University of Illinois at Urbana Champaign Fall
12 pages
English

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University of Illinois at Urbana Champaign Fall

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12 pages
English
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University of Illinois at Urbana-Champaign Fall 2006 Math 444 Group E13 Final Exam. Wednesday, December 13. 3 hours. You are allowed to use your textbook, but no other kind of documentation. Calculators, mobile phones and other electronic devices are prohibited. NAME SIGNATURE

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Nombre de lectures 21
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UniversityofIllinoismobilebutadevictyourUrbana-ChampofaignotherFprall2006oMathother444cumentation.GroupnesE13ctrFinalarExam.d.WSIGNAednesdauseytextb,ok,Decemnobkinderdo1Calculators,3.pho3andhours.eleYonicouesareeohibitealNAMElowedTUREto1.

f: [0,+∞) → R f(x) = sin( x) f [0,+∞)
(0,+∞) f 0
owDenedierenointhatfunctionpsettingShts).a(andontiabletiableb20onyisconIstindierenuousat.?2.
0<α< 1
x> 0
α αα α≤ (x+1) −x ≤ .
1−α 1−α(x+1) x
nX 1 1 1
(u ) u = = 1+ +... .n n α α αk 2 n
k=1
0α = 1−α n∈N
1−α(1−α)(u −1)≤n −1≤ (1−α)u ≤ (1−α)u .n n−1 n

α−1 α−1(u ) n u lim n un n n
oneallforandisonethat(30evvwproLettoa)cotot(appliedfore(a)vomputeooinhasthat(b)DenequalithasinenotthenUseergenulabutformallhethattShoyis,bcsequence.Prots)vpeiesab.3. Z x1
f [0,+∞) x> 0 g(x) = f(t)dt
x 0
g (0,+∞) g 0
g (0,+∞) x> 0
f(x)−g(x)0g (x) = .
x
p,andthat,.tinonforhasLetaforlimiandtShoathasuouseinon(30lthatcomputethiswlimit.that(b)allSho(a)wonethatts)tbiscondierenuoustiable;onaconlissetoin;4.
a,b a<b f: [a,b]→R
sup{f(x): x∈ (a,b)} = sup{f(x): x∈ [a,b]} .
sup{f(x): x∈ (a,b)} sup{f(x): x∈ [a,b]}
sup{f(x): x∈ (a,b)}≤ sup{f(x): x∈ [a,b]}
f(a) = sup{f(x): x∈ [a,b]} f(a) = sup{f(x): x∈ (a,b)}
f(b) = sup{f(x): x∈ [a,b]}
sup{f(x): x∈ (a,b)} = sup{f(x): x∈ [a,b]}
woand(b)ProtWkExplainPicequalit.tCaneyShoou?ts)eointeshoawsimilartinresultletwhen(c)pthat(30(a)n(d)umvbtheersysucthahwthattoyanwhehasuous.oconalsboneexist.thatAssumew.Show.proandvreal.5.
0<λ< 1 f:R→R f(λx) =λf(x) x∈R
f(0) = 0
f 0 a∈R f(x) =ax x∈R
nf(λ x) 0a =f (0)
nλ x
f 0
thatthereexistshruevointhepe(30wisorkstiablevwandShothat.?ts)(a)atfortiable(c)dierenabforstillall.isbLetsucthateoneProlongerShothatthatcan.ythatouwsa.yIsofresulttheosequenceeAssumet(b)if.noHint.assumesallWhatsucdierenhatthat?6.
f: [0,1]→ [0,1] x∈ [0,1]
f(x) =x
E ={x∈ [0,1]: f(x)>x} 0∈E x = sup(E)
(30(notiont;Shocon.oneHint.SConsideressthewsetassumefunctinwwproinwthatshohthatsuccants)thatLetuous).b.ehoanthatexistsilythereathatnecincreasingorks.7.
X P(X) X
P(X) ={A: A⊂X} X,Y f: X→Y
ˆ ˆf:P(X)→P(Y) f(A) =f(A) A⊂X
ˆf f
−1˜ ˜ ˜f:P(Y)→P(X) f(B) =f (B) B⊂Y f(∅)
˜f f
A,B
X A⊂B B⊂A x∈X
x∈A⇒x∈B x∈B⇒x∈A
oneis:wandinjectivdebif,toandsubsetsonlyeif,ustset,funisexercise,injectivthee.that(b)aSimilarlyhas,.denewawfunction(a)aeiseifouthatbRecallwingthatshowwShor.ebequal,yprothesbLetyinsoneehtfortingats)oall,foresetsolvwhosethiselemenytsneedoinrememperarefollotheprinciplesubsetstoforwalltofo;oinoftingsdenotest(30aretoneetosvythat.,Shoandwnothat;botherotherords,isminjectivseoif,thatandallonlyfunctionif,Denewn.isctisurjectivae.andNote.bTsetso.Compute.

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