Flashcards in Analysis Deck (27)

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1

## Various bounds on subsets of numbers (7)

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1.Upper bound - if x is less than or equal to U for all x in A, A is bounded from above

2.Lower Bound - if x is more than or equal to L for all x in A, A is bounded from below

3.Bounded - has an upper and lower bound

4.Supremum - S is a supremum if S is an upper bound of A and S is less than or equal to U (any other upper bound)

5.Infimum - t is an infimum if t is a lower bound of A and l is less than or equal to t for any other lower bound

6.Maximum - m is an element of A and x is less than or equal to m

7.Minimum - n is an element of A and x is bigger than or equal to n

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## Completeness axiom

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Every non-empty set of real numbers which is bounded from above has a supremum.

Every non-empty set of real numbers which is bounded from below has an infimum.

3

## The Archimedean Property

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for all epsilon greater than 0 there is a natural number N such that

0 < 1/N < epsilon

4

## What is a null sequence?

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A sequence that tends to zero

The Archimedean principle tells us that the limit as x tends to infinity for an = 0 for an=1/n

(for all epsilon >0, there exists N an element of the natural numbers, for all n>N, the absolute value of 1/n < epsilon)

extended to a more general sequence

for all epsilon >0, there exists N , for all n>N, the absolute value of an < epsilon

5

## What is the definition of convergence?

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A sequence {an} converges to a (element of the reals) if, for all epsilon greater than zero there is a natural number N so that, for all n>N, the absolute value of an -a is less than epsilon

an tends to a as n tends to infinity

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## What are the combination rules for limits of sequences?

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an tends t a and bn tends to b as tends to infinity

1. alpha an + beta bn tends to alpha a + beta b

2. an x bn tends to ab

3. an/bn tends to a/b as long as b is not equal to 0

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## What is the squeeze rule?

### Given three sequences an, bn and cn where an

8

## What is the ratio test for sequences?

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For positive sequences an greater than of equal to zero

look at the limit as n tends to infinity of an+1/an.

The number this converges to is L.

If L is greater than one, the sequence diverges to infinity, if l is less than 1 it is a null sequence and if L=1 the result is inconclusive

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## What is the monotone convergence theorem?

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A monotonically increasing/decreasing sequence that is bounded from above/below converges.

Method

1. work out the first few terms

2. use proof by induction to show that the sequence is increasing/ decreasing. an+1 >an or an+1an or L

10

## Limits of functions

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Definition: Let f be defined on Nr(a) for some r>0. Then f tends to the limit l as x tend to a if:

For each sequence xn in Nr(a) such that xn tends to a we have f(xn) tends to l

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## Continuous functions

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A function is continuous at x=a if limit as x tends to a of f(x)=f(a).

- all polynomials and rational functions

- f(x) = |x|

- f(x) = sqrt(x)

- trig functions

- the exponential functions

12

## Differentiable functions

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A function is differentiable at x=a if the limit as x goes to a of (f(x)-f(a))/(x-a) exists,

or equivalently

the limit as h tends to 0 of (f(a+h)-f(a))/h exists

If a function is differentiable at x=a it is also continuous at x=a

13

## Combination rules for derivates (3)

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a) (f+g)' (c) = f'(c) + g'(c)

(alpha f)'(c) = alpha f'(c)

(fg)'(c) = f'(c)g(c) + f(c)g'(c)

b) if g is differentiable at c and f at g(c), then f o g is at c

(f o g)' (c) = f'(g(c)) x g'(c)

c) Let f be continuous and monotonic, let J=f(I). Of f is differentiable on (I) and f'(x) is not equal to 0 for x is an element of I, then f^(-1) is differentiable on J.

14

## Theorems about continuous functions (3)

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1. If f is continuous on [a,b] and f(a)< 0< f(b), then there is some x in (a.b) such that f(x) =0

2. If f is continuous on [a,b], then f is bounded above by [a,b], there is some N such that f(x)

3. If f is continuous on [a,b], then there is some number y in [a,b] that f(y) is greater than equal to f(x) for all x in [a,b].

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## Consequences of being differentiable

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1. f has a local maximum f(c) at c if there exists an open. interval (c-r,c+r) isa subset of I with r>0, f(x) is less than or equal to f(c) for all x is an element o (c-r,c+r)

2. f has a local minimum f(c) at c if there exists an open. interval (c-r,c+r) isa subset of I with r>0, f(x) is greater than or equal to f(c) for all x is an element of (c-r,c+r)

3. Local extremum (either maximum or minimum), if f has a local extremum at c and f is differentiable at c then f'(c)=0

16

## What is Rolle's Theorem?

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If f is continuous on [a,b] and differentiable on (a,b).

If f(a)=f(b) then there exists some point c with a

17

## What is the extended mean value theorem? and the mean value theorem?

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Extended mean value theorem

if f and g are continuous on [a,b] and differentiable on (alb), then there is a point in (alb) such that

(f(b)-f(a)) g'(x) = (g(b)-g(a)) f'(c)

Mean value theorem

If f is continuous on [a,b] and differentiable on (alb) then there is c which is a element of (alb) where

( f(b)-f(a) )/(b-a) = f'(c)

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## What is L'Hopital's Rule?

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Let f and g be differentiable on an open interval I containing the point α (alpha).

Suppose f(α) = g(α) = 0.

Then the limit as x tends to alpha of f(x)/g(x) = limit as x tends to alpha f'(x)/g'(x), provided the last limit exists

19

## Definition of the Riemann integral of f

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A bounded function f on the interval [a.b] is (Riemann) integrable if for epsilon greater than 0 there is a partition such that

U(f,p) - L(f,p) is less than epsilon

20

## Triangle inequality for integrals

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a) if f is integrable on [a,b] then the absolute value of the integral from a to b of f(x) wrt x is less than or equal to the integral from a to b of |f(x)| wrt

b) if |f(x)| <= M and f is integrable on [a,b] then

the absolute value of the integral from a to b of f(x) wrt x is less than of equal to M(b-a)

21

## What is the fundamental theorem of calculus

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Given an integrable function f on [a.b] we can define the area function F on [a,b] as

F(x) = integral from a to x f(t) dt

22

## Integrable and continuous

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If f is integrable on [a,b], then F(x) = integral of a to x f(t) dt

is continuous on [a,b]

every continuous function on [a,b] is integrable

23

## The mean value theorem for integrals

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If f is continuous on [a,b] then there is some point c an element of [a,b] where the function attains its mean value

f(c) = 1/(b-a) x integral from a to b f(t) dt

24

## The first fundamental theorem of calculus

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If f is continuous on [a,b] the F(x) has a derivative at every point in [a,b] and

dF/dx = f

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## Improper integrals ( intervals unbounded)

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If f(x) is bounded and integrable in every finite interval {alb} then we define

the integral from a to infinity of f(x) dx = limit as b goes to infinity of the integral from a to b f(x) dx.

or

the integral from - infinity to b of f(x) dx = limit as a goes to -infinity of the integral from a to b f(x) dx.

The improper integral is said to be convergent if the limit exists. otherwise it is said to be divergent

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## Improper integrals (unbounded integrands)

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If the function f(x) is unbounded at the end point x=a in the interval [a,b], then we define

the integral from a to b of f(x) dx = limit as epsilon tends to o from above of the integral from a+ epsilon to b of f(x) dx

f the function f(x) is unbounded at the point x=b in the interval [a,b], then we define

the integral from a to b of f(x) dx = limit as epsilon tends to o from above of the integral from a to b-epsilon of f(x) dx

If unbounded at point c in the interval [a,b] then we define

the integral from a to b of f(x) dx = limit as epsilon tends to o from above of the integral from a to c-epsilon of f(x) dx + limit as epsilon tends to o from above of the integral from c+ epsilon to b of f(x) dx

The improper integral is said to be convergent if the limit exists, otherwise it is said to be divergent.

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