课程名称︰微积分乙 课程性质︰必修 课程教师︰李聪成教授 开课学院：管院 开课系所︰管院 考试日期（年月日）︰2009/11/12 考试时限（分钟）：2小时40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. Is there a number a such that lim 3x^2+ax+a+3 ------------------ x->-2 x^2+x-2 exsits? If so, find the value of a and the value of the limit. 2.Evaluate the lim it. lim √(9x^6-x) x->-∞ -------------- x^3+1 3.Evaluate the limit. lim (16+h)^1/4 -2 h->0 ------------------- h 4.Sketch the graph of the given function , making use of any sutiable information you can obtain from the function and its first and secong derivatives. x^3 f(x) = ------------ x^2 +1 5.Find all points on the graph of the function f(x)=2sinx +sin^2(x) at which the tangernt line (切线) is horizontal (水平) 6.Use implicit differentiation to find an equation of the tagent line to the curve x^3+y^3 =6xy at the point (3,3) 7.Let f(x)= (√x+c^2 )-c/x ,where c>0 is a real constant. (a) What is the domain (定义域) of f . (b) How can you define f at x=0 in order for f to be continuous there? 8. Show that the function f(x) = x^2sin(1/x) , if x≠0 0 , if x=0 is differentiable on (-∞ , +∞ ) and find f' . 9.Two cars moving from the same point . One travels sounth at 30 km/h and other travels west at 72 km/h . At what rate is the distance between the cars increasing two hours later? 10.Find the points on the ellipse(椭圆) 4x^2 +y^2 =4 that are closet (最近) to and farthest (最远) away from the point (1,0) . --

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