[試題] 100上 王藹農 微積分甲上 PreFinal
課程名稱︰微積分甲上
課程性質︰必修
課程教師︰王藹農
開課學院：工學院、管理學院、電機資訊學院
開課系所︰材料科學與工程學系、資訊管理學系、電機工程學系、資訊工程學系
考試日期（年月日）︰2012/01/04
考試時限（分鐘）：100分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. A martini glass in the shape of a right-circular cone of height h \ O /
and semivertical angle α is filled with liquid. Slowly a ball is \ /
lowered into the glass, displacing liquid and causing it to overflow. ||
Find the radius R of the ball that causes the greatest volume of liquid to
overflow out of the glass. (2/10)
2. Initially a tank contains 1,000 L of brine with 50 kg of dissolved salt.
Brine containing 10 g of salt per litre is flowing into the tank at a
constant rate of 12 L/min. If the contents of the tank are kept throughly
mixed at all times, and if the solution flows out at 10 L/min, how much
salt remains in the tank at the end of 40 min? (2/10)
3. A 45°notch is cut one quarter way (10 cm) into a cylindrical log having
radius 20 cm. One plane face of the notch is perpendicular to the axis of
the log. What volume of wood was removed from the log by cutting the
notch? (2/10)
4. Find the area bounded the loop of the parametric curve x = t^3 - t,
y = |t^3|, and its asymptotes. Sketch the curve. (2/10)
95 nπ π
5. Riemann sum Σ (sin -- ) -- = ? (2/10)
n=1 95 95
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