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）︰2010/09/08 考试时限（分钟）： 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1.(a) For each integer n and all γ≠0, show that ▽r^n = nr^(n-2)γ. Here r=│γ│ and γ=xi+yj+zk. Note that if n is positive and even, the result holds at γ=0. (b) Assume that ▽f(x) exist. n n-1 Prove that, for each integer n, we have ▽f(x)=nf(x) ▽f(x) 2.(a) Find the directional derivative of f(x,y)=ln(x^2+y^2) at P(0,1) in the direction of 8i+j (b) Find the directional derivative of f(x,y,z) at (1,2,-2) in the direction of increasing t along the path r(t)=ti+2cos(t-1)j-2e^(t-1)k 3.(a) Use the chain rule to find the rate of f(x,y,z)=x^2 y+zcosx with respect to t along the twisted cubic r(t)=ti+t^2j+t^3k (b) Find the rate of chang of f(x,y,z)=ln(x^2+y^2+z^2) with respect to t along the twisted cubic r(t)=sin(t)i+cos(t)j+e^2tk 4.(a) Calculate by double integration the area of the bounded region determind by the curves x^2=4y, 2y-x-4=0 (b) Calculate the volume within cylinder x^2+y^2=b^2 between the plane y+z=a and z=0 given that a≧b>0 5.(a) Use triple integration to find the volume of the tetrahedron T bounded by x+y+z=1 in the first octant. (Hint:0≦z≦1-x-y, 0≦y≦1-x, 0≦x≦1) (b) Calculate the mass of the solid 0≦x≦a, 0≦y≦b, 0≦z≦c, with the density function ρ(x,y,z)=xyz. 参考答案 1.略 2.(a)2/(65^0.5) (b)-7/(5^0.5) 3.(a)4t^3-t^3sint+3t^2cost (b)4e^(4t) / 1+e^(4t) 4.(a)9 (b)πab^2 5.(a)1/6 (b)a^2 b^2 c^2 /8 --

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