課程名稱︰微積分一
課程性質︰數學系大一必修
課程教師︰余正道
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2018.10.29
考試時限（分鐘）：180
試題 :
以下的ε均代表屬於
滿分125分
1.(a)[10%] Determine if the limit
lim(1/n+1/(n+1)+...+1/2n) converges;if it does, find the value.
n→∞
(b)[10%] Fix a1>b1>0. Define sequences(an), (bn) by
an+1=(an+bn)/2, bn+1=√anbn.
Show that lim an=lim bn.
n→∞ n→∞
(c)[10%] Prove the following: Let (an)∞ be a bounded sequence.
n=1
Then there exists a convergent subsequence of (an).
2. Consider the polynomial f(x)=x^n+an-1x^n-1+...+a1x+a0εR[x] of degree n>=1
(a)[10%] Show that lim f(x)=∞.
x→∞
(b)[5%] Suppose n is odd. Show that f(x) has at least one (real) root.
3.(a)[10%] Suppose f(x) is continuous. Let f(x) if f(x)≧0
f+(x)={
0 if f(x)＜0.
Show that f+(x) is continuous.
(b)[5%] Show that if f(x) on [a,b] has a continuous derivative, then
f(x) can be written as a (finite) sum of monotone functions.
4.(a)[10%] Determine the form of a rational function r(x) for which
xr'(x)
lim