Math 3001
Midterm Exam Solutions
Page 1
1. [20 points] Each of the following true/false questions is worth ve points. You will only
be graded on whether you write TRUE or FALSE, no other information will aect your
grade.
There are no tricks, but you must
Mathematics 3001
Practice Exam 1
February 21st, 2013
NAME
This exam consists of 4 problems, numbered 14. For partial credit you must present
your work clearly and understandably; no credit will be given for unsupported
answers.
Please write your name on e
NAME
Math 3001
Final Examination
Page 1
Math 3001 Section 2
Final exam solutions
1. [20 points] Each of the following true/false questions is worth ve points. You will only
be graded on whether you write TRUE or FALSE, no other information will aect your
Math 3001 Homework #4 Solutions
1. Section 2.2.4: 3. If x is a real number, show that there exists a Cauchy
sequence of rationals x1 , x2 , . . . representing x such that xn < x for all
n.
Solution:
Let n N be any integer; then by Theorem 2.2.5 (Density o
Math 3001 Homework #8 Solutions
1. Prove directly from the denition that the function
1
f (x) = 3x2 + 4 2
x
is continuous on D = (0, ).
Solution: Let x0 be any positive number; we want to prove that f is
continuous at x0 . So let > 0 be any positive toler
Math 3001 Homework #1 Solutions
1. Write the statement, For every action, there is an equal and opposite reaction, using
quantiers. Then write its negation.
Solution:
Let A denote action, let R denote reaction, let E(R) denote the statement that a
reactio
Math 3001 Homework #10 Solutions
1. For the function f (x) = x3 2x2 +4 and the partition P = cfw_2, 1, 1, 3
of [2, 3], compute S + (f, P ) and S (f, P ) explicitly.
Solution: Graph the function, as shown.
Knowing that the critical points, where f (x) = 3x
Math 3001 Homework #7
due Wednesday, March 10
1. Section 3.2.3: 1. Let A be an open set. Show that if a nite number of
points are removed from A, the remaining set is still open. Is the same
true if a countable number of points are removed?
Solution:
Supp
Math 3001 Homework #9
due Friday, April 16
1. Prove from the denition that f (x) =
You may use limit theorems.
1
x
is dierentiable at every x0 > 0.
Solution:
Since were allowed to use limit theorems, we just have to compute:
f (x0 ) = lim
xx0
= lim
xx0
=
Math 3001 Homework #6
due Wednesday, March 3
1. Section 3.1.3: 5. Prove lim supcfw_xn + yn lim supcfw_xn + lim supcfw_yn if
both lim sups are nite, and give an example where equality does not
hold.
Solution:
For each n N, let an = sup xm and bn = sup y
Math 3001 Homework #5 Solutions
1. Section 2.3.3: 2. Prove that every real number has a unique cube root.
Solution: We just have to imitate the method for square roots in the
textbook.
First we establish uniqueness: if y 3 = x and z 3 = x, then y 3 z 3 =
Math 3001: Analysis I, Section 002
Homework Assignment 3 Solutions
1. Recall that Z/nZ (or Zn ) is the set cfw_0, 1, . . . , n 1 together with addition and multiplication
modulo n.
Show that if n is not a prime number, then Z/nZ is not a eld.
Solution: Th
Analysis Spring 2015
CU Boulder Math 3001
worksheet 1
Exercise 1. Prove that |a + b| |a| + |b| by proving the statement in each of the following
cases:
(1) a 0 and b 0
(2) a < 0 and b < 0
(3) a 0, b < 0, and a + b 0
(4) a 0, b < 0, and a + b < 0
(5) a < 0
Analysis Spring 2015
CU Boulder Math 3001
1. Warm-up Proofs
Denition 1. An integer n is even if n = 2k for some integer k.
Denition 2. An integer n is odd if n = 2k + 1 for some integer k.
Exercise 3. Prove that the sum of two even integers is even.
Exerc
Analysis Spring 2015
CU Boulder Math 3001
worksheet 13
Read sections: 5.15.2
Denition. Let f : A R be a function dened on an interval A. The derivative of f at
c A is
f (x) f (c)
,
f (c) = lim
xc
xc
and we say that f is dierentiable at c if this limit exi
Analysis Spring 2015
CU Boulder Math 3001
worksheet 8
Read section: 3.2
Denition. Given a R and > 0, the -neighborhood of a is the open interval
V (a) = cfw_x R : |x a| < .
Denition. A set A R is open if for all points a A, there exists an such that V (a)
Analysis Spring 2015
CU Boulder Math 3001
worksheet 7
Read sections: 2.62.7
Denition. A series is an innite sum
an = a1 + a2 + a3 +
n=1
For any series, we dene the m-th partial sum by
m
an = a1 + a2 + + am .
sm =
n=1
The series an converges to a R if and
Analysis Spring 2015
CU Boulder Math 3001
worksheet 12
Read sections: 4.4
Denition. A function f : A R is uniformly continuous on A if for each
exists > 0 such that if |x y| < , then |f (x) f (y)| < .
Exercise 1. Prove that f (x) =
> 0, there
x is uniform
Analysis Spring 2015
CU Boulder Math 3001
worksheet 6
Read sections: 2.52.6
Denition. Let (an ) be a sequence of real number, and let n1 < n2 < n3 < be
any increasing sequence of natural numbers. The sequence (an1 , an2 , an3 , . . .) is called a
subseque
Analysis Spring 2015
CU Boulder Math 3001
worksheet 10
Read section: 4.2
Denition. Let f : A R, and let c be a limit point of A. We say that limxc f (x) = L
if, for each > 0, there exists a > 0 such that if 0 < |x c| < (and x A), then
|f (x) L| < .
Equiva
Analysis Spring 2015
CU Boulder Math 3001
worksheet 9
Read section: 3.3
Denition. A set A R is compact if every subsequence in K has a subsequence that
converges to a limit that is also in K.
Exercise 1. Use the denition of compact to determine whether or
Analysis Spring 2015
CU Boulder Math 3001
worksheet 4 supplement
1. logical statements
When trying to understand logical statements (such as the denition of convergence), it
often helps to think of the variables in these statements to real objects. Consid
Analysis Spring 2015
CU Boulder Math 3001
worksheet 4
Read sections: 2.12.2
Denition 1. A sequence is a function f whose domain is N. In other words, a sequence is
an ordered list of real numbers of innite length where the n-th item in the list is f (n).
Analysis Spring 2015
CU Boulder Math 3001
worksheet 5
Read sections: 2.32.4
Exercise 1. Prove the latter two cases of the Algebraic Limit Theorem:
Theorem (Algebraic Limit Theorem). Let lim an = a and lim bn = b. Then
(i) lim(can ) = ca, for all c R,
(ii)
Analysis Spring 2015
CU Boulder Math 3001
worksheet 3
Exercise 1. Let A R be a nonempty, bounded set. Prove that the supremum of A is
unique. That is, prove that A has exactly one supremum.
Exercise 2. Prove the following theorem.
Theorem. Let A R and let
ANALYSIS EXAM 1
MATH 3001 SPRING 2015
DUE WEDNESDAY 6PM, FEBRUARY 25
NAME
Instructions:
Please provide a complete and (to the best of your ability) well-written proof for each
of the problems below.
Arguments with inappropriate or incomplete justication
Math 3001 Homework #2 Solutions
1. Section 1.2.3: 7. Generalize Cantors diagonalization argument to show that 2A has
greater cardinality than A for any set A.
Solution: First lets rewrite the diagonal argument in set notation, when we prove that
2N is unc
Math 3001, Section 01
Quiz #1, solutions
Problem 1. Use proof by induction to prove Bernoullis inequality:
1 + nx (1 + x)n ,
n N, x R, x 0
Solution. First prove the inequality when n = 1: then it reduces to 1 + x 1 + x, which is
certainly true.
Now assume
Math 3001, Spring 2013, Section 1
Homework #8 Solutions
Problem 1. Let f : D R and let c D a limit point of D. Mark each statement True or False. Justify
each answer.
1. If f : R R is continuous at each rational number, then f is continuous on R.
False. T
Math 3001, Spring 2013, Section 1
Homework #9 Due April 10
Problem 1. Let f : D R. Is it true that, if f is continuous and (xn ) is a Cauchy sequence in D, then
(f (xn ) is a Cauchy sequence?
False. This is true for uniformly continuous functions but not