作者  StarryBlue (星藍)                                       看板  NTU-Exam 
 標題  [試題] 103-1 蔡雅如 微積分甲上 第一次小考                              
 時間  Tue Jan 20 14:28:19 2015                                               
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課程名稱︰微積分甲上
課程性質︰大一必修
課程教師︰蔡雅如
開課學院：電資院、管理院、社科院
開課系所︰資工系、資管系、經濟系
考試日期（年月日）︰2014/10/06
考試時限（分鐘）：1HR

試題 :

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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※ 編輯: StarryBlue (140.112.77.73), 01/20/2015 14:56:37