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, absolute
convergence, Cauchy criterion, comparison test,
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 everybody would agree to that. However, there are those people
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 to impact. For this Issue Essay,
you will :
-study an i
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/or supremum and nd out whether these inma/suprema
are a
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: Obviously, 13 = (1)2 .
n
(2)
Ind.Hyp.
1
n + 1: 13 + + n3 + (n +
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 supn (sn ).
Solution. We discard the case where lim inf n
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 =
Since n + 1 , it follows that the series diverges.
So
Homework 5 for MATH 104
Brief solutions to selected exercises
Problem 1
For x, y R, dene
d1 (x, y) = (x y)2
d3 (x, y) = |x2 y2 |
d2 (x, y) =
|x y|
1 + | x y|
d4 (x, y) = |x 2y|
| x y|
d5 (x, y) =
Determine, for each of these, whether it is a metric on R o
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.
Show that g is continuous at all irrational points, and di
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 converges at R.
Solution.
Obviously, since an
0 and R
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 fn gn does not converge uniformly to fg.
Solution.
Consi
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 have to show that
lim
n
f(x + hn ) f(x hn )
= f (x).
2hn
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 xed point.
Solution. Suppose f had two xed points, x an
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 explanation why.
(1)
If F is a eld and x, y F, then x y = 0 impl
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 explanation why.
(1)
If F is a eld and x, y F, then x y = 0 impl
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 exists q1 Q with a < q1 < a + 1.
Apply the density propert
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, and that
an bn for all n N.
Show that a b.
Problem 3
Le
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, absolute
convergence, Cauchy criterion, comparison test,
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 ruling on marriage equality or an employee ID card to
union