Fw: [試題] 98下 周青松 微積分甲下 期末考
※ [本文轉錄自 NTU-Exam 看板 #1CD0FJ03 ]
作者: SWW (鼠嗚嗚) 看板: NTU-Exam
標題: [試題] 98下 周青松 微積分甲下 期末考
時間: Wed Jul 7 12:34:24 2010
課程名稱︰微積分甲下
課程性質︰必修
課程教師︰周青松
開課學院：理學院
開課系所︰
考試日期（年月日）︰2010/6/21
考試時限（分鐘）：100分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
A.(a) Findｆwith the conditionｆ'(t)=2ｆ(t) and ｆ(0)=i-k.
(b) Letｆbe a differentiable vector-valued function. Show that whenver
∥ｆ(t)∥≠0.
d ｆ'(t) ｆ(t)‧ｆ'(t)
—(————— － ————————ｆ(t)
dt ∥ｆ(t)∥ ∥ｆ(t)∥^3
B.(a) Find the directional derivative of f(x,y,z)=Ax^2 + Bxyz+ Cy^2 at the
point P(1,2,1) in the direction of Ai+Bj+Ck.
(b) Assume that ▽f(x) exists. Prove that, for each integer n,
n n-1
▽f (x)=nf (x)▽f(x).
Dose the result hold of n is replaced by an arbitary real number?
C.(a) Use the chain rule to find the rate of change of
f(x,y,z)=x^2 y+zcosx
with repect to t along the twisted cubic ｒ(t)=ti+t^2 j+t^3 k.
(b) Set r=∥ｒ∥, whereｒ=xi+yj+zk. If f is a continuously differentiable
function of r, then
ｒ
▽〔f(r)〕=f'(r)—, where r≠0
r
D.(a) Evaluate the double intergral
__
∫∫ √xy dxdy, Ω:0≦y≦1, y^2≦x≦y.
Ω
(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.
E.(a) Evaluate
π/2 π/2 1
∫ ∫ ∫e^z cosxsiny dzdydx
0 0 0
(b) Evaluate the triple integral
∫∫∫2ye^x dxdydz,
T
where T is the solid given by 0≦y≦1, 0≦x≦y, 0≦z≦x+y.
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