课程名称︰微积分甲下 课程性质︰暑修 课程教师︰周清松 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2010/9/9 考试时限（分钟）：120分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : It's necessary to explain all the reasons in detail and show all of your work on the answer sheet; Or you will NOT get any credits. If you used any theorems in textbook or proved in class, state it carefully and explicitly. 1.(a)For each integer n and r≠0, we have ▽r^n = nr^(n-2)r. Here r=∥r∥ and r = xi+yj+zk. Note that if n is positive and even, the result holds at r=0. (b)Assume that ▽f(x) exists. Prove that, for each integer n, we have n n-1 ▽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)= xe^(y^2-z^2) 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 change of f(x,y,z)=x^2y+zcosx with respect to t along the twisted cubic r(t)=ti+t^2j+t^3k (b)Find the rate of change of f(x,y,z)=ln(x^2+y^2+z^2) with respect to t along the twisted cubic r(t)=sinti+costj+e^(2t)k 4.(a)Calculate by double integration the area of the bounded region determined by the curves x^2=4y, 2y-x-4=0. (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. 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 funtion ρ(x,y,z)=xyz. --

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