課程名稱︰微積分甲上
課程性質︰必修
課程教師︰周青松
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2014.7.9
考試時限（分鐘）：110
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Notice: There are 5 problems.
Each problem coonsists of 2 parts and counts 20pts.
(1) (a) For the function defined by setting (10pt)
1+x^2 , x<1
g(x)= 3 , x=1
4-2x , x>1
determine whether the limit lim g(x) exists and if g is continuous at
x→1 x=1.
(b) Show that lim f(x)=0 if and only if lim │f(x)│=0. (10pt)
x→c x→c
(2) (a) Determine the discontinuities, if any, of the following function:(10pt
)
2x+1 , x≦0
f(x)= 1 , 0<x≦1
x^2+1 , x>1
(b)Evaluate the limit if it exists. (10pt)
lim (sinx)^2
x→0 ---------
x(1-cosx)
(3) (a) Let f(x)=│x│.
Show that f is continuous but not differentiable at 0. (10pt)
(b) Find f'(1) where f is defined in the following: (10pt)
f(x)= x^2 , x≦1
2x-1 , x>1
(4) Differentiate the following functions (20pt)
(A) sec(x^2+1)
(B) (x/1+x^2)^1/2
(5) (a) Find the intervals on which g increases and intevals on which g
decreases. (10pt)
g(x)= x/2+2 , x<1
x^3 , x≧1
(b) Let f(x)=x^4-2x^3. Sketch the graph of f. (10pt)
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n > 2, X,Y,Z∈N, X^n + Y^n ≠ Z^n
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※ 編輯: yushenlin (140.112.25.121), 08/12/2014 21:48:25