課程名稱︰微積分甲下
課程性質︰必修
課程教師︰黃漢水
開課學院：理學院、生農學院、管理學院
開課系所︰地質系、生工系、生機系、工管科管組
考試日期（年月日）︰2012/6/18
考試時限（分鐘）：130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 Let f(x,y,z)=ln(x^2 -y+2z). Find the gradient and the maximum value of the
directional derivative of f(x,y,z) at the point (2,1,2). (15%)
二 Let S=｛(x,y,z)│x^2+y^2+z^2=20｝and the temperature of the point (x,y,z)
on S is T(x,y,z)=x^2+4yz+3z^2.
Find the higest and lowest temperatures on the surface. (20%)
三 Find the integral
4 1 3 6cos(x^2)
∫｛∫ [∫ ───── dx ] dy｝dz (15%)
1 0 3y √z
四 Let D be a thin plate outside the circle r=3 and inside the circle
r=6cosθ. Suppose the density at the point (x,y) is den(x,y)=
1
──────. Find the mass of D. (15%)
√(x^2+y^2)
五 Let K=｛(x,y,z)│x^2+y^2+z^2≦16, z≧√(x^2+y^2)｝be a solid with density
function den(x,y,z)=√(x^2+y^2).
Find the mass and the center of mass of K. (20%)
六 Find the line integral ∫cosydx+(xy-xsiny)dy,
c
where C is the boundary of the region bounded by the curves y=2x and
y=√(8x), oriented counterclockwise. (15%)
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