Fw: [試題] 97暑 周青松 微積分甲下 期末考
※ [本文轉錄自 NTU-Exam 看板 #1AfnpocO ]
作者: iwantowaylaw (我要成為太鼓達人) 看板: NTU-Exam
標題: [試題]
時間: Wed Sep 9 11:11:45 2009
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）︰2009/9/9
考試時限（分鐘）：110分鐘
是否需發放獎勵金：要
（如未明確表示，則不予發放）
試題 :
A.
(a)
Find the directional derivative of f(x,y,z) = Ax^2 + Bxyz + Cy^2 at P(1,2,1)
in the direction of Ai + Bj + Ck where A,B,C are given constant with
A^2 + B^2 + C^2 ≠ 0
(b)
Find the directional derivative of f(x,y,z)=xe^(y^2-z^2) at P(1,2,-2)in the
direction of the path γ(t)=ti+2cos(t-1)j-2e^(T-1)k.
B.
(a)
Find the rate of change of f(x,y,z)= (x^2)y + zcosx with respect to t along the
curve r(t)= t i + t^2 j + t^3 k
(b)
Find the rate of change of f(x,y,z)= ln(x^2+y^2+z^2) with respect to t along
the curve r(t)= sint i + cost j + e^2t k
C.
(a)
Use double integration to calculate the area of the region 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.
D.
(a)
Evaluate the double integral
3
∫∫ (3xy - y) dxdy ,Ω is the region between y = |x| and y = -|x|,
Ω
x in [-1,1].
(b)
Evaluate the trible integral
x
∫∫∫ 2ye dxdydz, where T is the solial
T
∕ given by 0 ≦ y ≦ 1 , 0 ≦ x ≦ y
﹨ and 0 ≦ z ≦ x + y.
E.
(a)
Find the mass of a solid right circular cylinder of radius r and height h given
that the mass density varies directly with distance from the lower base , that
is , the density function is λ(x,y,z) = kz where k > 0 is a given constant
(b) Find the volume enclosed by the ellipsed
2 2 2
x y z
— + — + — = 1
2 2 2
a b c
Remark : You may show that the area bounded by the ellipse
2 2
x y
— + — = 1 is πab first
2 2
a b
參考答案:
A.
(a) ( 2A^2 + B^2 + 2AB + 6BC ) / (A^2+B^2+C^)^(1/2)
(b) -7/5^(1/2)
B.
(a) 4t^3 - (t^3)sint + 3t^2cost
(b) 4e^(4t)/(1+e^(4t))
C.
(a) 9/2
(b) πab^2
D.
(a) 0
(b) 4e-29/3
E.
(a) πk(rh)^2 / 2
(b) 3abcπ/4
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◆ From: 140.112.24.223
※ 編輯: iwantowaylaw 來自: 140.112.4.109 (09/15 18:29)
※ sky2857:轉錄至看板 NTUBIME103HW 06/16 00:30
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※ 轉錄者: fanif (114.42.70.242), 時間: 06/15/2012 00:15:27
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