[試題] 971 蔡紋琦 高等微積分(一)期中考
課程名稱:高等微積分(一)
課程性質:必修
課程範圍:課本Ch1~Ch3
開課教師:蔡紋琦
開課學院:商學院
開課系級:統計二
考試日期(年月日):2008/11/18
考試時限(Mins):二234
試題本文:
1.(40 points) For each of the following statements, determine whether it is
true or false .
(a)___The set Z of integers is dense in R.
(b)___If the sequence {an^2} converges, then the sequence {an} also converges.
(c)___Let {an} be a sequence of real numbers. If for some ε﹥0, there is an
index N such that |an| ﹤εfor all indices n≧N, then {an} converges to 0.
(d)___A subsequence of a convergent sequence is convergent.
(e)___Suppose that the sequence {an} is monotone and that it has a convergent
subsequent subsequence. Then {an} converges.
(f)___The set [0,∞) is closed.
(g)___Every continuous function f:(0,1)→R has a bounded image.
(h)___Suppose that the function f:[0,1]→R is continuous and its image is con-
tained in set of rational numbers. Then f is a constant function.
(i)___Any Lipschitz function is uniformly continuous.(A function f:D→R is sa-
id to be Lipschitz function provided that there is a nonnegative number C
such that |f(u)-f(v)|≦C|u-v| for all u and v in D.
(j)___If f:[0,1]→R is continuous and one-to-one, then f is strictly monotone.
2.(6 points) Please state the Completeness Axiom which the set of real number
satisfies.
3.(6 points) Please state the Intermediate Value Theorem.
4.(10 points) Let A and B be compact sets. Show that the union A∪B is also
compact.
5.(10 points) Suppose that the function f:R→R is continuous at the point x0
and f(x0)>0. Then f(x) is positive around x0. Namely, please
show that there is an interval I = (x0 -1/N, x0 +1/N), where
N is a natural number, such that f(x)>0 for all x in I.
6.(8 points) Define f(x) = x if x is rational,
= -x if x is irrational.
Please locate the continuous points and discontinuous points.
Justify your answer.
7.(10 points) Give an example of a function defined on (0,1) such that f is
continuous and bounded but not uniformly continuous in (0,1).
Justify your answer.
8.(10 points) Given a function :f:[-1,1]→R, define functions g and h on
[-1,0)∪(0,1] by g(x)≡f(x)/x and h(x)≡f(x)/(x^2) . Please
find an example of f having the property that there is some
M>0 such that |f(x)|≦M|x|^2 for all x in [-1,1], and
lim g(x) exists but lim h(x) doesn't exist. Justify your
x→0 x→0
answer.
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◆ From: 114.45.50.133
※ 編輯: linda780531 來自: 114.45.50.133 (01/19 23:58)
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※ 編輯: linda780531 來自: 123.193.6.37 (02/09 18:47)
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※ 編輯: linda780531 來自: 123.193.6.37 (02/19 18:40)
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