※ [本文轉錄自 NTU-Exam 看板 #18oeEZE8 ]
作者: Julibea (Julia) 看板: NTU-Exam
標題: [試題] 96暑 周青松 微積分甲下期末考
時間: Fri Sep 12 22:58:42 2008
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）︰2008/09/09
考試時限（分鐘）：120分鐘 (8:10~10:10)
是否需發放獎勵金：yes, thanks
試題 :
I. A) Find F(t) given that F'(t)= 2cost i - t(sint^2) j + 2t k
F (0)= i + 3 k
B) Show that, if γis differentiable vector funtion of t,
where r =∥γ∥> 0,
d γ 1 dγ
then ── (──) = ── [(γ x ──) x γ]
dt r r^3 dt
II. Let γ= x i + y j + z k and r =∥γ∥> 0
A) Prove that, for each interger n, ▽r^n = nr^(n-2)γ
B) Find ▽(e^r)
III. A) Find the directional derivative of the function
f(x,y,z)= 2x(z^2) cosπy
at the point P(1,2,-1) toward the point Q(2,1,3)
B) Use the chain rule to find the rate of change of
f(x,y,z)= (x^2)y + zcosx
with respect to t along the twisted curve
γ(t)= t i + t^2 j + t^3 k
IV. A) Use double integration to calculate the area of the region
Ω enclosed by y=x^2 and x+y=2
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.
V. A) Find the mass of a solid riht circular cylinder of radius r
and height h given that the mass density varies directly
with distance from the lower base.
B) Evaluate the triple integral
∫∫∫ 2ye^x dxdydz , where T is the solid given by
T
0≦y≦1, 0≦x≦y, 0≦z≦x+y
(每大題均20分)
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※ 編輯: Julibea 來自: 140.112.222.196 (09/12 23:00)
※ u19901006:轉錄至看板 b982040XX 06/19 13:56
※ tom5707:轉錄至看板 b982040XX 06/20 00:28
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