课程名称︰微积分乙上 课程性质︰必修 课程教师︰李聪成 开课学院： 开课系所︰ 考试日期（年月日）︰2012年11月8日 考试时限（分钟）：160分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 请将每一步骤表达清楚,不可以只写答案。 1. Find numbers a and b such that √(ax+b)-2 lim ─────── = 1 x→1 √(x) 2. Evaluate the limit. lim [√(3x^2+8x+6)-√(3x^2+3x+1)] x→+∞ 3. Find a function f and a number a such that (2+h^6)-64 lim ─────── = f'(a) h→0 h and evaluate the limit. 4. Sketch the graph of the given function, making use of any suitable information you can obtain from the function and its first and second derivatives. x^2 f(x) = ───── x-1 5. Find the equations of the tangent line(切线) and normal line(法线) to the curve √(x) y = ───── 1+x^2 at the point (1,1/2). 6. Find the points on the curve 2(x^2+y^2)^2 = 25(x^2-y^2) where the tangent is horizontal. 7. Show that the function f(x) = x|x| has an inflection point(反曲点) at (0,0) but f''(0) does not exist. 8. Find the absolute maximum and absolute minimun valus of f(x) = 2cos(x)+sin(2x) on the interval (0,π/2). 9. A plane flying horizontally at an altitude(高度) of 2km and a speed of 800 km/h passes directly over a rader station. Find the rate at which the distance from the plane to the station is increasing when it is 3km away from the station. 10. If 1200cm2 of material is avilable to make a box with a square(正方形) base and an open top, find the largest possible volume of the box. --

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