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) -- n > 2, X,Y,Z∈N, X^n + Y^n ≠ Z^n --

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