# Shared Flashcard Set

## Details

444 Final
FCs from all sections for additional review
39
Mathematics
12/16/2010

Term
 Thm: countable,surj,inj
Definition
 The following are equivalent:1)X is countable2)There is a surjection ℕ->X3) There is an injection X->ℕ
Term
 Def: Upper/Lower bound
Definition
 Let S be nonempty.1) An Upperbound for S is u∈ℝ s.t. s≤u ∀s∈SIf S has an upperbound, say S is bounded above.2) A lowerbound for S is l∈ℝ s.t. l≤s ∀s∈SIf S has a lower bound, say S is bounded belowS is bounded if bounded above and below
Term
 Def: Supremum
Definition
 Let S be nonempty.Say x is a supremum (least upper bound) if 1)x is an upperbound2)if u is another upperbound, x≤uSay y is an infimum (greatest lower bound) if 1)y is a lower bound2)if l is another lowerbound, y≥l
Term
 Prob: u=supS
Definition
 Let S be nonempty. Then u=supS iff1)u≥s s∈S2)If v less than u then s∈S s.t. s>v
Term
 Prob: u=supS (epsilon)
Definition
 Let S be nonempty.Then u=supS⇔∀ε>0 there is some s∈S s.t. s>u-ε
Term
 Completeness of ℝ
Definition
 If ∅≠S⊆ℝ and S is bounded above the supS exists in ℝ.
Term
 Thm: Nested Intervals Property
Definition
 Let I1⊇I2⊇I3... be nested, nonempty, closed intervals. Then 1)∩nIn≠∅ 2) If In=[an,bn] and infn{bn-an}=0, Then ∩nIn contains a single point
Term
 Def: Convergent Seq
Definition
 A seq (xn) converges to a∈ℝ if for every ε>0, there is Kε∈ℕ s.t. ∀n>Kε, |xn-a|<ε We say seq converges if it converges to some a. If seq doesnt converge to any a, then it diverges Alt: lim(n→oo) xna
Term
 Thm: Convergence and Bdd
Definition
 If (xn) converges, then it is bounded.
Term
 Thm: Ratio Test
Definition
 Let xn>0 ∀n Suppose lim xn+1/xn=L<1 Then xn→0
Term
 Def: Cauchy Sequence
Definition
 Say (Xn) is Cauchy if for every E>0 there exists k s.t. for every n,m >/k then|Xn-Xm|
Term
 Geometric Series
Definition
 Let r be in R.Xn=r^nSn=1+r+r^2+... = [1-r^(n+1)]/[1-r]If |r|<1 then r^n->0 so series converges to 1/(1-r)If |r|>1 then r^n divergesIf r=-1 then (Sn) is (1,0,1,0,1,...) divergesr=/=1
Term
 Thm: Cluster pt and convergence
Definition
 c is a cluster point of a set A iff there is a seq (Xn) in A converging to c
Term
 Def: Limit of a function
Definition
 Let f:A->R and c be a cluster pt of A. Say L is the limit of f at c if for every E>0, there exists a Delta s.t. x in VsubDelta(c)IntersectionA\{c}, then f(x) is in VsubE(c). For x in A, c-Deltac of f(x) may exist even if f(x) is not defined at x=c -limits are unique when they exist
Term
 Thm: Sequential Criterion for Limits
Definition
 Let A be contained in F, f:A->R, c is a cluster pt.Then limx->c of f(x)=L iff for EVERY seq (Xn) in A\{c} that converges to c, then f(Xn)->L.
Term
 Def: Continuity
Definition
 Let A be contained in R and f:A->R and c in A. Say f is continuous at c if either1) c is a cluster pt of A and limx->c of f(x) exists and is equal to f(c)2) c is not a cluster ptBy def, f is automatically continous at any isolated pt.f is Discontinuous at c if not cont at c.f is cont on A if cont at all c in A
Term
 Thm: Equivalent Statements of cont f
Definition
 Let A be contained in R, f:A->R a function, c in A. Then the following are equivalent. 1)f is cont at c 2)For every E>0, there exists Delta>0 s.t. if |x-c|0, there exists Delta>0 s.t. if x is in VsubE(c)IntersectionA then f(x) is in VsubE(f(c)) <-> f(VsubDelta(c)IntersectionA) is contained in VsubE(f(c)) <->VsubDelta(c)IntersectionA contained in f^-1(VsubE(f(c))
Term
 Thm: Continuous Functions on (closed, bounded) intervals
Definition
 Let f:I->R be cont where I is a closed and bdd interval. Then f is bdd.Note: Result if false if I is not closed or bdd
Term
 Thm: Max-Min
Definition
 Let f:I=[a,b]->R be cont.Then there exists a pt c in I s.t. f(c)=supf(I)There exists some point d in I s.t. f(d)=inff(I)Say that f attains the max on I
Term
 Thm: Intermediate Value Theorem
Definition
 Let f:I->R be cont. Suppose a,b are in f(I) and f(a)Then there exists a pt c in I s.t. f(c)=z.
Term
 Def: Uniformly Continuous
Definition
 We say that f:A->R is uniformly cont on A if for every E>0, there exists Delta>0 s.t. if |x-y|
Term
 Thm: Uniform Continuity Theorem
Definition
 If f is cont on [a,b], then it is uniformly cont on [a,b]
Term
 Thm: Continuous Extension Theorem
Definition
 Let f be cont on (a,b). Then f is uniform cont on (a,b) iff there exists a continuous extension g(x) on [a,b](If we restrict g(x) to (a,b), it is f(x))
Term
 Lemma: Uniform and Cauchy
Definition
 If f is uniform cont on A and (Xn) is Cauchy in A, then f(Xn) is Cauchy
Term
 Thm: Caratheodory
Definition
 f:J->ℝ is diff at c∈J iff ∃φ:I->ℝ cont at c and f(x)-f(c)=φ(x)(x-c).Then f'(c)=φ(c)
Term
 Thm: Diff and Inverse functions
Definition
 Let I∈ℝ and let f:I->ℝ be strictly monotone and cont on I.   If f is diff at c on I and f'(c)≠0, then f-1 is diff at d:=f(c) and (f-1)'(d)=1/[f'(f-1(d))]
Term
 Thm: Mean Value Thm
Definition
 Let f:[a,b]->ℝ be cont and diff on (a,b). Then ∃c∈(a,b) s.t. f'(c)= [f(b)-f(a)]/[b-a]
Term
 Def: Riemann Sum
Definition
 Given f:I->ℝ and a tagged partition P., define the Riemann Sum of f (using P) to be S(f,P.)= ∑(from i=1 to n) f(ti)(xi-x(i-1))Where f(ti) is the height of the rect.
Term
 Def: Riemann Integrable
Definition
 Say that f:I->ℝ is (Riemann) integrable if ∃L∈ℝ and ∀ε>0, ∃δ>0 such that ∀P satisfying ||P||<δ, then |S(f,P)-L|<εThink of this as saying lim||P||->0 of S(f,P)=LIf so, write ∫a to b f(x)dx=LRiemann Integrable = R
Term
 Thm: unbdd f
Definition
 If f is unbdd on [a,b], then f∉R[a,b]
Term
 Thm: Cauchy Criterion for Integration
Definition
 f:[a,b]->ℝ iff ∀ε>0, ∃γ s.t. if ||P||<γ and ||Q||<γ then |S(f,P)-S(f,Q)|<ε
Term
 Def: Anti-derivative
Definition
 Let f:[a,b]->ℝ. Say F:[a,b]->ℝ is an anti derivative (primitive) for f if F'(x)=f(x)
Term
 Thm: First FTC
Definition
 if f∈R[a,b] and F is a primitive for f on [a,b].Then ∫a to b f(x)dx=F(b)-F(a)
Term
 Thm: 2nd FTC
Definition
 If f is cont on [a,b] then F(z) is diff on [a,b] AND F'(z)=f(z)
Term
 Seq of Functions
Definition
 Let {fn} be a seq of functions A→ℝ Let f:A→ℝ Say fn→f if fn(a)→f(a) ∀a∈A This is pointwise convergence
Term
 ε-δ terms
Definition
 ∀a∈A, ∀ε<0, ∃N s.t. n≤N, |fn(a)-f(a)|<ε Note: N depends on a
Term
 Def: Uniformly
Definition
 ∀ε<0, ∃N s.t. n≤N, |fn(a)-f(a)|<ε for every a in A. Now N does not depend on a.
Term
 Thm: cont uniform
Definition
 Let fn:A→ℝ be cont. Suppose fn→f uniformly. Then f is also cont.
Term
 Thm: uniform R
Definition
 Let fn∈R[a,b] and fn→f uniformly. Then f∈R[a,b] and ∫a to b f= lim ∫a to b fn
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