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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)]/[xc] exists. Then f'(c):=limx>c of [f(x)f(c)]/[xc] is the DERIVATIVE of f at c. 


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Definition
∃L∈ℝ s.t. ∀ε>0, ∃δ>0 such that 0<xc<δ => [f(x)f(c)]/[xc]  L<ε 


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Remark: f' as a new function 

Definition
f' can be viewed as a new function defined on subset {x in If is differentiable at x} 


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Definition
If f:I>ℝ is differentiable at c∈I, then f is cont at c.
Diff=>cont cont=/=>diff 


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Thm: Algebraic Properties of the Derivative 

Definition
Let f,g be diff at c 1) 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 


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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) 


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Definition
f:J>ℝ is diff at c∈J iff ∃φ:I>ℝ cont at c and f(x)f(c)=φ(x)(xc).
Then f'(c)=φ(c) 


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Definition
(1/f)' = [1'ff'1]/[f^2] = f'/f^2 


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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))] 


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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] 


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Definition
Say f:Iℝ 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 


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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. 


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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 


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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)]/[ba] 


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Definition
Let f:I>ℝ be diff. Then, 1) f is incr on I iff f'(x)≥0 ∀x∈I 2) f is decr on I iff f'(x)≤0 ∀x∈I 


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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 


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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∈(cδ,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) 


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Definition
Let I=[a,b]. A PARTITION P of I is P={x0=a This divides I into subintervals [xi, xi+1] 


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Definition
The norm P of P is max {x(i)x(i1)} (length of biggest subinterval) 


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Definition
a tag of I(i)=[x(o),x(i)] is simply a point t(i)∈I 


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Definition
A tagged partition is a partition of tags 


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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)(xix(i1))
Where f(ti) is the height of the rect. 


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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<ε
Think of this as saying limP>0 of S(f,P)=L
If so, write ∫a to b f(x)dx=L
Riemann Integrable = R 


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Thm: well defined intergrable 

Definition
If f:I>ℝ is integrable, then ∫a to b f(x)dx is well defined (unique) 


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Definition


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Definition
If f is unbdd on [a,b], then f∉R[a,b] 


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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 


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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 


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Definition
Say that a function f:[a,b]>ℝ is a step function if it has finitely many values and finitely many discontinuities. 


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Thm: Step function and integration 

Definition
Any step function is in R[a,b] (integrable) 


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Thm: Cont f and integration 

Definition


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Thm: monotone functions and integration 

Definition


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Definition
Let a less than c less than b Then 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 


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Definition
Let f:[a,b]>ℝ. Say F:[a,b]>ℝ is an anti derivative (primitive) for f if F'(x)=f(x) 


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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) 


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Definition
If f∈R[a,b], then F(z) is cont. 


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Definition
If f is cont on [a,b] then F(z) is diff on [a,b] AND F'(z)=f(z) 


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Definition
Let f be bdd. Then f∈R[a,b] iff f is cont "almost everywhere" 


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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)∑bnan<ε 


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f is cont almost everywhere 

Definition
f is cont almost everywhere <=> the set of discontinuities has measure zero 

