Math 409-502
Solutions Exam 1
October 9, 2003
1. (15) Dene or state the following
(a) lim an =
n
For any M there is an N such that if n > N , then an > M .
(b) lim+ f (x) = 4.
xa
For any
> 0, there is a > 0 such that if
0 < x a < , then |f (x) 4| < .
(c)
Math 409
Solutions Exam 1
March 2, 2009
1. (20)
(a) Dene the statement, f (x) is continuous at a E .
This means that > 0 , > 0 such that if x E and |xa| < , then |f (x)f (a)| < .
(b) Let A be a set. Dene sup(A).
The supremum of any set A is a number s, wh
Math 409-502
Solutions Final Exam
December 17, 2003
1. (25) Dene or state the following:
(a) the completeness property for real numbers
If E is a nonempty set of real numbers, which is bounded above, then there is a real
number that is a least upper bound
Math 409
Final Exam Solutions
May 9, 2005
1. (25) State and prove the monotone convergence theorem. Then use it to show that the
recursive sequence dened below converges to some number l. What must l equal?
a1 = 1, an+1 = 2 + an .
The monotone convergence
December 14, 2010
Solutions Final Exam
Math 409
1. (a) (5) State the completeness axiom for real numbers.
Let E be any set of real numbers, which is bounded above. Then E has a least upper
bound, which, since E is bounded, is nite.
(b) (15) Show that any
May 13, 1998
Math 409,
Final Exam
Each of the rst ve questions is worth 20 points, and each of the last three is worth 25 points. Work the
rst 5 problems, and only one of the last three problems.
No example no credit
1. Dene and give an example of each of
Math 409
Solutions Exam 2
March 30, 2005
1. (15) State and prove Rolles theorem
Rolles theorem: Let f be continuous on the closed bounded interval [a, b] and dierentiable on the open interval (a, b). If f (a) = f (b), then there is a point (a, b) such tha
November 29, 2010
Exam 2 Solutions
Math 409
1. (10) Let f be a real valued function dened on the closed bounded interval [a, b].
(a) Dene what it means to say that f is dierentiable at the point a.
f is dierentiable at a means that the following limit exi
April 8, 1998
Math 409,
Exam 2
Each problem is worth 20 points. You must work the rst two problems, and then any three of problems
3 through 7.
1. Dene the following terms and give an example of each. No example, no credit.
(a) Uniformly continuous.
Ans:
Math 409
Solutions Exam 1
February 23, 2005
1. (20) Dene or state the following:
(a)
lim an = l, Note, there are 3 cases here, l = or l nite, dene the limit for each
of these cases.
n
l nite For all > 0, N such that if n > N , then |an l| < .
l postive in
October 13, 2010
Exam 1 Solutions
Math 409
1. (15)
(a) State the completeness axiom of the real number system.
Every set of real numbers which is bounded above has a least upper bound.
(b) Let f (x) be a real valued function dened for all real numbers. Wh
Feb. 25, 1998
Math 409,
Exam 1
This exam consists of two parts, A and B. There are seven (7) questions in part A. Work all of them.
There are three (3) questions in part B. Work only one of them, and be sure to tell me which one you
work. The questions sh
Math 409-502
Solutions Exam 2
November 17, 2003
1. (20) Dene or state the following
(a) f is uniformly continuous on the interval I .
A function f is uniformly continuous on a set I if for any > 0, there is a > 0 such
that if x and y are in I and |x y | <