課程名稱︰微積分甲(下)
課程性質︰系定必修
課程教師︰黃漢水
開課學院：生機、生工、地質、工管
開課系所︰數學系
考試日期（年月日）︰2008/06/15
考試時限（分鐘）：170分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
一 Find the equation of the tangent plane to the surface
(e^x)yz + 3(z^2) - 2(y^2)z - 10 = 0 at the point (0,1,2) (15％)
二 Let D = {(x,y,z)︱x^2 + y^2 + z^2 ≦ 50 } be a thin plane and
the temperature at the point (x,y,z) on the D is
T(x,y,z) = 9(x^2) + 9(y^2) + 9(z^2) + 6xy + 8yz
Find the highest and lowest temperatures (25％)
2 4 _______
三 Find the integral ∫[∫ 6x√y^2 + 9 dy]dx (15％)
0 x^2
四 Let D = {(x,y)︱(x-2)^2 + y^2 ≦ 4 , y≦0 } be a thin plane (20％)
Suppose that the density at the point (x,y) is den(x,y) = -12y
(1) Find the mass of D
(2) Find the center of mass of D
五 Let D = {(x,y,z)︱x^2 + y^2 + z^2 ≦ 25 , z≧3 } Suppose that the
density at the point (x,y,z) is den(x,y,z) = 12z
Find the mass of D (15％)
六 Let D = {(x,y)︱x^2 + y^2 ≦ 16 , y≧0 } and C = σD be the bound of D
Find the intergal ∫(2xy + 3x)dx + (x^2 + 2x)dy (10％)
c
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