作者pingjung (谢排)
看板b002040XX
标题Fw: [试题] 98下 周青松 微积分甲下 期末考
时间Wed Jun 13 15:17:38 2012
※ [本文转录自 NTU-Exam 看板 #1CD0FJ03 ]
作者: SWW (鼠呜呜) 看板: NTU-Exam
标题: [试题] 98下 周青松 微积分甲下 期末考
时间: Wed Jul 7 12:34:24 2010
课程名称︰微积分甲下
课程性质︰必修
课程教师︰周青松
开课学院：理学院
开课系所︰
考试日期（年月日）︰2010/6/21
考试时限（分钟）：100分钟
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
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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※ 转录者: pingjung (140.112.240.73), 时间: 06/13/2012 15:17:38