課程名稱︰微積分甲下(暑修)
課程性質︰
課程教師︰周青松
開課學院：
開課系所︰
考試日期（年月日）︰2010/09/08
考試時限（分鐘）：
是否需發放獎勵金：
（如未明確表示，則不予發放）
試題 :
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.
參考答案
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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◆ From: 140.112.250.207
※ 編輯: bookh 來自: 140.112.250.207 (09/08 11:55)
※ 編輯: bookh 來自: 140.112.250.207 (09/09 00:17)