Solutions for Problems for Chapter 3.
8. Use both the definition of limit and a sequence approach to establish lim
x2
1 1 11 x
1 |x 2 |
1x
1x
1x
5
for x close to 2, say for |x 2 | 1
i. e. , 3
x
,
2
2
M 317 sec 2
Exam on Chapters 1 and 2
Solutions
1.Let a n denote a bounded sequence that does not converge. Prove that a n contains
at least two subsequences that converge to different limits.
Solution
Problem Set 2, M301
Problem 1 A binary string is a sequence of zeros and ones such as 00101101.
a) How many different binary strings of length 8 contain at most three 1s?
b) How many different binary
Transformation Methods
3
Fall 2017
1 / 35
J. A. Hoeting (CSU)
STAT400: Statistical Computing
Generate a random value u from Uniform(0,1).
Compute x = F 1 (u)
For each realization:
3
Write a command or
Example 3 MAHBHC) = P(A)P(B)P(C), A, B and C are paiiwise independent?
CWSIJW A. fair ovahqu' (K cfw_31%) ,.
At E C. 55NQ> 3C bW'KK S 721w
ZE(YM/M WAFt- M E 676/ Wilts/4C.
4" MN? rune - %
1W<>= V >
Example 11. There are 10 girls and 8' boys in the class. What is the probability that the
cleaning team will consist only of girls?
fm ox Lam W Ml WW SWW36/ld.
ywwwwawmw M WMWM
/ Example 1 A researc
[34: ? 1,13.
2.
74 ' \K \
WK X ' /%/ it
244* s a (T M!
Remark. Possible extension of the multiplication principle: \
We (id/iv WM eh Wlnumiw Ibvwilk/ its. LARA
/WM 997/, iw 014- m ii [P W/ Wu 75;, Wis
Example 3. For three oaes labeled by A,B and C, and three balls a,b and c. Put gm of
the three balls into 0 the three boxes. What is S .9 Let D = cfw_a is in A; b is in B, and
E = cfw_a e A; b 915 B;
Problem Set 1, M301
Problem 1 Let U = cfw_a N | 1 a 11. Thus U = cfw_1, 2, 3, . . . , 11.
/ S.
For any subset S U , let S = cfw_g U | g
Let |S| denote the number of elements in S.
Let A = cfw_1, 3, 7
dorgan s laws
B AB,AB=ALJBn
A
a = .S'
a : 0
m
f. M axe-la [:1
W:@ A= 64/28) U W3)
rel-9471]
1. Assume A, B, C are three events/sets. Express the followzng- 179094.111.
ppenEDB and C does not.
A
1. Denote 8 the set of all elements under consideration, then 8 is called the universal set
=
1 Review of set notation
1.1 Basic denitions in set theory
2. For any two sets A and B, Wis a called a
STAT 420, Fall 2017
Due on 4:45 P.M., Friday, September 15th
Homework 3
Part I: Problems to turn in.
(1) Two systems were designed to compile computational codes of same type. The probability
that the
Homework 1
Due: Aug 29, 2016 (all homework is due at the start of class)
This assignment is a warm-up on Mathematical Statistics and R.
Reading and software practice
Read Chapter 2.1-2.4 of our textb
M 317
Problems Chapter 5
1. For f x 1 x on I 0 x 2 , let P denote a uniform partition of I with N 4. Then
compute each of the quantities, m|I |, f, P , RS f, P , f, P , and M|I |, where the tag points
14. Suppose f
D R . Show that g x f x
Dhg x 1 f x h
h
1 f xh
h
2
fx
2
1 f xh
h
2
DR
f xh f x f xh f x
2
f x f xh 1 f xh
h
fx
2
fx fx
Dhf x f x h Dhf x f x
lim D h g x 2 f x f x
Then
h0
15. see exam
Sample Final Exam
Part A
1. State the completeness axiom. What is the point of this axiom; what is it designed to
achieve?
2. The nested interval theorem and the Balzano-Weierstrass theorem can be sta
Exam 2
Continuity
1. Use the
definition of continuity to show that that f x x 2 1 is continuous at x 1.
|f x
Then
f 1 | |x 2 1
0,
2 | |x 2
2 /5
1 | |x 1|x
such that |f x
1|
5
2
|x
f1 |
1|
if
1|
|x
Exam 4
Chapter 5
1. Suppose f x is defined and bounded on I 0, 1
a) State a condition on f which implies that f is integrable on I
f is continuous on I, or f is monotone on I
b) State a condition in t
Exam 3
Chapter 4
Solutions
1. Use the difference quotient to show that if f x is differentiable at all x, then
g x f x 1 f x 1 is also differentiable at all x.
Dhg x 1 f x h 1 f x h 1 f x 1 f x 1
h
1
Exam 2
Chapter 3
1.
For each of the following cases, either give an example or explain (using
definitions) why no such example is possible:
a. p is an accumulation point for A and is also an interior
Additional Problems
Sequences
3. Find an N such that |a n L | 10 3 for n N
2
2
a an
L 0 |a n 0 |
if 2
n1
n1
4 n1
i.e.,
Then |a n 0 |
if n 42 N
2
1
n3
n 1/3
1
b an 1
i.e.,
c an 2 2
n
1 | 13
n
Then
1. How is the number 0 defined by the axioms governing the reals? What do the axioms
imply about division by 0 ?
x0 x
Axiom 3 says:
x
Axiom 4 says:
x
Axiom 9 says:
x y
y
x0
x y
y
xy
Axiom 8 says:
x
Chapter 23: Amino Acids
Amino acids are molecules that contain both a carboxylic acid and an amino group
Of biological importance are -amino acids
small molecules that components of proteins (20 natu
MAT 142 Course Syllabus | Spring 2016 B
MAT 142: College Mathematics
Spring 2016 iCourse/oCourse Syllabus
Instructor: Beth Jones
E-mail: [email protected]
Office Hours:
In Person:
9 am 10 am Tuesday a