Foundations of Analysis MAT 319 Spring 2013 Midterm 1
There are ve problems on the exam. The exam is 80 minutes long. You
may not use your notes, book or calculator for the exam. If you have any
questions, ask me. Good luck!
Problem 1. 20 points. Prove by
Foundations of Analysis MAT 319 Spring 2013 Midterm 2
There are ve problems on the exam. The exam is 80 minutes long. You
may not use your notes, book or calculator for the exam. If you have any
questions, ask me. Good luck!
Problem 1. 20 points. For each
Review sheet for Final, MAT 319, Foundations of Analysis, Spring 2013
IN ADDITION to the material on the midterm review sheets, you need
to know the following.
1. Derivatives. Denition (28.1), theorems (28.2, 28.3, 28.4). Sample
problems: calculating the
Math 104: Homework 7 solutions
1. (a) The derivative of f ( x ) =
x is
1
f (x) =
2 x
which is unbounded as x 0. Since f ( x ) is continuous on [0, 1], it is uniformly
continous on this interval by Theorem 19.2. Hence for all > 0, there exists
> 0 such t
Math 104: Homework 3 solutions
1. Suppose tn is bounded so that |tn | M for all n, where M 0. If M = 0, then
|tn | 0 for all n implies that tn = 0 for all n. Thus sn tn = 0 for all n and hence
limn sn tn = 0.
Otherwise M > 0. Consider any > 0. Since sn co
MATH 104, SUMMER 2006, HOMEWORK 2 SOLUTION
BENJAMIN JOHNSON
Due July 5
Assignment:
Section 4: 4.1(d)(e)(h)(t)(u)(v); 4.7, 4.12
B1: Prove that an ordered eld F is Archimedean if and only if satises (a F)(n N)(n > a)
Section 5: 5.1, 5.2, 5.6
Section 6: 6.1,
Math 319/320 Worksheet 4 solutions
Problem 1. Find all x R that satisfy the following inequality:
(a) |x| + |x + 1| < 2
Method 1: draw a graph.
Method 2: Divide into cases, depending on the signs of x and x + 1.
Suppose x 0. Then we have 2x + 1 < 2, i.e.
MATH 104, SUMMER 2006, HOMEWORK 9 SOLUTION
BENJAMIN JOHNSON Due August 7
Assignment: Section 25: 25.3, 25.15 Section 26: 26.2, 26.5 Section 28: 28.2(a), 28.6(b), 28.8 Section 25 25.3 Let fn ( x) = 2n+cos2xx for all real numbers x. n+sin (a) Show that fn c
Math 104: Homework 9 solutions
1. (a) For x = 0,
1
1
cos
+ 2x sin
2
x
x
1
1
= cos
+ 2x sin
x
x
1
x
f ( x ) = x2
where the product rule and chain rule have been employed.
(b) At x = 0,
f (0) = lim
x 0
f ( x ) f (0)
= lim x sin
x 0
x0
1
x
= 0,
since | sin x
Math 104: Homework 10 solutions
1. (a) The rst derivative is
f (x) =
1
x+1
and the nth derivative is given by
f
(n)
(n 1)!(1)n1
=
.
( x + 1) n
Hence the Taylor series expansion at x = 0 is given by
n 1
f (x) =
f ( k ) (0)
+ Rn ( x )
n!
( n 1) !
(1)n1 + Rn
Consider the set SN = cfw_sn |n > N for any N N. Then 1 SN since there exists an even
number k > N and sk = 1.8: Since 1 + 1 n > 1 for all n, it follows that infSN = minSN = 1.
Thus uN = 1 for all N N. For any N in N, dene L to be the smallest odd number
Math 319/320 Worksheet 3
Problem 1. (i) Construct a function f : N N that is injective but not surjective.
Dene f (k ) = 2k .
(ii) Show that if S is an innite set then there is a function f : S S that is injective
but not surjective. You may use the fact
Math 319/320 Worksheet 2
Problem 1. Fill in the blanks in the following proof that
A (B C ) = (A B ) (A C ).
If x A (B C ) then either x A or x B C . If x A then x A B
and x A C and so x (A B ) (A C ). On the other hand, if x B C then
x B and x C . Hence
Foundations of Analysis MAT 319 Spring 2013 Final exam
Problem 1. 30 points. Use induction to prove that 11n 5n is divisible
by 3 for any natural number n.
Proof. For the base case, 111 51 = 6 is divisible by 3.
For the induction step, assume that 11n 5n
Semester Schedule
Date
Jan. 29
Jan. 31
Feb. 5
Feb. 7
Feb. 12
Feb. 14
Feb. 19
Feb. 21
Feb. 26
Feb. 28
Mar. 5
Mar. 7
Mar. 12
Mar. 14
Mar. 19
Mar. 21
Mar. 26
Mar. 28
Apr. 2
Apr. 4
Apr. 9
Apr. 11
Apr. 16
Apr. 18
Apr. 23
Apr. 25
Apr. 30
May 2
May 7
May 9
Mater
MAT 319 Syllabus
Course title: Foundations of Analysis, MAT 319, Spring 2013
Meeting times:
Lecture:
R01:
Tuesday and Thursday, 11:30-12:50, Mathematics 4130
Monday and Wednesday, 10:00-10:53, Physics P112
Course description: The purpose of this course is
Review sheet for Midterm 1, MAT 319, Foundations of Analysis, Spring
2013
1. Mathematical induction. You need to be able to clearly state the principle of mathematical induction and do a induction-based proof. Sample
problems: 1.2, 1.5, 1.6, 1.9.
2. Field
Review sheet for Midterm 1, MAT 319, Foundations of Analysis, Spring
2013
Here is a list of topics for the second midterm. You need to know all of
the denitions and the statements of all of the theorems. I wont ask you to
prove any of the theorems, but it
Practice problems for Midterm 1, MAT 319.
1.5. Prove that
n
i=0
1
1
1
1
= 1 + + + n = 2 n
i
2
2
2
2
for all natural numbers n.
We prove this by induction. We rst check the base case n = 1:
1+
1
1
=2 .
2
2
Note that the formula is also true for n = 0, but
Math 319/320 Solutions to Homework 1
Problem 1. Prove the identity: A (B C ) = (A B ) (A C ).
1. To show A (B C ) (A B ) (A C ):
let x A (B C ). Then x A or x B C . If x A then x A B and A C ,
Hence x (A B ) (A C ). On the other hand, if x B C then x is i
Math 319/320 Solutions to Homework 2
Problem 1. Show that the set Sodd of odd (positive and negative) integers is denumberable
by (a) enumerating them and (b) giving an explicit formula for the corresponding bijection
f : Sodd N.
An emumeration: Sodd = cf
Math 319/320 Homework 3
Due Thursday, September 22, 2005
revised version
Problem 1. Show that
2
1
1
(a + b) (a2 + b2 )
2
2
for all a, b R. Show that equality holds if and only if a = b.
We must show that (a2 + 2ab + b2 )/4 (a2 + b2 )/2. Multiplying both s
Math 319/320 Solutions to Homework 4
1
1
Problem 1. Let In := [0, n ] and Jn := (0, n ], for n N. Show that In = cfw_0.
n=1
Since 0 In for all n, 0 In . If x In for all n then 0 x 1/n for all n.
n=1
But if x > 0 then 1/x > 0 and there is by Archimedes pri