Higher Mathematics for Engineers and Scientists II
MATH 3351

Spring 2011
50 pts.
Problem 1. In each part you are given the augmented matrix of a system
of linear equations, with the coecient matrix in reduced row
echelon form. Determine if the system is consistent and, if it is consistent, nd
all solutions.
A.
B.
C.
1
0
0
0
0
March 1 Quiz # 5
Math 3351 Sec. 1,
Name
1. Find the radius of convergence of the power series
n=0
(n + 1)!3n
an
= lim
R = lim
n an+1
n
n!3n+1
3n
n!
(n + 1)
3
xn
=
y (n) (0)
) to nd
2. Use the Taylors series method (i.e., use the fact that an =
n!
a0 , a1
Feb. 15 Quiz # 4
Math 3351 Sec. 1,
Name
1. Classify the critical point at zero for
A=
.
y1 = 3y2
y2 = 3y1
.
03
= 3i, 2 = 3i , Center
3 0 1
3. Classify the critical point at zero for
A=
y2 = 2y1 2y2
2 2
= 2 + 2i, 2 = 2 2i , Stable Spiral
2 2 1
2. Classif
January 25, 2002 Quiz # 1
Math 3351 Sec. 1, Name
011
1. Find the rank of A = 1 2 1
110
1 2 1
2. Find the determinant of B = 0 2 1
2 0 1
3. Determine whether the vectors v1 =
dent or independent.
1
2
and v2 =
are linearly depen2
4
>
Quiz # 1
Math 3351 Spring 02
> with(linalg):
Warning, the protected names norm and trace have been redefined and unprotected
1. find the rank of A
> A:=matrix(3,3,[0,1,1,1,2,1,1,1,0]);
0 1 1
A := 1 2 1
1 1 0
> rref(A);
1 0 1
0 1
1
0 0
0
> rank(A);
2
Math 3351 Sec. 1, , Test # 3, Name
April 17
1. Find the eigenvalues and eigenfunctions for y + y = 0 on 0 < x < 1 with
y (0) = 0, y (1) = 0.
Case = 0: y = 0 y = ax + b 0 = y (0) = b y = ax. Then
0 = y (1) = a y 0 so = 0 is not an eigenvalue.
Case = 2 : y
Math 3351 Sec. 1, Sample Test # 2, Name
1. Determine the radius of convergence of the power series
ANSWER:
4n 2n
x=
n2
m=1
we have
4n 2n
x
n2
m=1
n
(4x2 )
so set t = 4x2 to obtain
n2
m=1
R = lim
n
tn
. Thus, for t,
n2
m=1
(n + 1)2
an
=1
= lim
n
an+1
n2
so
Math 3351 Sec. 1, Sample Test # 1, Name
021
1. Find the rank of A = 1 1 1
102
ANSWER: 3
12
2 2 1
2. If A =
and B = 1 1.
023
2 3
Find det(AB ) and det(BA)
ANSWER: 62
1
1
1
3. Determine whether the vectors v1 = 1, v2 = 1 and v3 = 1 are linearly dependent
1
March 1 Quiz # 5
Math 3351 Sec. 1,
Name
1. Find the radius of convergence of the power series
n=0
(n + 1)2 + 1
an
= lim
lim
n an+1
n
n2 + 1
n
xn
2+1
n
n
(n + 1)
2. Find the radius of convergence of the power series
n=0
=1
3
2
n
x2n
First set t = x2 and nd
Higher Mathematics for Engineers and Scientists II
MATH 3351

Spring 2011
50 pts.
Problem 1. In each part you are given the augmented matrix of a system
of linear equations, with the coecent matrix in reduced row
echelon form. Determine if the system is consistent and, if it is consistent, nd
all solutions.
A.
B.
C.
70 pts.
1
0
Higher Mathematics for Engineers and Scientists II
MATH 3351

Spring 2011
PROBLEM SET
Problems on Span, Independence, and Matrix Spaces
Math 3351, Fall 2010
Sept. 27, 2010
ANSWERS
i
Problem 1. Consider the matrix
15 80 145
89 19
51
A=
66 62 190
77
81
85
22
181
50
132
.
96
150
78
8
A Find a basis of the nullspace of A.
B Fin
Higher Mathematics for Engineers and Scientists II
MATH 3351

Spring 2011
EXAM
Exam #1
Math 2360, Second Summer Session, 2002
April 24, 2001
ANSWERS
i
50 pts.
Problem 1. In each part you are given the augmented matrix of a system
of linear equations, with the coecent matrix in reduced row
echelon form. Determine if the system i
Higher Mathematics for Engineers and Scientists II
MATH 3351

Spring 2011
EXAM
Exam #1
Math 2360, Second Summer Session, 2002
April 24, 2001
ANSWERS
i
50 pts.
Problem 1. In each part you are given the augmented matrix of a system
of linear equations, with the coecent matrix in reduced row
echelon form. Determine if the system i
Higher Mathematics for Engineers and Scientists II
MATH 3351

Spring 2011
40 pts.
Problem 1. Let A be the matrix
2
2
6
5
A=
.
Find the characteristic polynomial of A and the eigenvalues of A by hand
computation.
70 pts.
Problem 2. In each part, you are given a matrix A and the eigenvalues of
A. Find a basis for each of the eige
Higher Mathematics for Engineers and Scientists II
MATH 3351

Spring 2011
EXAM
Exam 2
Math 3351, Spring 2010
April 11, 2011
Write all of your answers on separate sheets of paper,
do not write on the exam questions handout. You
can keep the exam questions when you leave. You
may leave when nished.
You must show enough work to
Higher Mathematics for Engineers and Scientists II
MATH 3351

Spring 2011
EXAM
Exam 3
Math 3351, Spring 2010
April 22, 2011
This is a Takehome exam.
Write all of your answers on separate sheets of paper.
Do not write on the Exam questions sheets. You can
keep the exam questions.
The use of a TI89 (or similiar) calculator i
Higher Mathematics for Engineers and Scientists II
MATH 3351

Spring 2011
NOTES ON SOLVING LINEAR SYSTEMS OF DIFFERENTIAL
EQUATIONS
LANCE D. DRAGER
1. Introduction
A problem that comes up in a lot of dierent elds of mathematics and engineering is solving a system of linear constant coecient dierential equations. Such a
system l
Higher Mathematics for Engineers and Scientists II
MATH 3351

Spring 2011
PROBLEM SET
Problems on Span, Independence, and Matrix Spaces
Math 3351, Fall 2010
Sept. 27, 2010
Write all of your answers on separate sheets of paper.
You can keep the question sheet.
You must show enough work to justify your answers.
Unless otherwise
Higher Mathematics for Engineers and Scientists II
MATH 3351

Spring 2011
PROBLEM SET
Problems on Matrix Exponentials
Math 3351, Spring 2011
March 29, 2011
Write all of your answers on separate sheets of paper.
You can keep the question sheet.
You must show enough work to justify your answers.
Unless otherwise instructed, giv
Higher Mathematics for Engineers and Scientists II
MATH 3351

Spring 2011
PROBLEM SET
Problems on Matrix Exponentials
Math 3351, Spring 2011
March 29, 2011
ANSWERS
i
These problems assume the use of a TI89 (or similar) calculator. To nd the
eigenvalues of a matrix, use the calculator to nd the characteristic polynomial
and use
Higher Mathematics for Engineers and Scientists II
MATH 3351

Spring 2011
EXAM
Exam #1
Math 2360
Summer II, 2000
Morning Class
Oct. 11, 2000
ANSWERS
i
50 pts.
Problem 1. In each part, you are given the augmented matrix of a linear
system. The coecient matrix is already in Reduced Row Echelon Form. Determine if the system is con
Math 3351 Sec. 1, , Final , Name
February 22
Part I (You must show all work for credit)
1 2 1
1. Find (a) the Rank, (b) the Determinant of A = 0 2 1
2 0 1
1
1
1
2. Determine whether the vectors v1 = 1, v2 = 1 and v3 = 1 are
1
1
1
linearly dependent or in