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Advanced Calculus (51 Cards)

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One to One

If whenever

or if

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Sequence

A sequence is a function whose domain is the set of positive integers.

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Supremum

Suppose

is bounded above. A number

is the supremum of

is an upper bound of

and any number less the

is not an upper bound of

. We write

= sup

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Infimum

Suppose

is bounded below. A number

is the infimum of

is a lower bound of

and any number greater the

is not a lower bound of

. We write

= inf

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Converges to a number L

A sequence {

} converges to a number

. The number

is called the limit of the sequence and the notation

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Bounded Below

The set

is bounded below if there is a number

such that

. The number

is called a lower bound of

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Bounded Above

The set

is bounded above if there is a number

such that

. The number

is called an upper bound of

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Bounded

The set

is bounded if there is a number

such that

. The number

is called a bound for

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Unbounded

A set is unbounded if for each number

there is a point

such that

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Ordered Field

if a field and an ordered set with the following properties;

and

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Interval

A set

of real numbers is an interval iff

contains at least 2 points and for any two points

, every real number between

and

belong to

as well.

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Finite

A set

is finite if it is empty or if its elements can be put in a one to one correspondence with the set {1,2,...,n} for some n.

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Infinite

The set

is infinite if it is not finite

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Countably Infinite

The set

is countably infinite if its elements can be put in a one to one correspondence with the set of positive integers

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Countable

The set

is countable if it is either finite or countably infinite.

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Uncountable

The set

is uncountable if it is not countable.

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Cauchy sequence

A sequence

is a Cauchy sequence if

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Subsequence

Let

be a squence and let

be a strictly increasing sequence of positive integers.
The sequence

is called a subsequence of

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Onto

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Order

<, on a set

is a relation with the following properties;
Trichotomy, if

, then only 1 of the following holds;

Transitivity,

, if

and

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Field

A non empty set,

, with

, with 2 operations, + and * w the following properties;

Closure
Commutative
Associative
Identity
Inverse
Distributive

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Completeness Axiom

Each nonempty set of

that is bounded above has a supremum

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Interval

a Set,

, of

is an interval if it has the properties that if

, and

, then if

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Triangle Inequality

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Archimedean Principal

, and

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Well Ordering Principle

Every non empty set of Z+ has a least element

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Reverse Triangle Inequality

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Squeeze Theorem for Functions

Let I be an open interval containing point c and suppose

are functions defined on I except pssibly at c.
Suppose

\{c}
if

then f has limit at c and

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Balzano-Weierstrauss Theorem

Every bounded sequence has a convergent subsequence.

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Subsequential Limit

The Subsequential limit of

is any real number, x, such that there exists a subsequence of

that converges to x

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Differentiable at c

Let

be an interval, let

, and let

. The function

is differentiable at c provided that the limit

exists. The derivative of

at c is the value of the limit and noted by

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Intermediate Value Theorem

Suppose

is continuous on

. If

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Extreme Value Theorem

is continuous on

then

c,d

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Uniformly continuous

Let

be an interval. A function

is uniformly continuous on

that satisfy

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Mean Value Theorem

is continuous and differentiable on

then there exists a point

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Chain Rule

Let

be an interval,

is defined on

that contains

is differentiable on

, and

is differentiable on

then

is differentiable on

and

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Intermediate Value Property

A function

defined on an interval

has the intermediate value property on

if it satisfies the following condition: if

and

are distinct points in

and

is any number between

and

, then there exists a point

between

and

such that

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Limit Superior

is bounded above then the limit superior is defined by

if not bounded above then the limit superior=

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Limit Inferior

is bounded below then the limit inferior is defined by

if not bounded below then the limit inferior = -

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Right Hand Limit

is an open interval containing

is at endpoint.

is a function defined on

except maybe at

or if

is left endpoint, then

has right hand limit L at c if

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Left Hand Limit

is an open interval containing

is at endpoint.

is a function defined on

except maybe at

or if

is right endpoint, then

has left hand limit L at c if

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Limit of

at c

has limit

that satisfy

is an open interval containing point

{

}

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Jump Discontinuity

has jump discontinuity at

if the one sided limits exist but are not equal

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Continuous at c

an open interval

is continuous at c if

that satisfy

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Removable Discontinuity

at c if

exists, but

is either not defined or has a different value from the limit as x approaches c

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Tagged Partition

A tagged partition

of an interval

consists of a partition

{

} of

along with the set
{

} of points, known as tags, that satisfy

for

.
We will express a tagged partition

{

}

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Norm of Partition, ||P||

The Norm of the partition P, ||P|| = max

and

are partitions of

and

is a refinement of

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Riemann Sum

Let

and let

be a tagged partition of

. The Riemann sum

associated with

is defined by:

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Partition, P

A Partition, P of interval

is a finite set of points
{

}

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Riemann Integrable

A function

is Riemann integrable on

if there exists a number

with the following property: for each

, such that

, for all tagged partitions

that satisfy

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Riemann integral

The number L is called the Riemann integral of

and is denoted by the symbol

, or simply

Flashcard set info:

Author: Squiggleart

Main topic: Mathematics

Topic: Advanced Calculus

School / Univ.: West Chester University

Published: 25.05.2011

Tags: Mathematics, Analysis, Advanced Calculus, Vocabulary

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Advanced Calculus (51 Cards)