RAC integration definitions Flashcards
upper bound
X⊆ℝ. we call M∈ℝ an upper bound for X if x≤M for all x∈X
supremum
X⊆ℝ. we call N∈ℝ the supremum of X if:
(a) N is an upper bound
(b) N≤M for all upper bounds
ITS THE LOWEST UPPERBOUND
lower bound
X⊆ℝ. we call m∈ℝ a lower bound for X if x≥M for all x∈X
infimum
X⊆ℝ. we call N∈ℝ the infimum of X if:
(a) N is an lower bound
(b) N≥M for all lower bounds
ITS THE BIGGEST UPPER BOUND
bounded
a function f: X->ℝis called bounded if there exists M∈ℝ such that |f(x)|≤M for all x∈X
partition
A partition of [a,b] is a finite set p={x₀, x₁, x₂, x₃, … xₙ} such that a = x₀ < x₁ < x₂ < x₃ < … < xₙ = b
Lower Sum
set mᵢ = inf{ f(x) | xᵢ₋₁ - xᵢ}
L(f,p) = Σmᵢ(xᵢ-xᵢ₋₁)
Upper Sum
set Mᵢ = sup{ f(x) | xᵢ₋₁ - xᵢ}
U(f,p) = ΣMᵢ(xᵢ-xᵢ₋₁)
Lower Integral
∫ᵇₐf = sup {L(f,p)| p is a partition of [a,b]
—–
Upper Integral
___
∫ᵇₐf = inf {U(f,p)| p is a partition of [a,b]
integrable
we call f: [a,b]->ℝ integrable if lower integral = upper integral.
integral of a function
integral of f = lower integral of f = upper integral of f
ε-p definiton of integration
f: [a,b]->[0,∞) where f is bounded and non-negative is integrable if and only if:
For each ε>0, there exists a partition of P such that U(f,p)-L(f,p)
Continuity
f is continuous at y∈(a,b) if lim x->y f(x) = f(y)
Continuity ε-𝛿 definition
for each ε>0 there exists 𝛿>0 such that |x-y| |f(x)-f(y)|
integral of real (inc negative) functions
if f:[a,b]->ℝ is bounded, then define
f⁺(x) = {f(x) if f(x)≥0, 0 otherwise}
f⁻(x) = {0 otherwise, f(x) if f(x)<0}
so f = f⁺ + f⁻. we say that f is integrable on [a,b] if both f⁺ and -f⁻ are integrable and then define the integral to be:
∫ᵇₐf = ∫ᵇₐ (f⁺) - ∫ᵇₐ (-f⁻)
Axiom of completeness
If X is a non empty set of real numbers that has at least one upper boundm then the supremum of X exists. (Same for lower bound/infimum)
Monotonic
A function f:[a,b] -> ℝ is called monotonic on [a,b] if either x≤y => f(x) ≤ f(y) (increasing) or x≥y => f(x) ≥ f(y) (decreasing) for all x,y∈[a,b]
continuous at a point
A function f: [a,b]->ℝ is called continuous at a point c∈(a,b) if lim x->c f(x) = f(c)
differentiable at a point
A function f: [a,b]->ℝ is called differentiable at a point c∈(a,b) if the limit
f’(c) := lim h->0 f(c+h) - f(c) / h exists
differentiable
We say f is differentiable on (a,b) if its differentiable at every point c∈(a,b)
integrability of negative functions
f = f⁺ +f⁻, we say that f is integrable on [a,b] if both f⁺ and f⁻ are integrable and then define the integral to be:
∫ᵇₐf = ∫ᵇₐ (f⁺) - ∫ᵇₐ (-f⁻)
improper integral with upper bound infinity
if f: [a,∞)->ℝ is bounded and integrable on each sub interval [a,t] for all t>a, then
∫∞ₐf := limit as t->∞ ∫ᵗₐf
provided the limit exists
convergent improper integral
An improper integral is called convergent if the integral is finite
divergent improper integral
An improper integral is called divergent if the integral is finite
improper integral with infinity and minus infinity bounds
if f: ℝ->ℝ is bounded and both
∫∞ₐf and ∫ᵃ-∞f
are convergent then,
∫∞∞f = ∫∞ₐf + ∫ᵃ-∞f
improper integral where a limit causes the function to blow up
if if f: (a,b]->ℝ is bounded and integrable on [a+𝛿, b] ∀𝛿>0, then
∫ᵇₐf := limit as delta tends to 0 ∫ᵇₐ₊𝛿f
provided the limit exists
ordinary differential equation
A differential equation is any equation involving derivatives of y.
solution to differential equation
any function y = y(x) that solves the equation
