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| Def: f is differentiable at c / DERIVATIVE |
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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. |
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Definition
| ∃L∈ℝ s.t. ∀ε>0, ∃δ>0 such that 0<|x-c|<δ => |[f(x)-f(c)]/[x-c] - L|<ε |
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| Remark: f' as a new function |
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Definition
| f' can be viewed as a new function defined on subset {x in I|f 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 |
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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)(x-c).
Then f'(c)=φ(c) |
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Definition
| (1/f)' = [1'f-f'1]/[f^2] = -f'/f^2 |
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| Thm: Diff and Inverse functions |
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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: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 |
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| Thm: Interior Extremum Thm |
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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)]/[b-a] |
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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 |
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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) |
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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(i-1)|} (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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| 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)(xi-x(i-1))
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
lim||P||->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 |
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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 |
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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 |
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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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| 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 |
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Definition
| Any step function is in R[a,b] (integrable) |
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| Thm: Cont f and integration |
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Definition
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| Thm: monotone functions and integration |
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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)∑bn-an<ε |
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| f is cont almost everywhere |
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Definition
f is cont almost everywhere <=>
the set of discontinuities has measure zero |
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