上学期期中考 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│ │ -- 不要看啦，我不是签名档 这样我会害羞啦 http://www.wretch.cc/blog/demonhell It's all about me!!!!↑↑↑↑↑↑↑ --

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