Math 504 505
Miscelaneous Problems
Jerry L. Kazdan
In the following, when we say a function is smooth, we mean that all of its derivatives exist
and are continuous.
1. Prove that
(1 + x)n > 1 + nx
for
Math 504 505
1. a) Show that b) Show that
Algebra Problems
2 is not a rational number. 3 is not a rational number.
Jerry L. Kazdan
2. a) Prove that there are innitely many prime numbers. b) Prove tha
Exam
Math 504
December 8, 2005
Jerry L. Kazdan
12:00 1:20
Directions This exam has two parts, the rst has four short computations (5 points each ) while
the second has seven traditional problems (10 p
Math 504
December 8, 2005
Exam Solutions
Jerry L. Kazdan
12:00 1:20
Directions This exam has two parts, the rst has four short computations (5 points each ) while
the second has seven traditional prob
Printed Name
Signature
Math 504
October 22, 2009
Exam 1
Jerry L. Kazdan
10:30 11:50
Directions: Part A has 5 shorter problems (5 points each) while Part B has 6 traditional problems
(10 points each).
Printed Name
Signature
Math 504
December 10, 2009
Exam 2
Jerry L. Kazdan
10:30 11:50
Directions: Part A has 5 shorter problems (5 points each) while Part B has 6 traditional problems
(10 points each).
Algebra Problems
Math 504 505
1. a) Show that
b) Show that
Jerry L. Kazdan
2 is not a rational number.
3 is not a rational number.
2. a) Prove that there are innitely many prime numbers.
b) Prove that
Analysis Problems
Penn Math
Jerry L. Kazdan
sn the followingD when we sy funtion is smoothD we men tht ll of its derivtives exist
nd re ontinuousF
hese prolems hve een rudely sorted y topi ut this sho
Calculus Problems
Math 504 505
Jerry L. Kazdan
IF keth the points (x ; y) in the plne R2 tht stisfy jy xj
PF e ertin funtion f (x) hs the property tht
nd the onstnt g F
Zx
0
f (t ) dt
=
PF
ex os x + g
Linear Algebra Problems
Math 504 505
Jerry L. Kazdan
Although problems are categorized by topics, this should not be taken very seriously since
many problems t equally well in several dierent topics.
Math 504 505
Miscelaneous Problems
Jerry L. Kazdan
In the following, when we say a function is smooth, we mean that all of its derivatives exist and are continuous. 1. Prove that
(1 + x)n > 1 + nx
for