Name:
Student ID:
TA Section:
I have read and understand the directions below, and I pledge that I have neither given nor received any
unauthorized assistance on this exam.
(signature)
Math 6A, Spring 2015
Practice Midterm 2
Directions:
1. Show all your w
MATH 8 PRACTICE MIDTERM 2
Summer 2015
1. Prove that for every real number x, there exists a unique real number y such that x2 y = x y.
2. Let A, B, C, and D be sets.
(a) Prove that if A B and C D, then A C B D.
(b) Give an example where (AC)(B D) is a non
The Pigeonhole Principle
Rosen 4.2
Pigeonhole Principle
If k+1 or more objects are placed into k boxes, then there is
at least one box containing two or more objects.
Generalized Pigeonhole Principle
If N objects are placed into k boxes, then there is at
Tiffany Mann
6364087
March 09, 2016
Math 104A Final Project
Professor Atzberger
Operations Project
I will be considering two firms that are competing over a common pool of
customers. Each month the firms can adjust their advertising and other incentives t
Tiffany Mann
6364087
Motivation & Problem Statement
Operations Project: considering 2 firms that are competing over a
common pool of customers. Each month the firms can adjust their
advertising & other incentives to attract more customers.
Questions: What
Math 34B Lecture 4
April 7, 2016
velocity is the rate of change of distance.
velocity v (t) = 2t + 5 cm/sec
t = time in seconds
How far does object move between t = 0 and t = 3 ?
A = 20 B = 22 C = 24 D = 26 E = need hint
NUN x(t) = distance object has tra
Math 34B Lecture 2
March 31, 2016
page 59 Area of a trapezium is width average height
A trapezium is a rectangle with a triangle stuck on top
b-a
b
a
W
W = Width of trapezium
a and b are lengths of 2 vertical sides
a+b
average height =
2
a+b
area = W
2
A
Math 34B Lecture 3
April 5, 2016
Can use antiderivative to find
Rt
f (x) dx
Rb
How can we use antiderivative to find a f (x) dx ?
answer
Rb
a
f (x) dx =
0
Rb
0
f (x) dx
Ra
0
f (x) dx
reason
(area between a and b) = (area between 0 and b) - (area between
Math 34B Lecture 1
(Primarily Plagarized from Professor Daryl Cooper)
April 1, 2016
Today:
What to expect, when youre expecting to take Math 34B from me
Quick review of 34A
Log onto GauchoSpace https:/gauchospace.ucsb.edu/ to find
Syllabus
Information
Name:
Sample Problems:
Professor: Paul J. Atzberger
MATH 3B
Scoring:
Problem 1:
Problem 2:
Problem 3:
Problem 4:
Directions: Answer each question carefully and be sure to show all of your
work. Be sure to state your final solution in the indicated locatio
Name:
TA / Section:
Practice Problem Set 1 : Preparation
for the Final Exam:
Professor: Paul J. Atzberger
MATH 3B
Directions: Answer each question carefully and be sure to show all of your work. Be sure
to state your final solution in the indicated locati
Math 4A Study Guide
Linear Combonitation
Linear Independent
Linear Dependent
Onto
One-to-One
Null Space
Column Space
Subspace of vectors
Span of vectors
Linear maps defined by matrices
Reflections, Rotations, Dilations
Application of Inverse matrix
Theore
Practice Test 3
July 28, 2015
Definitions
A semi-randomly chosen 6 of the following definitions will appear on the exam.
Solution to Inhomogeneous Diffusion on the whole real line
Solution to Inhomogeneous Wave equation on the whole real line
Separated So
Wenjian Liu (Instructor)
PSTAT 120A Summer 2015Solutions
of Homework 2
Problem 2.1
What is the probability of rolling an even number with a single die, given the die roll is 3 or less?
Proof. Write A = cfw_the rolling number is even and B = cfw_the rollin
PSTAT 120A
Probability and Statistics
Summer 2015
Midterm Examination
P#1
20
P#2
20
P#3
20
P#4
20
P#5
20
P#6
30
Total 100
Please note that the full credit of this midterm is 100, although these 6 questions are worth 130 points in all. And you total score
Wenjian Liu (Instructor)
PSTAT 120A Summer 2015
Homework 1
Due Thursday, July 2nd 2015
Problem 1.1
Suppose that a number x is to be selected from the real line R, and let A, B, and C be the
events represented by the following subsets of R, where the notat
\
1. (4 points) SET LIP BUT DO NOT EVALUATE aa integral which calculates the area
bonaded by A: rt, a = x - 2 and the c-axis.
(.(,1
v
/ellow AYeu )
I
5
3
,
19
7
7
ISr
*
q
T
Z
u/)'on
Jy =
x-Z
a,.J 1 2s
+
vyho4
cfw_=Z
\=*
dte
^ra,.
I-
Ji
y-1:-O
is
+
dx
J
z
3.3 Proofs Involving Quantifiers
1. In exercise 6 of Section 2.2 you use logical equivalences to show that
x(P (x) Q(x) is equivalent to xP (x) xQ(x). Now use the
methods of this section to prove that if x(P (x) Q(x) is true,
then xP (x) xQ(x) is true. (N
Math 8 Winter 2016: Homework 8
Due Fri, March 11, 2016, 5pm, in mailbox of professor (South Hall 6th floor mailroom).
(1.) Let Cm be the congruence mod m relation defined in the text, for a positive integer m.
(a) Give a complete proof that Cm is an equiv
Math 8 Winter 2016: Homework 6
Due Wed, February 24, 2016, 5pm, in mailbox of professor (South Hall 6th floor mailroom).
(1.) Suppose A, B, C, and D are sets.
(a) A x (B C) = (A x B) (A x C).
(b) (A x B) (C x D) = (A C) x (B D).
(2.) Suppose cfw_Ai | i I
Math 8 Winter 2016: Homework 5
Due Wed, February 17, 2016, 5pm, in mailbox of professor (South Hall 6th floor mailroom).
(1.) Prove that there is a unique real number x such that for every real number y, xy+x4 =
4y.
(2.) (a) Prove that for all a, b R, |a|
Math 8 Winter 2016: Homework 4
Due Wed, February 3, 2016, 5pm, in mailbox of professor (South Hall 6th floor mailroom),
but this material will be on the exam, so it is HIGHLY recommended that you complete
this assignment before the exam.
(0.) Again, it is
Math 8 Winter 2016: Homework 2
Due Wed, Jan 20, 2016, 5pm, in mailbox of professor (South Hall 6th floor mailroom)
(1.) What are the truth sets of the following statements? List a few elements of the truth
set if you can.
(a) x is a real number and x2 4x
Math 8 Winter 2016: Homework 1
Due Wed, Jan 13, 2016, 5pm, in mailbox of professor (South Hall 6th floor mailroom)
(1.) Analyze the logical forms of the following statements.
(a) Either John and Bill are both telling the truth, or neither of them is.
(b)
Math 8 Winter 2016: Homework 7
Due Fri, March 4, 2016, 5pm, in mailbox of professor (South Hall 6th floor mailroom).
(1.) For each of the following theorems from the book, write out the proof in your own
words and filling in the steps left out.
(a) Theore
Math 8 Winter 2016: Homework 3
Due Wed, Jan 27, 2016, 5pm, in mailbox of professor (South Hall 6th floor mailroom)
(0.) It is recommended that you spend some time looking at examples in the book. Some
of the material covered in lecture is from Sections 3.
Math 8 Winter 2016: Homework 2
Due Wed, Jan 27, 2016, 5pm, in mailbox of professor (South Hall 6th floor mailroom)
(0.) It is recommended that you spend some time looking at examples in the book. Some
of the material covered in lecture is from Sections 3.
PSTAT 120A: HW3 - Solutions
Problem 1.
Urn I contains 25 white and 15 black balls. Urn II contains 15 white and 25 black balls. An urn is
selected at random and ve balls are drawn randomly from this urn without replacement. If exactly
ve of these balls ar
Midterm Exam Outline
Math 3B: Calculus II with Applications
Professor: Paul J. Atzberger
Review properties of derivatives.
o Definition of derivative.
o Review derivatives of common functions.
o Review derivatives of trigonometric functions.
o Derivatives