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 every integer n 2 and all x > 1, x = 0. (Hint: Inducti
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 that there are innitely many primes of the form 4n + 3 (th
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 points each ).
Part A: Short Computations (4 problems, 5
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 problems (10 points each ).
Part A: Short Computations (4 p
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). To receive full credit your solution should be clear an
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). To receive full credit your solution should be clear a
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 there are innitely many primes of the form 4n + 3 (the
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 should not e tken seriously
sine mny prolems t in vriety o
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 F pind oth f
os x 1=x2
QF gompute lim
F
x30 osPx
RF k
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.
Notation: We occasionally write M (n, F) for the ring o
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 every integer n 2 and all x > 1, x = 0. (Hint: Inducti