[試題] 99上 周青松 微積分甲上 期中考
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：管理學院、生農學院、理學院
開課系所︰工管系科管組、生機系、生工系、地質系
考試日期（年月日）︰2010/11/8
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
試題 :
Warning : Each part from I - V is of 20 points. Please write down your answer
in the answer sheets in detail as well as possible.
I.
｜1+x^2 , x<1
A. Evaluate lim f(x) if it exists, where f(x)=｜ 3 , x=1
x→1 ｜4-2x , x>1
B. Give that g(x)=x^2-4x, evaluate the following limit if that exists.
g(x)-g(2)
lim －－－－－－.
x→3 x-3
II.
A. Determine the discontinuties, if any, of the following function
｜2x+1 ,x<=0
f(x)=｜ 1 ,0<x<=1.
｜x^2+1 ,x>1
B. Prove that
sinx
lim－－－－ = 1.
x→0 x
III.
x 1
A. Find the drivative of the function f(x)=（－－－－）^－ .
1+x^2 2
B. Find the second derivative of the function y=x^1/2(tanx^1/2).
IV.
A. Suppose that f is differentiable on the interior of an interval I and
continuous on all of I. Prove that
(i)If f'(x)>0 for all x in the interior of I, then f increases on all of I.
(ii)If f'(x)<0 for all x in the interior of I, then f decreases on all of I.
(iii)If f'(x)=0 for all x in the interior of I, then f is constant on all of I.
B. For the function
｜x^3 ,x<1
f(x)=｜ , determine the intervals on which f decreases and the
｜1/2x+2 ,x=>1
intervals on which f decreases.
V.Sketch the graph of f(x)=1/4x^4-2x^2+7/4 on [-5,5].
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