Sample Midterm 2 for MATH 104
Problem 1
[Review all important denition and results of the relevant material. You will be asked
to state a few of them precisely. Among those are: series, convergence, a
Cavazos 1
Sarah Cavazos
ENGL 104
Elizabeth Mobley
10-11-15
Keep Calm and Set Your Privacy On
When people say privacy is a given and we should take full advantage of it, you would think
almost everybod
English 105
Phase One Project
Researched Position Essay
Because this Engl 105 course builds to a Writing to Change the World project, students-that is you-will choose and study an issue that you wish
Homework 1 for MATH 104 Solutions
Due: Tuesday, September 12, 9:40am in class
Problem 1
Determine whether the following sets are bounded (from below, above, or both). If so,
determine their inmum and/
Homework 2 for MATH 104
Due: Tuesday, September 19, 9:30am in class
Problem 1
Verify the following statements by induction: For all n N,
(1)
13 + 23 + + n3 = (1 + 2 + + n)2 ,
Solution.
n = 1: Obviousl
Homework 3 for MATH 104
Due: Tuesday, September 26, 9:30am in class
Problem 1
Given a sequence (sn )nN , let (sn ) be the sequence dened as (s1 , s2 , s3 , . . . ).
Show that lim inf n (sn ) = lim sup
Homework 4 for MATH 104
Brief solutions to selected exercises
Problem 1
Determine if the following series converge. Justify your answer.
n=1 ( n
(a)
+1
n),
This is a telescoping sum. It holds that Sn
Homework 6 for MATH 104
Brief solutions to selected exercises
Problem 1
Dene the function g : R R by
1 if x = 0,
g(x) =
1
q
0
if x =
p
q
with p Z, q Z \ cfw_0 relatively prime,
if x is irrational.
Sho
Homework 7 for MATH 104
Brief solutions to selected exercises
Problem 1
(a)
Suppose
an xn has nite radius of convergence R and that an
0 for all n.
Show that if the series converges at R, then it also
Homework 8 for MATH 104
Brief solutions to selected exercises
Problem 1
Find two sequences of real functions (fn )nN and (gn )nN from some S R into R such that fn f
uniformly and gn g uniformly, but f
Homework 9 for MATH 104
Brief solutions to selected problems
Problem 1
(a)
Suppose f : R R is dierentiable at x R. Show that
f(x + h) f(x h)
= f (x).
h0
2h
(*)
lim
Solution. Assume hn 0, hn = 0. We ha
Homework 10 for MATH 104
Solutions to selected exercises
Problem 1
Suppose f : R R. Call x a xed point of f if f(x) = x.
(a)
If f is dierentiable and f (t) = 1 for all t, prove that f has at most one
Sample Final for MATH 104
Problem 1
[Review all important denition and results of the relevant material.]
Problem 2
If the followings statements are true, answer "TRUE". If not, give a brief explanati
Sample Final for MATH 104
Problem 1
[Review all important denition and results of the relevant material.]
Problem 2
If the followings statements are true, answer "TRUE". If not, give a brief explanati
Sample Midterm 1 for MATH 104
Brief sketches of solutions
Problem 1
Show that every real number is the limit of a sequence of rational numbers.
Given a R, by the density of rational numbers, there exi
Sample Midterm 1 for MATH 104
Problem 1
Show that every real number is the limit of a sequence of rational numbers.
Problem 2
Let (an ) and (bn ) be two sequences. Assume that lim an = a, lim bn = b,
Sample Midterm 2 for MATH 104
Problem 1
[Review all important denition and results of the relevant material. You will be asked
to state a few of them precisely. Among those are: series, convergence, a
Focusing Guide
Take each of your artifacts and connect them to an article to in the NYT in a
different context. For example, an engagement ring might connect to an article
about the Supreme Court ruli