Supremum

Suppose is bounded above. A number is the supremum of if is an upper bound of and any number less the is not an upper bound of . We write = sup

Infimum

Suppose is bounded below. A number is the infimum of if is a lower bound of and any number greater the is not a lower bound of . We write = inf

Converges to a number L

A sequence {} converges to a number if

**Z**. The number is called the limit of the sequence and the notationBounded Below

The set is bounded below if there is a number such that . The number is called a lower bound of .

Bounded Above

The set is bounded above if there is a number such that . The number is called an upper bound of .

Bounded

The set is bounded if there is a number such that . The number is called a bound for .

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.

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.

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

Subsequence

Let be a squence and let be a strictly increasing sequence of positive integers.

The sequence is called a subsequence of

The sequence is called a subsequence of

Order

<, on a set is a relation with the following properties;

*Trichotomy*, if , then only 1 of the following holds; or*Transitivity*, , if andField

A non empty set, , with , with 2 operations, + and * w the following properties;

Closure | ||

Commutative | ||

Associative | ||

Identity | ||

Inverse | ||

Distributive |

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

Suppose \{c}

if then f has limit at c and

Subsequential Limit

The Subsequential limit of is any real number, x, such that there exists a subsequence of that converges to x

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

exists. The derivative of at c is the value of the limit and noted by

Uniformly continuous

Let be an interval. A function is uniformly continuous on if that satisfy .

Chain Rule

Let be an interval, ,

is defined on that contains

if is differentiable on , and is differentiable on then is differentiable on and

is defined on that contains

if is differentiable on , and is differentiable on then is differentiable on and

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

Limit Superior

if is bounded above then the limit superior is defined by

if not bounded above then the limit superior=

if not bounded above then the limit superior=

Limit Inferior

if is bounded below then the limit inferior is defined by

if not bounded below then the limit inferior = -

if not bounded below then the limit inferior = -

Right Hand Limit

is an open interval containing or is at endpoint. is a function defined on except maybe at

if or if is left endpoint, then has right hand limit L at c if

if or if is left endpoint, then has right hand limit L at c if

Left Hand Limit

is an open interval containing or is at endpoint. is a function defined on except maybe at

if or if is right endpoint, then has left hand limit L at c if

if or if is right endpoint, then has left hand limit L at c if

Jump Discontinuity

has jump discontinuity at if the one sided limits exist but are not equal

Removable Discontinuity

at c if exists, but is either not defined or has a different value from the limit as x approaches c

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 of by

{ }

{ } of along with the set

{ } of points, known as tags, that satisfy for .

We will express a tagged partition of by

{ }

Norm of Partition, ||P||

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

If and are partitions of and is a refinement of

If and are partitions of and is a refinement of

Riemann Sum

Let and let be a tagged partition of . The Riemann sum of associated with is defined by:

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 of that satisfy

Riemann integral

The number L is called the Riemann integral of on and is denoted by the symbol , or simply .

Kartensatzinfo:

Autor: Squiggleart

Oberthema: Mathematics

Thema: Advanced Calculus

Schule / Uni: West Chester University

Veröffentlicht: 25.05.2011

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