课程名称︰微积分甲 课程性质︰ 课程教师︰朱桦 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2013/11/2 考试时限（分钟）：150分钟(两个半小时) 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : (1) (20%) Find the limits. (a) lim x^(2/3)*(sqrt(x+2)-2*sqrt(x+1)+sqrt(x)) as x to inf (b) lim [x+x^2*ln(1-2/x)] as x to inf (c) lim (tanx)^cosx as x to (pi/2)- {|x|^x if x!=0 (2) (15%) Let f(x) = { { 1 if x=0 (a) Determine whether f(x) is continuous at 0. (b) Determine whether f(x) is differentiable at 0. (3) (15%) (a) Show that the function f(x)=x^3+x+1 is strictly increasing on R. (b) If g(x) is the inverse function to the function of f(x) of part(a). Find g'(5) and g''(5). (4) (10%) Prove the identity / x-1 \ arcsin|-----| = 2*arctan(sqrt(x)) - pi/2. \ x+1 / (5) (10%) The minute hand on a watch is 8mm long and the hour hand is 4mm long. How fast is the distance between the tips of the hands changing at two o'clock. (6) (15%) Two vertical poles PQ and ST are secured by a rope PRS as shown in the picture. S /| P / | |\/ | Q R T Given that PQ = 1 m, ST = 3 m, QT = 2 m, find the position of R such that (a) the length of the rope PRS is maximized. (b) the angle θ= PRS is maximized. (7) (25%) Let f(x) = (x-1)^(5/3)(x^2-1)^(-1/3). (a) What is the domain of f(x). (b) Find all the asymptotes of f(x). (c) Determine the intervals of increase or decrease. (d) Determine the intervals of conccavity and the inflection points. (e) Find the local maximum and minimum values. (f) Sketch the graph of f(x). --

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