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※ [本文轉錄自 NTU-Exam 看板]
作者: cawaiik (5/1台大熱舞成發在舊體) 看板: NTU-Exam
標題: [試題] 93下 張秀瑜 微乙 期末考題
時間: Fri Aug 19 17:03:34 2005
課程名稱︰微積分乙
課程性質︰
課程教師︰張秀瑜
開課系所︰管院
考試時間︰93下
試題 :
1. Find dz/dz if z is defined implicitly as a function of x
and y by the equation sin(xyz) = x + 2y + 3z .
2. Show that f(x,y) = x√y is differentiable at (1,4) and find
its linearization there. Then use it to approximate (1.1,3.9) .
3. If z = f(x,y) has continuous second-order partial derivatives
and x = r^2 + s^2 and y = 2rs , find (a) dz/dr and (b) d^2z/drds .
4. Consider the curve xsin(x+y) + 2x^2 = 0 in xy-plane. Find a
vector which is orthogonal to the curve at (1/2, 3π-1 / 2)
5. (a) Find the volume of the solid under the plane x + 2y - z = 0
and above the region bounded by y = x and y = x^4
1 1 ________
(b) Evaluate ∫ ∫ √x^3 + 1 dxdy by reversing the order of
0 √y
integration.
6. Evaluate ∫∫ xdA , where D is the region in the first quadrant
D
that lies between the circles x^2 + y^2 =4 and x^2 + y^2 = 2x .
7. Evaluate ∫∫∫ (x^2 + y^2 + z^2)dV , where E: 0 ≦ z ,
E
x^2 + y^2 + z^2 ≦ 1 .
8. Evaluate ∫∫ sin(9x^2 + 4y^2)dA , where R is the region in the
R
first quadrant bounded by the ellipse 9x^2 + 4y^2 = 1 .
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