※ 引述《sky2857 (楷中)》之铭言： : ※ [本文转录自 NTU-Exam 看板 #18oeEZE8 ] : 作者: Julibea (Julia) 看板: NTU-Exam : 标题: [试题] 96暑 周青松 微积分甲下期末考 : 时间: Fri Sep 12 22:58:42 2008 : 课程名称︰微积分甲下(暑修) : 课程性质︰ : 课程教师︰周青松 : 开课学院： : 开课系所︰ : 考试日期（年月日）︰2008/09/09 : 考试时限（分钟）：120分钟 (8:10~10:10) : 是否需发放奖励金：yes, thanks : 试题 : : I. A) Find F(t) given that F'(t)= 2cost i - t(sint^2) j + 2t k : F (0)= i + 3 k : B) Show that, if γis differentiable vector funtion of t, : where r =∥γ∥> 0, : d γ 1 dγ : then ── (──) = ── [(γ x ──) x γ] : dt r r^3 dt : II. Let γ= x i + y j + z k and r =∥γ∥> 0 : A) Prove that, for each interger n, ▽r^n = nr^(n-2)γ 以上好无聊 : B) Find ▽(e^r) 这好像要算一下吧 e^r * γ/r吧 : III. A) Find the directional derivative of the function : f(x,y,z)= 2x(z^2) cosπy : at the point P(1,2,-1) toward the point Q(2,1,3) : B) Use the chain rule to find the rate of change of : f(x,y,z)= (x^2)y + zcosx : with respect to t along the twisted curve : γ(t)= t i + t^2 j + t^3 k 好烦XD : IV. A) Use double integration to calculate the area of the region : Ω enclosed by y=x^2 and x+y=2 9/2吗? = = : 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. ab^2 pi : V. A) Find the mass of a solid riht circular cylinder of radius r : and height h given that the mass density varies directly : with distance from the lower base. 圆筒 密度应该是 kz吧 然後就开始积分了 1/2 k h^2 再乘上圆面积吧 所以 1/2 k h^2 pi r^2 : B) Evaluate the triple integral : ∫∫∫ 2ye^x dxdydz , where T is the solid given by : T : 0≦y≦1, 0≦x≦y, 0≦z≦x+y 重复 : (每大题均20分) --

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