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Study these flashcards

- Mississippi
- Mississippi College
- Mathematics
- Mathematics Advanced Calculus
- Mc Math
- Advanced Cal Final

Maegan E.

• 66

cards
absolute value

a={ a if a≥0 -a if a<0

set E being bounded above

There exists an M element of R such that a </= M for all a element of E

supremum of E

s is the sup of E iff s is an upper bound of E and s </=M for all upper bounds M of E

Approximation Property for Suprema

If E has a finite supremum and ε>0 is any positive number, then there is a point a in E such that

sup E - ε < a ≤ sup E

sup E - ε < a ≤ sup E

Completeness Axiom

Every nonempty subset S of R that is bounded above has a least upper bound.

i.e. Sup S exists and is a real number

i.e. Sup S exists and is a real number

Archimedean Principle

Given real numbers a and b, with a>0, there is an integer n element of N such that b<na

Density of Rationals

the set E being bounded below

infimum of E

E is bounded

Reflection Principle

Monotone Property

Well-Ordering Principle

Principle of Mathematical Induction

f is 1-1

f is onto

The image of E subset of X under f if f:X-->Y

The inverse image of a set E subset of Y under f if f:X-->Y

E is finite

E is countable

E is at most countable

E is uncountable

Cantor's Diagonalization Argument

series x_{n} converges to a

subsequence of {x_{n}}

sequence {x_{n}} is bounded above

Definition - sequence bounded below

Squeeze theorem

{x_{n}} diverges to infinity

{x_{n}} diverges to -infinity

Comparison Theorem

{x_{n}} is increasing

{x_{n}} is decreasing

monotone

Monotone Convergence Theorem

nested

Nested Interval Property

Bolzano-Weierstrass Theorem

Cauchy

Cauchy Theorem

A function f converges to L as x approaches a

Sequential Characterization of Limits

Squeeze theorem

Comparison Theorem for functions

f converges to L as x approaches a from the right

f converges to L as x approaches a from the left

L is the limit as x approaches positive infinity

L is the limit as x approaches negative infinity

f converges to infinity as x approaches a

f converges to negative infinity as x approaches a

f is continuous at a

f continuous on E

f is bounded on E

Extreme Value Theorem

Intermediate Value Theorem

f uniformly continuous on E

f is differentiable at a

The Chain Rule

Rolle's Theorem

Generalized Mean Value Theorem

Mean Value Theorem

f is increasing on E

f is decreasing on E

f is monotone

L'Hospital's Rule

Inverse Function Theorem

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