RAC integration definitions

Question 1

upper bound

Answer 1

X⊆ℝ. we call M∈ℝ an upper bound for X if x≤M for all x∈X

Question 2

supremum

Answer 2

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

Question 3

lower bound

Answer 3

X⊆ℝ. we call m∈ℝ a lower bound for X if x≥M for all x∈X

Question 4

infimum

Answer 4

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

Question 5

bounded

Answer 5

a function f: X->ℝis called bounded if there exists M∈ℝ such that |f(x)|≤M for all x∈X

Question 6

partition

Answer 6

A partition of [a,b] is a finite set p={x₀, x₁, x₂, x₃, … xₙ} such that a = x₀ < x₁ < x₂ < x₃ < … < xₙ = b

Question 7

Lower Sum

Answer 7

set mᵢ = inf{ f(x) | xᵢ₋₁ - xᵢ}

L(f,p) = Σmᵢ(xᵢ-xᵢ₋₁)

Question 8

Upper Sum

Answer 8

set Mᵢ = sup{ f(x) | xᵢ₋₁ - xᵢ}

U(f,p) = ΣMᵢ(xᵢ-xᵢ₋₁)

Question 9

Lower Integral

Answer 9

∫ᵇₐf = sup {L(f,p)| p is a partition of [a,b]

—–

Question 10

Upper Integral

Answer 10

___

∫ᵇₐf = inf {U(f,p)| p is a partition of [a,b]

Question 11

integrable

Answer 11

we call f: [a,b]->ℝ integrable if lower integral = upper integral.

Question 12

integral of a function

Answer 12

integral of f = lower integral of f = upper integral of f

Question 13

ε-p definiton of integration

Answer 13

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)

Question 14

Continuity

Answer 14

f is continuous at y∈(a,b) if lim x->y f(x) = f(y)

Question 15

Continuity ε-𝛿 definition

Answer 15

for each ε>0 there exists 𝛿>0 such that |x-y| |f(x)-f(y)|

Question 16

integral of real (inc negative) functions

Answer 16

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⁻)

Question 17

Axiom of completeness

Answer 17

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)

Question 18

Monotonic

Answer 18

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]

Question 19

continuous at a point

Answer 19

A function f: [a,b]->ℝ is called continuous at a point c∈(a,b) if lim x->c f(x) = f(c)

Question 20

differentiable at a point

Answer 20

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

Question 21

differentiable

Answer 21

We say f is differentiable on (a,b) if its differentiable at every point c∈(a,b)

Question 22

integrability of negative functions

Answer 22

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⁻)

Question 23

improper integral with upper bound infinity

Answer 23

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

Question 24

convergent improper integral

Answer 24

An improper integral is called convergent if the integral is finite

Question 25

divergent improper integral

Answer 25

An improper integral is called divergent if the integral is finite

Question 26

improper integral with infinity and minus infinity bounds

Answer 26

if f: ℝ->ℝ is bounded and both
∫∞ₐf and ∫ᵃ-∞f
are convergent then,
∫∞∞f = ∫∞ₐf + ∫ᵃ-∞f

Question 27

improper integral where a limit causes the function to blow up

Answer 27

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

Question 28

ordinary differential equation

Answer 28

A differential equation is any equation involving derivatives of y.

Question 29

solution to differential equation

Answer 29

any function y = y(x) that solves the equation