课程名称︰微积分甲上 课程性质︰必修 课程教师︰王蔼农 开课学院：工学院、管理学院、电机资讯学院 开课系所︰材料科学与工程学系、资讯管理学系、电机工程学系、资讯工程学系 考试日期（年月日）︰2012/01/04 考试时限（分钟）：100分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 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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