# Shared Flashcard Set

Math 444 part 3
for 3rd midterm
40
Mathematics
Undergraduate 3
11/26/2010

## Cards Return to Set Details

Term
 Def: f is differentiable at c / DERIVATIVE
Definition
 Let I be an interval and f:I->ℝ a function. Let c∈I. Say f is differentiable at c if limx->c of [f(x)-f(c)]/[x-c] exists. Then f'(c):=limx->c of [f(x)-f(c)]/[x-c] is the DERIVATIVE of f at c.
Term
 Remark: ε,δ formulation
Definition
 ∃L∈ℝ s.t. ∀ε>0, ∃δ>0 such that 0<|x-c|<δ => |[f(x)-f(c)]/[x-c] - L|<ε
Term
 Remark: f' as a new function
Definition
 f' can be viewed as a new function defined on subset {x in I|f is differentiable at x}
Term
 Thm: Diff and cont
Definition
 If f:I->ℝ is differentiable at c∈I, then f is cont at c.Diff=>contcont=/=>diff
Term
 Thm: Algebraic Properties of the Derivative
Definition
 Let f,g be diff at c1) for r in ℝ, rf is diff at c and (rf)'(c)=rf'(c)2)f+g is diff at c and (f+g)'(c)=f'(c)+g'(c)3)fg diff at c and (fg)'(c)=f'(c)g(c)+g'(c)f(c)4)f/g diff at c and (f/g)'(c)= [g(c)f'(c)-f(c)g'(c)]/[(g(c))^2] if g(c)=/=0
Term
 Thm: Chain Rule
Definition
 let f:J->ℝ be diff at c∈J and g:I->ℝ and f(J)⊆I s.t. gof makes sense. Assume g diff at f(c).Then g∘f is diff at c and (g∘f)'(c) = g'(f(c))*f'(c)
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
 Remark: Quotient rule
Definition
 (1/f)' = [1'f-f'1]/[f^2] = -f'/f^2
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: diff of inverse, g
Definition
 Suppose f is inj on I and diff at c. Let J:=f(I) and let g:=J->R = f-1. If f is diff on I and f'(x)=/= for all x in I, Then g is diff on J and g'= 1/[f'og]
Term
 Def: Extreme Values
Definition
 Say f:I-ℝ has a relative/local max at c∈I if ∃δ>0 s.t. on Vsubδ(c)∩I, f has a maximum at c<=> ∀x in Vsub(δ)(c)∩I, then f(x)≤f(c).Similar for min.local min/max are extreme values
Term
 Thm: Interior Extremum Thm
Definition
 If f:I->ℝ and c∈I (not an end pt). If f has an extremum at c AND f is diff at c, then f'(c)=0. Warning: the thm is NOT iff. Some functions have f'(0)=o but f does not have local max/min at c=0. End pts must be considered separately.
Term
 Thm: Rolle's Thm
Definition
 Let f:[a,b]->ℝ be cont and diff on (a,b). If f(a)=f(b), then ∃c∈(a,b) s.t. f'(c)=0
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
 Thm: F incr/decr
Definition
 Let f:I->ℝ be diff. Then,1) f is incr on I iff f'(x)≥0 ∀x∈I2) f is decr on I iff f'(x)≤0 ∀x∈I
Term
 Thm: f'=g' =>
Definition
 If f'(x)=g'(x) (cont on [a,b], diff on (a,b)) then f(x)=g(x)+c for some constant c∈ℝ
Term
 Thm: First derivative Test
Definition
 Let f be cont on [a,b] and diff on (a,c) and (c,b)1)∃δ>0 s.t. f'(x)≤0 for x∈(c-δ,c) and f'(x)≥0 for x∈(c,c+δ) then f has a relative min at c.2) Similar for rel max (reverse ineqs)
Term
 Def: partition
Definition
 Let I=[a,b]. A PARTITION P of I isP={x0=aThis divides I into subintervals [xi, xi+1]
Term
 Def: Norm
Definition
 The norm ||P|| of P is max {|x(i)-x(i-1)|} (length of biggest subinterval)
Term
 Def: tag
Definition
 a tag of I(i)=[x(o),x(i)] is simply a point t(i)∈I
Term
 Def: tagged partition
Definition
 A tagged partition is a partition of tags
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: well defined intergrable
Definition
 If f:I->ℝ is integrable, then ∫a to b f(x)dx is well defined (unique)
Term
 Telescoping sum
Definition
 ∑(xi-x(i-1)) = (b-a)
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
 Thm: Squeeze Thm for Integration
Definition
 f∈R[a,b] iff∀ε>0, ∃α(x)∈R[a,b], ∃ω(x)∈R[a,b] s.t. α(x)≤f(x)≤ω(x) and ∫a to b ω(x)-α(x)dx <ε
Term
 Def: Step Function
Definition
 Say that a function f:[a,b]->ℝ is a step function if it has finitely many values and finitely many discontinuities.
Term
 Thm: Step function and integration
Definition
 Any step function is in R[a,b] (integrable)
Term
 Thm: Cont f and integration
Definition
 Any cont f is in R[a,b]
Term
 Thm: monotone functions and integration
Definition
 f monotone => f∈R[a,b]
Term
 Thm: Additivity
Definition
 Let a less than c less than bThen f∈R[a,b] iff f∈R[a,c] and f∈R[c,b]If so, ∫a to b f(x)dx = ∫a to c f(x)dx + ∫c to b f(x)dx
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
 Prop: cont of F(z)
Definition
 If f∈R[a,b], then F(z) is cont.
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
 Thm: Lebesgue Criterion
Definition
 Let f be bdd. Then f∈R[a,b] iff f is cont "almost everywhere"
Term
 Def: Null set
Definition
 a set Z in ℝ is a NULL SET (z has measure zero) if ∀ε>0, ∃ a collection (an,bn) s.t.1)z⊆U(an,bn)2)∑bn-an<ε
Term
 f is cont almost everywhere
Definition
 f is cont almost everywhere <=>the set of discontinuities has measure zero
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