12/10/2013
Unit 2
Limits and Continuity
Unit 2.1
Limit of a Function
1
Consider
2
x2 4
f x
x2
Dom f 2
y
6
5
4
2, 4
3
f x
x2 4
x2
x 2 x 2
2
1
x2
x 2, x 2
-1
1
2
3
4
5
6
x
-1
3
4
We can make the values of f x closer to 4
by taking x sufficiently clos
3/18/2014
Consider the shaded region.
Section 5.3
The Riemann Integral
1
2
Sigma Notation
Example
Evaluate the following.
5
n
i
1.
f i f 1 f 2 . f n
2
1 4 9 16 25 54
i 1
i 1
3
2i 3 1 1 3 3
2.
i 1
3
4
Theorems
4.
n
1.
c nc
i 1
n
n n 1
i 1
2
i
n
5.
i
2
3/4/2014
Absolute Extrema
A function f has an absolute maximum value
on an interval I if there is some number c in
Section 4.5
Optimization
I such that f c f x for each x in I .
A function f has an absolute minimum value
on an interval I if there is some
3/13/2014
Section 5
Integrable Functions
Section 5.1
Antiderivatives
1
Antiderivatives
2
Example
Consider f x 3 x 2 2 x 4.
If f is the derivative of F , that is F ' x f x ,
If F x x 3 x 2 4 x 5 then
F is an antiderivative of f .
then F is an antiderivativ
2/27/2014
Relative Extrema
A function f has a relative minimum at c if
f c f x for all x in some open interval
Section 4.4
Curve Sketching
containing c.
A function f has a relative maximum at c if
f c f x for all x in some open interval
containing c.
1
2
12/12/2013
Let f be a function defined on some open interval
containing a, except possibly at a.
Unit 2.2
One-Sided Limits
The limit of the function as x approaches a is L,
written as lim f x L,
xa
if for every 0, there exists a 0, such that
if 0 x a , th
3/20/2014
Definite Integral
If f is a function defined on a, b , then the
definite integral of f from a to b is given by
Section 5.4
The Definite Integral
f x dx lim f x * x
n
b
n
a
i 1
i
if the limit exists.
1
2
Properties of Definite Integral
1. If t
3/20/2014
Section 5.5
Area of a Plane Region
b
n
A lim f xi * x f x dx
n
i 1
a
Example
Find the area of the plane
region bounded above by
y 1 x 2 , bounded below
by y x, and bounded on
the sides by x 1 and
AS f x dx g x dx
b
b
b
a
a
a
f x g x dx
x 3
12/4/2013
y
Section 1.6
Rotation of Axes
x
r
y
sin
r
cos
P x, y
r
x r cos
y r sin
y
x
x
1
2
P x, y
y
P x, y
P x, y
y
r
y
x
y
r
x
x
x r cos
x
y r sin
3
x r cos
y r sin
x r cos
y r sin
x r cos cos sin sin
r cos cos r sin sin
x cos y sin
3/11/2014
Differentials
If y f x , where f is a differentiable
Section 4.6
Differentials and
Approximation
function, then
the differential of x is given by dx x
and the differential of y is given by
dy f ' x dx.
Example
Find the differential of the follow
11/21/2013
Standard Equation
Example
A parabola with vertex at h, k has standard equation
For each of the given equation, find the
i.
y k
2
4 p x h if principal axis is horizontal
opens to the right/left
vertex
ii. focus
iii. directrix
iv. principal ax
Questions to guide your review:
1. What exactly does lim f x L mean?
x x0
2. What function behaviors might occur for which the limit does not exist? Give examples.
3. What theorems are available for calculating limits? Give examples of how the theorems ar
D I S C R E T E -T I M E
M A R KOV C H A I N
( C O N T I N U AT I O N )
THE GAMBLERS RUIN
PROBLEM (ROSS)
Consider
a gambler who at each play of the game has
probability of winning one unit:
Probability of losing one unit:
Assume that successive plays
STOCHASTIC
PROCESSES:
AN OVERVIEW
M AT H 1 8 2 2 N D S E M AY 2 0 1 6 - 2 0 1 7
STOCHASTIC PROCESS
Suppose we have an index set
We usually call this time
where is a stochastic or random
process
The set of random variables indexed by
time is called a s
D I S C R E T E -T I M E
M A R KOV C H A I N
( C O N T I N U AT I O N )
CONVERTING SOME NONMARKOV CHAINS TO
MARKOV CHAINS
Let us explain this using an
example:
Suppose we have a stochastic process
and .
CONVERTING SOME NONMARKOV CHAINS TO
MARKOV
CHAINS
D I S C R E T E -T I M E
M A R KOV C H A I N
( C O N T I N U AT I O N )
CHAPMAN-KOLMOGOROV
EQUATIONS
IN
MATRIX
FORM
Starting from the transition matrix , we
have
In general,
CHAPMAN-KOLMOGOROV
EQUATIONS
IN MATRIX FORM
Recall our example with
Refer to
D I S C R E T E -T I M E
M A R KOV C H A I N
( C O N T I N U AT I O N )
FIRST PASSAGE TIME
First passage time: The length of time (or the
number of transitions) made by the process in
going from state to state for the first time.
When , this first passa
D I S C R E T E -T I M E
M A R KOV C H A I N
( C O N T I N U AT I O N )
0.4
DEFINITION
Stat
eA
State C is
accessible
from state A
State B is
NOT
accessible
from state C
0.6
0.2
Stat
eC
0.8
Stat
eB
0.4
DEFINITION
Stat
eA
State C is
accessible
from state
D I S C R E T E -T I M E
M A R KOV C H A I N
( C O N T I N U AT I O N )
PROBABILITY OF
ABSORPTION
If state j is an absorbing state, what is the
probability of going from state i to state j?
Let us denote the probability as .
Finding the probabilities is