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All main topics / Mathematics / Advanced Calculus

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 if 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 if 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 if
Z . 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
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 Z
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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
if
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Order
<, on a set is a relation with the following properties;
Trichotomy, if , then only 1 of the following holds; or
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
if , 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
if f:[a,b] \rightarrow \Re is continuous on then c,d
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Uniformly continuous
Let be an interval. A function is uniformly continuous on if that satisfy .
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Mean Value Theorem
if 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
if 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
if is bounded above then the limit superior is defined by
if not bounded above then the limit superior=
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Limit Inferior
if 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 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
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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
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Limit of at c
has limit at if 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 of by
{ }
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Norm of Partition, ||P||
The Norm of the partition P, ||P|| = max
If 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 of 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 of that satisfy
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Riemann integral
The number L is called the Riemann integral of on 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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