課程名稱︰ 微積分甲下
課程性質︰ 系必修
課程教師︰ 周青松
開課學院： 理學院
開課系所︰ 數學系(但是現地理、地質、生工、生機、工管等系修習)
考試日期（年月日）︰96.06.22
考試時限（分鐘）：120
是否需發放獎勵金：是，感謝
（如未明確表示，則不予發放）
試題 :
I.
2
A) Find F(t) given that F'(t)=2costi-tsint j+2tk and F(0)=i+3k
B) Let γ be a differentiable vector function of t and r=||γ||.
d γ 1 dγ
Show that ---- --- = ---[(γ╳----)╳γ], when r≠0.
dt r r^3 dt
II.
A) Let γ=xi+yj+xk and r=||γ||. Show that, for each integer n and
n n-2
all γ≠0, ▽r =nr γ.
B) Find a function f with the gradient F:
2
F(x,y,z) = yzi+(xz+2yz)j+(xy+y )k.
III.
2 2
y -z
A) Find the directional derivative of f(x,y,z)=xe at (1,2,-2)
t-1
in the direction of the path γ(t)=ti+2cos(t-1)j-2e k.
B) Use the chain rule to find the rate of change of
2
f(x,y,z)=x y+zcosx with respect to t along the tuisted cubic
2 3
γ(t)=ti+t j+t k.
IV.
A) Use double integration to calculate the area of the region Ω
2
enclosed by y=x and x+y=2.
2
-y /2
B) Evalute the double integral ∫∫e dxdy, Ω the triangle
Ω
fomed by y-axis, 2y=x, y=1.
V.
A) Find the mean of the solid ight cicular cylinden of radius r
and height h given that the mass density is directly proportional
to the distance from the lower base.
x
B) Evaluate the triple integral ∫∫∫2ye dxdydz, where T is the
T
solid given by 0≦y≦1, 0≦x≦y, 0≦z≦x+y
每大題均20分
(每一小題10分)
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