Fundamentals of Analysis I ( MATH 321) Spring 2010
Homework Assignment
Number
1
Date Assigned
8/23/2010
Date Due
8/30/2010
Total Points
50
1. (10 pts) Prove that for all real numbers x and y, |(|x| |y|)| |x y|. Hint: enumerate the dierent
cases.
2. (30 pt
3- Ll paw " (.0an $691 Fundamentals of Analysis I ( MATH 321) — Spring 2010
Five—minute quiz
Reading: Marsden 8: Hoffman, Elementary Classical Analysis (Q/E), Section 3.4 (Path—Connected Sets).
A map gr) : [(1,1)] —- llu’ of an interval [(1, 3)] into a me
Example of uniform convergence.
In[1]:=
f@x_, k_D = x H1 + k ^ 2 x ^ 2L
x
Out[1]=
1 + k2 x2
Plot the first function in the sequence.
In[5]:=
Plot@f@x, kD . k 81<, 8x, - 1, 1<D
0.4
0.2
Out[5]=
-1.0
-0.5
0.5
1.0
-0.2
-0.4
Plot the first three functions in t
Fundamentals of Analysis I ( MATH 321) Fall 2010
Question Bank for Second Midterm
Note: this will be a closed-book exam, closed-notes exam.
1. Dene what it means for a subset A of a metric space M to be compact. Dene what it means for a
set to be closed.
MATH 321, Fall 2014
(this is a preliminary syllabus, when further changes are made, those will be announced in class and on
the Blackboard course webpage).
Instructor: Julia Dobrosotskaya
email: jxd365@case.edu
oce: Yost 227
Class webpage: please, see CWR
Fundamentals of Analysis I ( MATH 321) Fall 2010
Homework Assignment
Number
13
Date Assigned
11/22/2010
Date Due
12/01/2010
Total Points
100
1. (20 pts)
(a)
(b)
(c)
(d)
Prove
Prove
Prove
Prove
that x/n 0 uniformly, as n , on any closed interval [a, b].
th
Fundamentals of Analysis I ( MATH 321) Spring 2010
Homework Assignment
Number
11
Date Assigned
10/25/2010
1. (10 pts) Let f : R R be given by f (x) =
Date Due
11/1/2010
Total Points
120
0, x Q
Prove that f is continuous at exactly
x, x R\Q.
one point.
2.
Fundamentals of Analysis I ( MATH 321) Spring 2010
Homework Assignment
Number
12
Date Assigned
11/1/2010
Date Due
11/8/2010
Total Points
100
1. (20 pts) Give an , proof showing that f (x) = 3 x is continuous on all of R. If f is uniformly
continuous, give
Fundamentals of Analysis I ( MATH 321) Spring 2010
Homework Assignment
Number
8
Date Assigned
9/22/2010
Date Due
9/29/2010
Total Points
100
1. (10 pts) Let (M, d) be a nonempty metric space with the discrete metric. Show that a sequence xk M
converges to
Fundamentals of Analysis I ( MATH 321) Spring 2010
Homework Assignment
Number
10
Date Assigned
10/13/2010
Date Due
10/20/2010
Total Points
100
1. (30 pts)
(a) (10 pts) Let A = D(0, 1) R2 be the closed unit disk, i.e. cfw_x R2 | |x| 1. Prove that A is
tota
Fundamentals of Analysis I ( MATH 321) Spring 2010
Homework Assignment
Number
9
Date Assigned
9/29/2010
Date Due
10/6/2010
Total Points
100 pts
1. (10 pts) Marsden & Homans Proposition 2.8.4 claims for a general metric space (M, d) three results
that were
Fundamentals of Analysis I ( MATH 321) Spring 2010
Homework Assignment
Number
7
Date Assigned
9/15/2010
Date Due
9/22/2010
Total Points
50 pts
1. (10 pts) Let A R2 be the set
A = cfw_(x1 , x2 ) R2 |x1 > 0 and x2 > 0 and x1 + x2 1.
(a) Determine whether A
Fundamentals of Analysis I ( MATH 321) Spring 2010
Homework Assignment
Number
4
Date Assigned
9/1/2010
Date Due
9/8/2010
Total Points
50 pts
1. (30 pts) Prove the following: If cfw_xn is a sequence bounded below and xn does not diverge to +,
then there i
Fundamentals of Analysis I ( MATH 321) Spring 2010
Homework Assignment
Number
6
Date Assigned
9/13/2010
Date Due
9/20/2010
Total Points
20 pts
1. (20 pts) Let d be the standard (Euclidean) metric on R2 and let d be the Manhattan or taxicab
metric, d (x, y
Fundamentals of Analysis I ( MATH 321) Spring 2010
Homework Assignment
Number
5
Date Assigned
9/8/2010
Date Due
9/15/2010
Total Points
60
1. (20 pts) M&H give the following example of a metric space (M, ) for which the metric is bounded
(Example 1.7.2.d).
Fundamentals of Analysis I ( MATH 321) Spring 2010
Homework Assignment
Number
3
Date Assigned
8/30/2010
Date Due
9/8/2010
Total Points
50
1. (10 pts) Prove Proposition 1.2.3 in Marsden & Homan, i.e. prove that if a xn b for all elements
of a sequence cfw_
Fundamentals of Analysis I ( MATH 321) Fall 2010
Question Bank for First Midterm
Note: this will be a closed-book exam.
1. Let 0.99 represent the innitely repeating decimal expansion 0.9999 .
9
Prove that 0.99 = 1.
9
2. Let cfw_xn be an innite sequence o
Claim: Sin[x]=x+o(x). That is, for every e>0 there is a d>0 such that |Sin[x]-x|<e|x|. In fact any straight line will bound the
difference near zero, so you can choose d=me for any m>0 (for instance, m=1).
In[3]:=
Plot@HSin@xD - xL, 8x, - 1, 1<D
0.15
0.10