課程名稱︰微積分乙下
課程性質︰必修
課程教師︰黃漢水
開課學院：醫學院
開課系所︰牙醫、物治、職治、公衛、農化、藥學、醫技
考試日期（年月日）︰2012/06/19
考試時限（分鐘）：120min.
是否需發放獎勵金：yes
（如未明確表示，則不予發放）
試題 :
一、Let f(x,y,z)=ln(x^2 + 2y + 3z). Find the gradient and the maximum value of
the directional derivative of f(x,y,z) at the point (2,1,2). (20%)
二、Let D={(x,y)│x^2 + y^2 ≦40, y≧0} and the temperature of the point (x,y)
on D is T(x,y)=2x^2 + 6xy - 6y^2.
Find the higest and lowest temperatures on D. (20%)
0 ┌ √(4-x^2) 12 ┐
三、Find the integral ∫ │ ∫ ──────── dy │ dx. (20%)
-2└ - √(4-x^2) 1 + x^2 + y^2 ┘
四、Let D = {(x,y)│(x-2)^2 + y^2 ≦4, x^2 + y^2 ≧ 4. Suppose the density at
the point (x,y) is den(x,y) = √(x^2 + y^2). Find the mass of D. (20%)
五、Let K = {(x,y,z)│x^2 + y^2 + z^2 ≦25, z≧3} 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%)
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