E9 R is I -
3 1. Dene the following: (18 Pts.)
a)fbeingabsohztelyoontinnouson [11, b] d)1heuppahrtegraloffon [11,11]
b) The nthTaylorpolynominl off ml 12-.) Thelowa-sum off overpartion P
c) fsasfyhaganLoondionon [a,b] DfRiunannintegmbleon [11, b]
2.Iffisa

Solutions to the Problems on the Third Exam
g ( x) A. Multiply both sides of the
3) Let f ( x ) f ( x) g ( x ) and note that lim
x
x
x
previous equation by e x . Thus we have e f ( x ) f ( x) e g ( x) so that
F ( x)
e x f ( x) e x g ( x). Let F ( x) e x

Real Anal sis I - Exam 3
1. Dene the following: (18 Pts.)
,aj/f being absolutely continuous on [a, 1)] K The mesh of a partition P
The upper Riemann integral of f on [(1, b] ,e/ f satisfying an L condition on
a, b]
c) f being Riemann-Stieltj es integrab

1
Solutions to the Problems on the Final
5) We need to show that the derivative of
f ( x)
is nonnegative on (0, a). Note that
x
f ( x ) xf ( x) f ( x) Let x be a point in (0, a) and use the Mean Value
.
x2
x
f ( x) f (0)
f ( ) for some in (0, x). Si

Solutions to some of the problems on the first exam
3. This is problem 11 on page 18 of the text and was done in class.
4. Let 0 be given. Then form
2 2n 2 5n 2 2
19 n 4
19n 4
19n 1
an
.
2
2
2
5 5n 3n 7 5 5(5n 3n 7) 5(5n 3n 7) 25n 2 n
2
2
1
1
2
when

Finite mathematics
Dr. LaBuz
Exam
Chapter 1 Sets and logic
1. Let A = cfw_1, 2, 3, 4, 5, B = cfw_1, 6, and C = cfw_2, 3, 4. Find the following sets.
(a) A B
(b) A B
cfw_1, 2, 3, 4, 5, 6
cfw_1
(c) B C
= empty set
(d) A C
cfw_ 1, 5
(e) C A
= empty set
2.

Math 111
Dr. LaBuz
Exam
Chapter 2 Counting and probability
Suppose we spin the spinner 25 times.
1. Find n(S), the total number of possible outcomes. Give the exact answer (you should copy and paste the
number from WolframAlpha). Also give the name of the

Math 111
Dr. LaBuz
Exam
Chapter 2 Counting and probability
Suppose we spin the spinner 20 times.
1. Find n(S), the total number of possible outcomes. Give the exact answer (you should copy and paste the
number from WolframAlpha). Also give the name of the

Real Analysis I - Final
1. Dene the following:
a) f being absolutely continuous on [0, b].
b) f being Riemann integrable on [a, b]
c) f being RiemannStieltjes integrable with respect to g on [a, b]
d) f being Lebesgue integrable on [0, b]
e) f being Riema

_ Real Analysis I - Exam 3
1. Dene the following: (18 Pts.)
a) E being a set of measure zero c) f satisfying an L condition on [0, b]
b) f being absolutely continuous on [a, b] d) The upper integral of f on [a, b]
c) The nth Taylor polynomial of f at a e)

8.1
1. False
3. True
5. True
7. True
9. False
11. Zero is a number and 0means there is no answer
13.
15.
17.
19.
21. 8
23. 128
25. xlx 0 or 8
27. An intersection contains all elements of both sets while a union has all elements in at least one set.
29

Real Analysis I - Exam 2 Name:
1. Dene the following: (18 Pts.)
21) Vliminf f (x)
16%ch
3 _, J? 4
$5M) " gigt (526x44 Edd
b) P being a partition of [a, b]
Fig aWWM53Za] cfw_6 athXF3<~ (291365
c) C being a full cover of E g [a,b]
(0/ f5 aMmm/cqu 5 ~10 W

Real Analysis I - Exam 2
1. Dene the following: (18 Pts.)
51) lim inf f(x)
b) P being a partition of [a, b]
. c) C being a full cover of E"; [a,b]
d) f being uniformly continuous on D
e) 5f(xo)
f) E being a set of measure zero
i 2- If C is a ll! cover of

Study Guide for the Second Exam
1. Know all definitions covered in class from page 52 up to and including what is
done on Monday, October 20th.
2. Be able to prove the following theorems: (One will be chosen.)
a) Theorem 3.4a, page 60
b) Theorem 3.5a, pag

Solutions to Problems on the Second Exam
3. This is problem 8(i) on page 74 and was done in class. Please see your notes.
4. This is problem 19 on page 75 and was done in class. Please see your notes.
5. This is part of problem 26 on page 76 and was also

Math 111
Dr. LaBuz
Exercise
Chapter 3 Module 5 Commutativity
Verify that matrix multiplication is not commutative by calculating the following products. You may use
WolframAlpha or another calculator.
2
1
2
5
0
1
9
8
0
1
9
-2
=
8
5
9
2 2
= 10
1
5
2
49
45

Math 111
Dr. LaBuz
Exercise
Chapter 2 Module 6 Factorials
Calculate the following factorials. Give the exact answer (you should copy and paste the number from
WolframAlpha).
1. 9!
2. 50!
3. 100!
= 362880
= 3041409320171337804361260816606476884437764156896

Math 111
Dr. LaBuz
Exercises
Module 4 Describing sets in words
1. Let S be the set of all students enrolled in this course. Let A be the set of students in this course that are taking
an online class for the first time and B be the set of students in this

Math 111
Dr. LaBuz
Exercise
Chapter 3 Module 6 Associativity
Verify that matrix multiplication is associative by calculating the left and right side of the following equation. You
may use WolframAlpha or another calculator.
2
1
2
5
0
1
9
8
3
4
2
3
=
Lef

Math 111
Dr. LaBuz
Exercise
Chapter 1 Module 5 The algebra of sets
Verify associativity of intersection using Venn diagrams. You may shade the regions using the pen tool or put an x
is the regions that should be shaded. Be sure to label your diagrams. To

Exercises
Module 4 Describing sets in words
Math 111
Dr. LaBuz
2. Suppose there is a bowl containing red, yellow, and green marbles with at least two of each color. Suppose two
marbles are drawn from the bowl, one after another.
Let A be the set of all ou

Math 111
Dr. LaBuz
Exercise
Chapter 3 Module 7 Distributivity
Verify that distribution of matrix algebra holds by calculating the left and right side of the following equation. You
may use WolframAlpha or another calculator.
2
1
2
5
0
1
9
3
+
8
4
2
3
0
1

Math 111
Dr. LaBuz
Exercise
Chapter 2 Module 4 Finding the probability of an or event
1. Suppose the spinner is spun ten times. Find the probability of getting no green or no red. Be sure to define
events, use correct notation, and explain the meaning of