Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Homework Assignment 6
Due Wednesday, November 22
1. Show that there does not exist a strictly increasing function f : Q R such that
f (Q) = R.
Solution. [Yash Malik] Suppose f : Q R is stric
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
MATH 142A
Homework 3
Jack Geller
Exercise 1. Let (an ) be a sequence of real numbers. Determine whether each
statement below implies that (an ) converges. Then determine whether each statement below i
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
SOLUTION OF HOMEWORK 2
YINGJIA FU
Solution 1:
A
B
C
D
E
It
It
It
It
It
converges to 1.
converges to 0. (Notice n + 1 n =
does not converge.
converges to 3.
converges to 0.
1
)
n+1+ n
Solution 2:
A Fa
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
MATH 102 HOMEWORK ASSIGNMENT 3
Due Friday, October 20th, 2017 before the first mideterm.
Handwritten submissions only.
Exercise 1 (2+2 points).
Let A, B, C Rmn be matrices. Show the following statemen
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Homework Assignment 2
Selected Solutions
2. Let cfw_ an and cfw_bn be sequences. Prove the following statements.
(a) If an 4 L and an S K for every index n, then LS K.
Solution. Given an 4
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
MATH 102 HOMEWORK ASSIGNMENT 2
Due Friday, October 13th, 2017 before the beginning of the lecture.
Handwritten submissions only.
Exercise 1 (4 points).
Let R. Consider the matrix A R33 and the vector
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Assignment 1
Due: October 6th, 2017
1. The Tower of Hanoi
The Tower of Hanoi is a puzzle invented by douard Lucas in 1883. It consists of
three pegs and several wooden discs of various sizes
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Please check the website for your room assignment!
Please check the website for your room assignment!
Please check the website for your room assignment!
General remarks:
You will be asked to write dow
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Assignment 3
Due: October 20th, 2017
1. Let (a_n) be a sequence of real numbers. Determine whether each statement below implies that (a_n) converges. Then
determine whether each statement be
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
MATH 102 HOMEWORK ASSIGNMENT 4
Due Friday, October 27th, 2017 before the lecture.
Handwritten submissions only.
Exercise 1 (1+1+1+1 points).
Bring the following complex numbers into the form z = a + b
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Octember 1st , 2101
Math 123A, Assignment #1
Your Name
Question 1
We proceed by induction on the number of discs in the tower. If the tower has only a single disc,
the puzzle can be solved in one move
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Homework Assignment 6
Due Wednesday, November 22
1. Show that there does not exist a strictly increasing function f : Q R such that
f (Q) = R.
2. Suppose f : [a, b] R is continuous and onet
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Homework Assignment 3
Due Wednesday, October 25
1. Let cfw_an be a monotone sequence.
(a) Prove that cfw_an converges if and only if cfw_a2n converges.
(b) Show that the result in part (a
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Homework Assignment 7
Due Wednesday, December 6
2. Let h : R R be a strictly increasing differentiable function with h (x) > 0 for all x and
h (R) = R. Given f : R R differentiable, define g
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Assignment 5
Due: November 8th, 2017
1. Prove that, at any given instant, there are two antipodal points on the equator with the same temperature. (You may
assume that temperature varies con
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Assignment 4
Due: October 30th, 2017
1. Suppose that (a_n) is a bounded sequence.
A. Prove that for any \epsilon > 0 there are only finitely many indices j for which a_j >
\displaystyle\lims
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Homework Assignment 3
Due Wednesday, October 25
1. Let cfw_an be a monotone sequence.
(a) Prove that cfw_an converges if and only if cfw_a2n converges.
Solution. =) Suppose lim an = a. Th
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Homework Assignment 2 (Corrected)
Due Wednesday, October 18
1
= 0.
n an
1. Given a sequence cfw_an . Prove that lim an  = if and only if lim
n
2. Let cfw_an and cfw_bn be sequences. Prov
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Homework Assignment 5
Due Wednesday, November 15
1. Show that f : [1, ) R given by f (x) =
Conclude that f is uniformly continuous.
x satisfies the criterion on [1, ).
2. Show that if f : (a
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Homework Assignment 7
Due Wednesday, December 6
1. Consider the function f : R R given by
(
x2 sin
f (x) =
0
1
x
if x 6= 0
if x = 0
Assuming knowledge of the sine and cosine functions from y
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Assignment 7
Due: December 1st, 2017
Answers to these problems are now available here. The \LaTeX file used to produce them is here.
1. For each of the following functions, compute the deriv
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Assignment 8
Due: December 8th, 2017
Answers to these problems are now available here. The LATEX file used to produce them is here.
1. Suppose that f : [a, b] R is continuous, x 0
(a, b), an
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Assignment 6
Due: November 17th, 2017
Answers to these problems are now available here. The \LaTeX file used to produce them is here.
1. For each of the following, evaluate the limit indicat
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Assignment 2
Due: October 13th, 2017
1. Determine which of the following sequences in n converge. Find the limits of those which do.
A.
B.
C.
D.
n
( n+1
)
(n + 1
n)
(n)
(3)
(1)n
E. ( 2 )
n
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 142A Homework Assignment 1
Selected Solutions
1.1.19 Define S cfw_x R  x2 < x. Prove that sup S = 1.
Solution If x S, then x(x 1) < 0 which implies 0 < x < 1. Thus, 1 is an upper
bound for S. By
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
MATH 102 HOMEWORK ASSIGNMENT 1
Due Friday, Oct. 6th 2017 at the beginning of the lecture.
Handwritten handins only.
Exercise 1 (4 points).
Let R be a real number and let A Rmn be a matrix with entrie
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Solutions to HW1
MATH 180A, Winter 2017, UCSD
Book: A First Course in Probability, Ninth Edition, Sheldon Ross
1
Problem 1, Page 48
Let the red marble be represented by R, green by G and blue by B.
E
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Matchbox problem:
Suppose we have two match boxes A and B, each containing 10 matches.
Assume that we pick matches one at a time, each time choosing at random the
box from which we take the match. Wha
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Math 20F
March 17, 2015
Final Examination
Version A
Name:
PID:
Instructions
1. Write your Name, PID, Section, and Exam Version on the front of your Blue Book.
2. No calculators or other electronic dev
Political Science Civic Engagement Internship Course
POL SCI 195CE

Spring 2016
Notes for Math 18: Linear Algebra
prepared by Guoce Xin
This note is a supplement for the textbook. I hope to give more examples for the most fundamental concepts, algorithms, etc.
1. Solving a linear