課程名稱︰微積分甲上
課程性質︰數學系必修
課程教師︰林紹雄
開課學院：理學院
開課系所︰數學系
考試日期︰2006年11月02日
考試時限：未知
是否需發放獎勵金：是
試題 :
The total of the following problems is 50 points.Please write down your
computational, or proof steps clearly on the answer sheets.
A. Let f(x) be defined on the interval (-1,0)
(a) (5 points) Write down the "definition" of the limit lim x→0- f(x)=α.
(b) (8 points) Use the above definition to prove the limit lim x→0- f(x)=1
for the function f(x)=x[1/x], where [1/x] denotes the greatest integer
not greater than 1/x.
sin(1-cos x)
B.(7 points) Find the limit lim x→0 ─────── .
x^2
C.Define the function
f(x)={cos x+(1/2)cos 2x, if x≧0}
{a|x+1|+b, if x＜0}
where a and b are two constants.
(a) (5 points) Find conditions on a and b so that f(x) is continuous
at every point.
(b) (5 points) Find conditions on a and b so that f(x) is differentiable
at x=0.
(c) (10 points) Under the conditions of (b), find all the critical points
of f(x), and find the maximum and minimum of f(x)
on the interval [-2,2π].
D.(10 points) Let f(x) be defined by
f(x)={1/x, if 0＜x≦1}
{0, if x=0 }
Give sufficient reasoning to show that f(x) is not Riemann integrable on [0,1]
試題結束
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