课程名称︰微积分甲上 课程性质︰大一必修 课程教师︰蔡雅如 开课学院：电资院、管理院、社科院 开课系所︰资工系、资管系、经济系 考试日期（年月日）︰2014/10/06 考试时限（分钟）：1HR 试题 : 1.Find the limits (Please show your work): x^⅓-1 (a) lim ───── x→1 √(x^3) -1 １ (b) lim sin(─)‧arctan([x]) x→∞ ｘ １ (c) lim e^(───) x→0+ x^2 -x (d) lim √x^2+ax + √x^2+b+2x , where a,b are constants. x→-∞ ＿＿ 1-√1-x 2. Let f(x) = {a(────) + [x] , for x≠0. | |x| { b , for x＝0. (a) Find values of a and b such that f is continuous at x=0. f(x)－f(0) f(x)－f(0) (b) For the a, b found in part (a), compute lim ───── & lim ───── x→0+ ｘ x→0- ｘ Is f differentiable at x=0? 3.Let f(x)=tan(arccos x). (a) Simplify f(x) and state the domain and the range of f(x). (b) Show that the equation f(x) = x has a root insite the interval (½,１). --

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