EE/TE 3341
Aria Nosratinia
Assignment 0
Math Review
Due Date: January 20, at the start of the class.
The problems in this homework set reflect basic calculus and algebra. If you have any problems
with this homework set, you may have deficiencies in the pr
Final Test outline and practice questions AP Calculus
You should be able to:
1.
2.
3.
4.
5.
Determine the limit of a function form its graph or using algebra
Use the concepts of continuity or differentiability to describe a function
Determine the derivati
Math Review
Aria Nosratinia Probability and Statistics
11
Sequences and Limits
A sequence is just an infinite string of numbers, e.g.
1
2
3
4
5
6
0
0
0
0
0
0
1
1
1
1
1
1
1
1
2
1
4
1
8
1
16
1
32
We often show a sequence as x(n) or x[n] or xn .
We say a
THE MIN H. KAO DEPARTMENT OF
ELECTRICAL ENGINEERING AND COMPUTER SCIENCE
Ph.D. QUALIFYING EXAMINATION
Monday, August 16, 2010
Exam Packet Number: _
You are allowed 4 hours to complete this exam.
This exam is closed book and closed notes. No calculators or
Multivariate Distributions
Aria Nosratinia Probability and Statistics
61
Why Random Vectors?
Going from two to multiple variables is a logical step.
Often systems have multiple random parameters.
Another application of random vectors is modeling rando
Review of Integration Techniques
January 13, 2010
1
Antiderivatives and Indefinite Integral
Definition 1.1. Let I denote an interval in R and f : I R be a function. We say that a function
F : I R is an antiderivative of f (x) on I if
F (x) = f (x)
for all
MAT 21C22 SSH  Jason Barnett  Exam 2
a. (was
You have 60 minutes to complete this exam. No calculators, notes, or books may be used.
Please clear everything off of your desk. Make sure to write legibly and show all work unless
otherwise stated. Anythi
MAT 21C22 SSH ~ Jason Barnett  Exam 1
Name: 50 i M71
You have 50 minutes to complete this exam. No calculators, notes, or books may be used.
Please clear everything off of your desk. Make sure to write legibly and show all work unless
otherwise stated.
MAT 21022 SSH  Jason Barnett  Quiz 2 ‘ .
Name: Score: I 6 6
1. (2 points) Find the sums of the following series if they exist. If they don’t exist, write
” Series diverges”.
00 3 A \_
(80 i3“); Z 3 (If Geemw‘
":0 4n A=0 7 3 1 7’ I“,
a g 4’ " iv 1’
EE/TE 3341
Aria Nosratinia
Assignment 5
Due Date: March 1 before start of the class
Reading Exercise: Yates and Goodman, Chapter 2, Sections 2.2 through 2.5
1. Problem 2.3.3.
2. Problem 2.3.7.
3. Problem 2.3.11.
4. Problem 2.3.13.
5. Problem 2.4.1.
6. Pro
PSHI
A
(
10.1
is
sequent
ie
Sequences
1,35
.
7
[
firstaterm
of
list
a
)
7,9
,
numbers
,
ie
ie
,
2h

1
,
.
.
nthterm
An
rules
by
,
,
th "
are
n,a 's ,
,
formula for
a
ties
order
given
a
naff
,n ?
.
:
.
.
Ex Find
(a)
.
onadzkrm
be described
can
.
T
an=nI
S
PSI
Exl
y
(a)
Plot
vector called
F
=
"
,
+4J
Zi
6k
"
+
mini
"
size
v
"
"
length
(b) magnitude of =
x
L
IIFH
(c)
unit
vector
Fytho
(d)
the
=
.
express
in
=D
the direction of 8
=
,
8
^2
iii
:p
:ei!Ii:II"
:iEH:in
:
as
hitting =ri+j+k
a
of
product
length & di
PS
4
#
Root Test
For
any
where
cfw_
series
3
"
Determine whether each
where
an
at
=P
series
:cfw_
hfI= # testify
Note
:
this
nz
fsi
or
,
't
.
diverges
ftp.t.it#ntHntY
it
:
to behave
start
series
r=
Consider
him
.
by
EK 'Y
2)
fH= I
tfti
3)
me
E
II
needn ,f
PS #
Ex
:
2
Consider
an
cfw_E

an
,
sequences
/ series
.
.
,
.
3m
=
tennis
(b)
I ,z
,
infinite
following
diverge
or
converge
(a)
the
whether
an
=
sequence
converges
3M
,
nn# =3
Since lingua
(c)
n+#+o
nt#
=
,
Consider
3nnIT
to
of Partial
sequence
must
E
PS
Ex
I
# 7
flxtk
the Taylors series for
Find
:
f(x )
nf
=
(9) ( x
"
Need formula for nth derivative at
f(x)
no

c=q
"
9)

at
9
e
na
"
'ztHf3stxHlth
W=

=
'
ftp.tzttzttahx#=HIz3Dx
n=
note
this
ftp.fhn#p
be expressed
ftp.tzx#n=zfYx)=tzftz)53n=3f'
:
can
PS #6
We 've
to
how
Seen
find
you
1
but what if
+
x
x2
+
certain
x3
+
Series
.
.
still
we
formula
fk )
fYxe
+
.
formula
sum
:
=
.
ox
find
"
the
sum
series
power
a
formula for
for it ?
c)
+
f"Hc5+
"
Centered at
condensed
form
:
"
C
.
"
expanded
form
ftp.#2
ftp.EIHFg#
see
?aI
:[
we

PS3
If
an
If
an
<
<
for all
bn
for all
bn
n
n
Ebn
If
,
If Ean
,
cfw_
converges
an
converges
Ebn
diverges
diverges
comparison test
Determine
if
each series
converges
au
or
diverges
#n=kH
&
=
413 '
nn
, s9lYeH
Evatt
=
En
:ttaEo
2)
PS 5
Infinite
Series
Find the infinite
9+3+1
(a)
(b)
8 4
's
+
.
+
'
.
=
.
Ytkrrmn
geometric )
(
's
r=
of
sun
2
+
+
I
8
.
.
=
.
r=
Q
:
Can
I
find
we
+
+
+
.
f
+
+
.
method
Using this
adds to
1)
1st
choose
I
+
+
,
rat
3
10 ?
10
.
.
we
can
!
+
+
5
find
an
3. (6 points) Determine whether the following sequences converge or diverge. If they converge,
ﬁnd the limit. If they diverge, say whether they diverge to 00, —o<>, or neither. You do NOT
need to show any of your work.
(a)
an 2 n cos(n).
n «700 aqua C03 0
MAT 21C—22 SSH — Jason Barnett — Quiz 3
Name; 56” m
1. (2 points) Let u" be a sequence. Consider the series
00
Z(_1)n+1un_
11:0
What are the 3 requirements to use the Alternating Series Test?
6) (M70 19% 07M ‘ .
~  ‘1 ' 7N
one rgjlxﬁ/ (A non v— m News 1
MAT 210—22 8811  Jason Barnett  Quiz 4
Name: 1
. I
’ (9‘): 1452773
1. (4 points) Find the Taylor polynomlal of order 3, P3(ac) for the functlons
<a>f<$>eﬁmbom=1~ H mm 0001 + f/“T'Mw 03 gm
?.L°<3:n@+£f‘x°*u 6‘ V); I
3 I \ 2. _ "I, (x P v/
0' . ,7" " (WI
L7 Uses of
Taylor series
and
Taylor polynomials
The partial sums
N
SN (x) =
k=0
1 (k)
f (a)(x a)k
k!
of the Taylor series for f at a form a sequence of functions that approximate
f at a.
For example we have found that the Taylor series of
ln(x)
at 1 is
n=
L2 Convergence of Series
Some more depth on sequences
Denition: A sequence is a function with domain the set of natural numbers.
Notation: The string of symbols V : N R is shorthand forV is a function
with domain N and codomain R. This does not give all o
L10 Vectors
Vectors
This is an introduction to vectors. We will cover a more advanced notion
of vectors toward the end of 21C and 21D will use extremely sophisticated
vector concepts. Linear algebra, 22A or 67, is about functions that take vectors to vect