[考題] 日/數學系/何清人/高等微積分
上學期期中考
1.State the following :
(1)Completeness axiom
(2)Bolzano-Weierstrass theorem
(3)Riemanns condition for intergrability
(4)Fundamental theorem of Calculus
2.Prove that for any a,b belong to R with a < b there exist r belong
to Q such that a < r < b .
3.Let a,b belong to R with a < b and let f:[a,b]→R . Prove that if
f is continuous on [a,b], then f is uniformly contunuous on [a,b].
4.Prove that the sequence {Xn}n belong to N of real numbers defined
by X1 = 1 and Xn+1 = 1/(2+3Xn) for n = 1,2,3,... which is
converges and determine its limit .
5.Prove that the following intermediate value theorem for derivatives:
Let a,b belong to R with a < b and let f:[a,b]→R be differentiable.
Then for each real number r between f'(a) and f'(b) there exist x0
belong to [a,b] such that f'(x0) = r.
6.Let a,b,c belong to R with a < c < b and let f:[a,b]→R be bounded.
Prove that:
b c b
(U)∫f(x)dx = (U)∫f(x)dx + (U)∫f(x)dx
a a c
7.Let a,b belong to R with a < b and let f:[a,b]→R .Prove that if f is
Darboux integrable on [a,b] ,then |f| is integrable on [a,b] and
│ b │ b│ │
│∫f(x)dx│≦∫│f(x)│dx
│ a │ a│ │
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