MATH 104, SUMMER 2008, REVIEW SHEET FOR FINAL EXAM
The nal will be on Thursday, August 13, 2:10-4:00pm. It is cumulative (covering the whose semesters materials) but with an emphasis on 2034. Approximately 50% of the exam problems are based on these secti
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 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 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 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 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 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
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
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
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
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,
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,
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 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 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
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
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
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