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 B) Find ▽(e^r) 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 IV. A) Use double integration to calculate the area of the region Ω 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. 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. 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 (每大题均20分) --

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