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）︰2015.11.16 考试时限（分钟）：160 分钟 试题 : 请将每一步骤表达清楚，不可以只写答案。 1. Evaluate the limit or show that it does not exist. x^2+x lim ─────── x→0 √(x^4+2x^2) 2.Evaluate the limit or show that it does not exist. _ ___ ___ lim √x (√x+4 - √x+2) x→+∞ 3. Express the limit as the derivative of some function f at some point a. Then evaluate the limit. sin(3+x)^2－sin9 lim ──────── x→0 x 4. Sketch the graph oh the given function, making use of any suitable information you can obtain from the function and its first and second derivatives. x^2+1 f(x)= ───── 3x^2-1 5. Find the absolute maximum and absolute minimum values of f(x)=2x^3-3x^2-12x+5 on the interval [-2,4]. 6. Use implicit differentiation to find an equation of the tangent line(切线) to the curve x^4=4(x^2-y^2) _ at the point (1,√3/2). 7. Show that the function f(x)=x|x| has an inflection point at (0,0) but f"(0) does not exist. 8. Find a value of c satisfying the conclusion of the Mean Value Theorem for f(x)=x^3-3x^2+2x+2 on the interval [0,1]. 9. A 10-foot ladder leans against the side of a building. If the top of the ladder begins to slide down the wall at the rate of 2ft./sec, how fast is the bottom of the ladder sliding away from the wall when the top of the ladder is 8 feet off the ground? 10. A farmer has 1200m of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimension (长,宽) of the field that has the largest area? --

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