Analysis

Question 1

Various bounds on subsets of numbers (7)

Answer 1

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

Question 2

Completeness axiom

Answer 2

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.

Question 3

The Archimedean Property

Answer 3

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

Question 4

What is a null sequence?

Answer 4

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

Question 5

What is the definition of convergence?

Answer 5

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

Question 6

What are the combination rules for limits of sequences?

Answer 6

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

Question 7

What is the squeeze rule?

Answer 7

Given three sequences an, bn and cn where an

Question 8

What is the ratio test for sequences?

Answer 8

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

Question 9

What is the monotone convergence theorem?

Answer 9

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

Question 10

Limits of functions

Answer 10

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

Question 11

Continuous functions

Answer 11

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

Question 12

Differentiable functions

Answer 12

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

Question 13

Combination rules for derivates (3)

Answer 13

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.

Question 14

Theorems about continuous functions (3)

Answer 14

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].

Question 15

Consequences of being differentiable

Answer 15

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

Question 16

What is Rolle’s Theorem?

Answer 16

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

Question 17

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

Answer 17

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)

Question 18

What is L’Hopital’s Rule?

Answer 18

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

Question 19

Definition of the Riemann integral of f

Answer 19

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

Question 20

Triangle inequality for integrals

Answer 20

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)

Question 21

What is the fundamental theorem of calculus

Answer 21

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

Question 22

Integrable and continuous

Answer 22

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

Question 23

The mean value theorem for integrals

Answer 23

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

Question 24

The first fundamental theorem of calculus

Answer 24

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

Question 25

Improper integrals ( intervals unbounded)

Answer 25

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

Question 26

Improper integrals (unbounded integrands)

Answer 26

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.

Question 27

Comparison test for improper integrals

Answer 27

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