 開課學院： 開課系所︰ 考試日期（年月日）︰2009/9/9 考試時限（分鐘）：110分鐘 是否需發放獎勵金：要 （如未明確表示，則不予發放） 試題 : A. (a) Find the directional derivative of f(x,y,z) = Ax^2 + Bxyz + Cy^2 at P(1,2,1) in the direction of Ai + Bj + Ck where A,B,C are given constant with A^2 + B^2 + C^2 ≠ 0 (b) Find the directional derivative of f(x,y,z)=xe^(y^2-z^2) at P(1,2,-2)in the direction of the path γ(t)=ti+2cos(t-1)j-2e^(T-1)k. B. (a) Find the rate of change of f(x,y,z)= (x^2)y + zcosx with respect to t along the curve r(t)= t i + t^2 j + t^3 k (b) Find the rate of change of f(x,y,z)= ln(x^2+y^2+z^2) with respect to t along the curve r(t)= sint i + cost j + e^2t k C. (a) Use double integration to calculate the area of the region enclosed by y=x^2 and x+y=2 (b) Calculate the volume within the cylinder x^2 + y^2 = b^2 between the planes y+z=a and z=0 given that a≧b＞0. D. (a) Evaluate the double integral 3 ∫∫ (3xy - y) dxdy ,Ω is the region between y = |x| and y = -|x|, Ω x in [-1,1]. (b) Evaluate the trible integral x ∫∫∫ 2ye dxdydz, where T is the solial T ∕ given by 0 ≦ y ≦ 1 , 0 ≦ x ≦ y ﹨ and 0 ≦ z ≦ x + y. E. (a) Find the mass of a solid right circular cylinder of radius r and height h given that the mass density varies directly with distance from the lower base , that is , the density function is λ(x,y,z) = kz where k > 0 is a given constant (b) Find the volume enclosed by the ellipsed 2 2 2 x y z — + — + — = 1 2 2 2 a b c Remark : You may show that the area bounded by the ellipse 2 2 x y — + — = 1 is πab first 2 2 a b 參考答案: A. (a) ( 2A^2 + B^2 + 2AB + 6BC ) / (A^2+B^2+C^)^(1/2) (b) -7/5^(1/2) B. (a) 4t^3 - (t^3)sint + 3t^2cost (b) 4e^(4t)/(1+e^(4t)) C. (a) 9/2 (b) πab^2 D. (a) 0 (b) 4e-29/3 E. (a) πk(rh)^2 / 2 (b) 3abcπ/4 --

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