Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
MATH 308 Homework 9
9.1. Find the fundamental matrix (t) = eAt satisfying (0) = I for the system
x=
3 2
2 2
x.
4 2
8 4
x.
9.2. Find the general solution of the system
x=
9.3.
MATH 409.200/501
Spring 2009
Exam #1 Solutions
1. (a) Every nonempty subset of R which is bounded above has a supremum.
(b) An example is the interval [0, 1).
(c) First we show that sup(A + B ) sup A + sup B . If a A and b B then
a + b sup A + sup B since
MATH 409.200/501
Spring 2009
Exam #2 Solutions
1. (b) Since a sequence is Cauchy if and only if it converges, it suces to show that
cfw_xn + (1)n fails to be Cauchy. Since cfw_xn is Cauchy there is an N N such
n=1
n=1
that if n, m N then |xn xm | < 1. T
MATH 409.501
Final Exam
May 4, 2012
Name:
ID#:
The exam consists of 11 questions, the rst 5 of which are multiple choice. The point
value for a question is written next to the question number. There is a total of 100 points.
No aids are permitted.
For que
MATH 409.501
Spring 2012
Exam #1 Solutions
1. (a) false, (b) true, (c) true, (d) false, (e) true, (f) true, (g) true, (h) false.
2. (b) We proceed by induction. In the case n = 1 we have 211 = 20 = 1 = 1!, and so
the inequality holds. Now suppose that the
Misprints for the fourth edition of
An Introduction to Analysis
by W.R. Wade
p. 3,
p. 57,
p. 89,
p. 150,
p. 108,
p. 158,
p. 158,
p. 324,
p. 366,
p. 369,
p. 398,
p. 404,
p. 405,
p. 413,
p.413,
p. 413,
p. 419,
p. 426,
p.
p.
p.
p.
432,
457,
458,
529,
p.
p.
p
MATH 409, TEST 1, FALL 2011
Show all steps for credit. Q1-Q4, 12 pts., Q5-8 13 pts.
Q1. (12 pts) Prove by induction that
n
X
k=1
k2 =
n(n + 1)(2n + 1)
.
6
Q2. (12 pts) Prove that
N N = cfw_(m, n) : m, n N
is countable.
Q3. (i) (4 pts) Define what it means
Wade, An Introduction to Analysis, Fourth Edition
Errata and Addenda
A list of errata from the first printing can be found at http:/www.math.utk.edu/~wade . The errata below
remain after these have been corrected in current printings.
p. 23 Theorem 1.22:
Generating Functions
1
Definition and first examples
Generating functions offer a convenient way to carry the totality of the information about a
sequence in a condensed form. Precisely, the (ordinary) generating function of the sequence
(an )n0 is define
Texas A&M University, Department of Mathematics, Fall 2016
MATH 409 (502) Advanced Calculus I
TR 9:35am-10:50am, Blocker Building, Room 121
Instructor: Simon Foucart, 608L Blocker Building, foucart@tamu.edu
Office hours: TWR 11:00am-11:30am and by appoin
Lecture 0: Review
This opening lecture is devised to refresh your memory of linear algebra. There are some
deliberate blanks in the reasoning, try to fill them all. If you still feel that the pointers are too
sketchy, please refer to Chapters 0 and 1 of t
Notes
on
Compressed Sensing
for
Math 394
Simon Foucart
Spring 2009
Foreword
I have started to put these notes together in December 2008. They are intended for a
graduate course on Compressed Sensing in the Department of Mathematics at Vanderbilt
Universit
MATH 409.200/501
Examination 1
February 19, 2009
Name:
ID#:
The exam consists of 5 questions. The point value for a question is written next to the
question number. There is a total of 100 points. No aids are permitted.
1. [20] (a) State the completeness
MATH 409.501
Fall 2012
Exam #1 Solutions
1. (a) true, (b) true, (c) true, (d) false, (e) true, (f) true, (g) false, (h) true.
2. (b) We proceed by induction. In the case n = 1 we have 1 =1 k 2 = 12 = 1 = (1 2 3)/6,
k
and so the formula holds. Now suppose
MATH 409
Fall 2012
Solutions to some assignment problems
1.3.0. (a) True under the assumption that A B = (otherwise sup(A B ) is undened).
Indeed let A and B be nonempty bounded subsets of R such that A B = . Note
that sup(A B ) exists since A B , being a
Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
MATH 308 Homework 10
10.1. Determine the radius of convergence of the power series
2n xn .
n=0
10.2. Find the rst ve terms of the power series solution y (x) = an xn of the in
Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
MATH 308 Homework 11
11.1. Find the general solution of the system
x=
2
1
1 2
x.
11.2. Find the general solution of the system
10
0
x = 2 1 2 x .
32
1
Express it in terms of r
Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
MATH 308 Homework 1
1.1. Sketch a direction eld for the dierential equation
y = 1 + 2y.
Based on the direction eld, determine the behavior of y as t . If this behavior
depends
Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
MATH 308 Homework 2
2.1. Find the general solution of the equation
y + y 2 sin x = 0.
2.2. Find the general solution of the equation
y=
x ex
.
y + ey
2.3. Find the solution of
Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
MATH 308 Homework 3
3.1. Determine whether the equation
(3x2 2xy + 2) dx + (6y 2 x2 + 3) dy = 0
is exact. If it is exact, nd the general solution.
3.2. Find the value of b for
Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
MATH 308 Homework 4
4.1. Consider the initial value problem
3
y y = 2et ,
2
y (0) = y0 .
Find the value of y0 that separates solutions that grow positively as
t from those tha
Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
MATH 308 Homework 5
5.1. Find the solution of the initial value problem
y 2y + y = tet + 4,
y (0) = 1,
y (0) = 1.
Plot it using MatLab. (Try to choose a nice interval [0, T ]
Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
MATH 308 Homework 6
6.1. Find the inverse Laplace transform of F (s) =
2s3
.
s2 +2s+10
6.2. Use the Laplace transform to solve the initial value problem y 4y + 4y = 0,
y (0) =
Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
MATH 308 Homework 8
8.1. Find the general solution of the system
x1 = x1 + x2
x2 = 2 x2
8.2. Find the general solution of the system
x=
2 1
3 2
x.
8.3. Find the general soluti
Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
MATH 308 Homework 7
7.1. Verify that the functions y1 and y2 satisfy the corresponding homogeneous equation,
and nd a particular solution of the non-homogeneous equation.
t2 y
Exam #1
If you use any Theorems from the text in working a problem identify the Theorem either by name or
number.
1. (10 points) Prove that the sequence cfw_ n2 + n n converges or prove that it diverges. Find the
limit if the sequence converges.
Solution:
Exam #2
If you use any Theorems from the text in working a problem identify the Theorem either by name
or number. Follow the guidelines for submitting exams given in the Exams Section of our Web site.
1. (15 points) Suppose f is uniformly continuous on (a
Exam #3
If you use any Theorems from the text in working a problem identify the Theorem either by name
or number. Follow the guidelines for submitting exams given in the Exams Section of our Web site.
1. (15 points) Suppose f is continuous on [a, b]. Prov