課程名稱︰微積分一

課程性質︰數學系大一必修

課程教師︰余正道老師

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 ------=0.
x→∞ r(x)

(b)[10%] Find the formula of the functions fn(x) on x＞0, n=1,2,...defined by
x fn-1(t)
f0(x)=1, fn(x)=∫ ---------dt
1 t
5.[10%] Show that the function f(x)=e^x is transcendental: For any n, if
ai(x)εR[x] such that anf^n+...+a1f+ao=0,then ai=0 for all i=0,1,...,n.

6. Let f(x) be defined on [a,b]. Recall that f(x) is "integrable" if the
Riemann sums converge.

(a)[5%] Give an example of "non"-continuous f(x) which is integrable.
Justify your answer.
(b)[15%] Let a＜c＜b. Prove that f(x) is integrable if and only if it
is integrable on [a,c] and [c,b].

7.[15%] Suppose f(x) has f''(x) on [a,b]. Show that f''(x)≧0 on [a,b]
if and only if the line segment connecting(x1,f(x1)) and(x2,f(x2))
lies above the graph y=f(x) for any a≦x1,x2≦b.

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※ 編輯: momo04282000 (140.112.25.100), 01/05/2019 15:37:05
※ 編輯: momo04282000 (140.112.240.53 臺灣), 06/14/2019 13:42:19