[轉錄][試題] 98暑修 周青松 微積分甲下 期末考
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※ [本文轉錄自 NTU-Exam 看板 #1CXl86h6 ]
作者: sr333444 (欸啥) 看板: NTU-Exam
標題: [試題] 98暑修 周青松 微積分甲下 期末考
時間: Wed Sep 8 10:16:34 2010
課程名稱︰微積分甲下
課程性質︰暑修
課程教師︰周清松
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2010/9/9
考試時限（分鐘）：120分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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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