Math 331
Babylonian Math
N. Emerson
The past is fixed. However, our interpretation of the past is not. Our interpretation of
Babylonian mathematics has changed over time. The current interpretation is that Babylonian mathematics was more geometric in char
Math 150 Exam I Review
Study all homework problems and lecture notes!
6.3 Volume by Slicing
6.4 Volume by Shells
Know the formulas for the washer and the shell methods. Be able to set up
the integrals for some bounded regions. For practice, look at proble
Extra Credits
1. Evaluate the integral
ln x
x 1 + (ln x)2
dx.
2. Test the series for convergence or divergence
n=1
n
n+1
n2
.
3. Find the Maclaurin series for the function
f (x) =
x
.
4 + x2
4. Identify the functions represented by the power series
(1)k
Math 151 Final Review
Review all techniques of integration (chapter 7) and all tests on testing innite
series (chapter 8).
9.1 Approximating Functions with Polynomials
Let f be an n-times dierentiable function. We can approximate f by an n-th
order Taylor
|<E 9
MATH 150, Fall 2013
Instructor: J. Brown
Practice Exam 1
This exam has four parts: True/False, Multiple Choice, Examples, and Short Answer.
Please read the specic instructions given for each section.
You may not use a calculator, and you may not con
Name:
MATH 150 Sections 1 and 2, Fall 2013
Practice Exam 2
This exam has four parts: True/False, Multiple Choice, Examples, and Short Answer.
Please read the specic instructions given for each section.
You may not use a calculator or consult books, notes,
Name:
MATH 150, Fall 2013
Instructor: J. Brown
Practice Exam 1
This exam has four parts: True/False, Multiple Choice, Examples, and Short Answer.
Please read the specic instructions given for each section.
You may not use a calculator, and you may not con
Math 105 Practice Final
1. Find the difference quotient
2.
Name:
f ( x +h ) - f ( x )
, h 0 ,of the function f(x) = x2 3x + 2
h
Given the following two points: P1 = (-5, 2) and P2 = (1, -2)
a) Find the mid point between P1, and P2
b) Find the distance bet
HW6
P143: 2, 7, 8, 10, 14; 6, 7
P133: 6, 7, 10, 22; P161: 9; 6
P170: 4, 5, 6, 9
ALEX FLURY
batch 1
3.5.2. Suppose A, B , and C are sets. Prove that A (B C ) (A B ) C .
Proof. Let A, B , and C be sets, and let x be an arbitrary element of A (B C ). Then
ei
Homework Set 3 Solutions
Kyle Chapman
January 27, 2013
72.2
a
It is not true that there is someone in the freshmen class who doesnt have a roomate.
Everyone in the freshmen class has a roomate.
b
It is not true that both everyone likes someone and noone l
P25: 10, 12, 13, 18; P33: 3, 4; P42: 2
P42: 4, 5, 8, 9; P53: 2, 4, 6
P54: 5, 10; P63: 2, 3, 5, 7
batch 1
1.2.10. Use truth tables to check these laws.
(a). The second DeMorgans law.
P
F
F
T
T
T
T
T
F
Q
F
T
F
T
(P
F
F
T
T
Q)
FF
FT
FF
TT
T
T
F
F
P
F
F
T
T
P13: 2, 3, 4, 6, 7
P24: 2, 4, 5, 6, 8
batch 1
1.1.2. Analyze the logical forms of the following statements.
(a). Either John and Bill are both telling the truth, or neither of them is.
P = John is telling the truth.
Q = Bill is telling the truth.
(P Q) (P
Homework Set 7 Solutions
Kyle Chapman
March 5, 2013
178.2
a
The domain of this function is the set of people who are brothers. The range is the set of people who have
brothers.
b
To nd the domain, we allow y to be any real value, and so y 2 can be any non
Homework Set 4 Solutions
Kyle Chapman
February 6, 2013
93.1
a
The hypothesis is that n is an integer larger than one and that n is not prime. The conclusion is that 2n 1
is not prime. The hypotesis is satised for n = 6 since 6 is greater than one and not
HW9
P243: 8, 9, 12, 15, 16; P253: 8, 9, 11
P254: 13; P265: 2, 4, 5, 9
P266: 8, 12, 16, 17
ALEX FLURY
batch 1
5.2.8. Suppose f : A B and g : B C .
(a). Prove that if g f is onto, then g is onto.
Proof. Suppose g f is onto. Suppose c C . Since g f is onto,
HW5
P122: 2, 4, 8
4, 5
P122: 18, 20, 21, 23
ALEX FLURY
batch 1
3.3.2. Prove that if A and B \ C are disjoint, then A B C .
Proof. Suppose A and B \ C are disjoint, and let x be an arbitrary element of A B .
Since x A, therefore x B \ C . This means either
Homework Set 8 Solutions Dai
Kyle Chapman
March 11, 2013
223.4
a
Not an equivalence relation because (1, 0) R but (0, 1) R.
/
b
This is an equivalence relation. The equivalence classes are of the form x + Q for each real number x.
c
This is an equivalence
Page 296: 3a:proof: Contradiction. Suppose 6 is a rational number. This means
that there exist q Z + and q Z + such that p = 6. So the set S = cfw_q Z + |( p =
q
q
6) is nonempty. By the well ordering principle we can let q be the smallest element
2
of S