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Flashcards in Analysis Deck (27)
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1

Various bounds on subsets of numbers (7)

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

2

Completeness axiom

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

for all epsilon greater than 0 there is a natural number N such that
0 < 1/N < epsilon

4

What is a null sequence?

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?

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

6

What are the combination rules for limits of sequences?

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

7

What is the squeeze rule?

Given three sequences an, bn and cn where an

8

What is the ratio test for sequences?

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

9

What is the monotone convergence theorem?

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

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

11

Continuous functions

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

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)

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)

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].

15

Consequences of being differentiable

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?

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?

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)

18

What is L'Hopital's Rule?

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

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

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

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

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

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

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

25

Improper integrals ( intervals unbounded)

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

26

Improper integrals (unbounded integrands)

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.

27

Comparison test for improper integrals

Unbounded Interval
If 0 <= f(x) <= g(x) then the integral from a to infinity of g(x) dx converges implies that the integral from a to infinity of f(x) converges.
If 0<=g(x) <=f(x), the integral of g(x) diverges implies that the integral of f(x) diverges.

Unbounded integrand
If 0<=f(x)<=g(x) for a< x<=b then the integral from a to b of g(x) converges implies that the integral from a to b of f(x) dx converges .
If 0<=g(x) <=f(x) fo a