课程名称︰微积分乙 课程性质︰必修 课程教师︰王振男 开课学院：医学院 开课系所︰医学系 考试日期（年月日）︰2010/11/09 考试时限（分钟）：120 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1. Find the following limits. (a) (x^2)cos(x) lim ─────── x→0 1-sec(x) (b) (x^2) ∫ te^(-t^2)dt (0) lim ───────── x→∞ e^(-x^4) 2. For these two functions y=e^(x^x) and y=x^(e^x), the value of y approaches infinity as x approaches infinity. Determine which function grows faster and justify your answer. 3. Graph y=e^x-2e^(-x)-3x. Find out all the extremal points, where the function is increasing or decreasing, the concavity of the function, and the point of inflection, etc. 4. The function f(x) is continuous on [a,b]. 1, 2, ..., n are postive integers and x1, x2, ...., xn are points on [a,b]. Show that there exists a point c on [a,b] such that f(x1)+f(x2)+...+f(xn) f(c)= ──────────── n 5. The function f(x) is a strictly increasing function on [a,b]. Suppose its inverse function f^(-1)(x) is continuous on [f(a),f(b)]. Now define xj=a+(j/n)(b-a), j=1, 2, ..., n. Using the result from the last problem, we know that there exists a number cn in [a,b] such that f(x1)+f(x2)+...+f(xn) f(cn)= ──────────── n Does the limit of cn as n approaches infinity exist? If yes, find the limit. 6. 2y=x+4, y=x, and y-axis together form a triangular area. Find the volume of rotation of the area under the circumstances below. (a) Use the disk(washer) method, rotaion about x-axis. (b) Use the shell method, rotation about y-axis. (c) Use the shell method, rotation about the line x=4. (d) Use the disk(washer) method, rotation about y=8. (因为是凭记忆，所以有些用词不精确...欢迎补充) --

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