Math 3220
Fall 2013
Dr. Lamb
Exam 1
September 25, 2013
Name (Print):
This exam contains 6 pages and 5 problems. Check
to see if any pages are missing. Enter all requested
information on the top of this page, and put your last
name on the top of every page
Solutions of Fourth Homework
Solution of 7.4 Ex. 3:
Let U be an open set in Rd and
T Kn , n N, a nested family of
compact sets in Rd . Assume that
n=1 Kn U .
Put
Zn = Kn U,
n N.
Since Kn are closed and the complement of U is closed, they are all
closed s
Solutions of Twelfth Homework
Solution of 10.1 Ex. 3: Let P be a partition of R. For a rectangle
Ri in the partition we have
|f (x) f (y)| K|g(x) g(y)| K| sup g(x) inf g(y)|
yRi
xRi
for each x, y Ri . Therefore, we have
| sup f (x) inf f (y)| K| sup g(x)
Solutions of Tenth Homework
Solution of 9.3 Ex. 8: Define the function H(s, t) = (st, s + t),
(s, t) R2 . Then, we have G(s, t) = (F H)(s, t) for (s, t) R2 . By
the chain rule, we have
#
"
f1
f1
(st,
s
+
t)
(st,
s
+
t)
t s
x
y
dG(s, t) = dF (H(s, t)dH(s,
Solutions of Eighth Homework
Solution of 9.2 Ex. 4: Consider the matrix of partial derivatives
for G(x, y) = (y ln x, xey , sin(xy) for x R+ and y R:
y
ln
x
x
ey
xey .
y cos(xy) x cos(xy)
Since all coefficients of that matrix are continuous, the function
HOMEWORK 9 SOLUTIONS
8.3.7 Prove that if cfw_Fn is a sequence of bounded functions from a set D
Rp Rq and if cfw_Fn converges uniformly to F on D, then F is also bounded.
Proof. Let > 0. Uniform convergence of the sequence cfw_Fn means that there exis
MATH 3220 HW 2 SOLUTIONS
7.1.3. Using only the properties listed in Theorem 7.1.1, prove that if u is a
vector in a vector space, a is a scalar, and au = 0, then either a = 0 or u = 0.
Either a is zero or it is not. If a is 0, then were done. So lets supp
HOMEWORK 6 SOLUTIONS
2. Give a simple reason why the function : R R4 defined by (t) =
(t, sin t, et , t2 ) is continuous on R.
Theorem 8.1.5 says that a function is continuous at a point a if and only if each of its
component functions is. The component f
MATH 3220 HW 5 SOLUTIONS
11. Prove that if X is a compact metric space, then every sequence in X has a
convergent subsequence.
Proof. Let cfw_xn be a sequence in X that has no convergent subsequences. For each x X,
let x > 0 be chosen such that Bx (x) ha
MATH 3220 HW 4 SOLUTIONS
1. If K is a compact subset of Rd and U1 U2 Uk is a nested upward
sequence of open sets with K k Uk , then prove that K is contained in one of
the sets UK .
Proof. By the definition of compactness, every open cover of K has a fini
Math 3220 HW 3 solutions
7.3.3 Find the interior, closure, and boundary for the set
cfw_(x, y) R2 : 0 x < 2, 0 y < 1.
The interior is cfw_(x, y) R2 : 0 < x < 2, 0 < y < 1. The closure is cfw_(x, y) R2 : 0 x
2, 0 y 1.
The boundary is cfw_(x, y) R2 : x = 0
HOMEWORK 7 SOLUTIONS
8.1.12 Is it true that every continuous function f : B1 (0, 0) R takes Cauchy
sequences to Cauchy sequences?
1
Nope. Consider the function f : B1 (0, 0) R de ned as f (x, y) = 1y
. This function
1
is de ned and continuous on all of B1
Math 3220
Fall 2013
Dr. Lamb
Exam 2
November 4, 2013
Name (Print):
This exam contains 6 pages and 5 problems. Check
to see if any pages are missing. Enter all requested
information on the top of this page.
8 12
This is a closed book test, but you may use
Solutions of Sixth Homework
Solution of 8.2 Ex. 2:
Assume that F is continuous. Let Z be a closed subset of Rq . The
complement Z c is open. Hence, since F is continuous, F 1 (Z c ) is a
relatively open set in D. On the other hand, F 1 (Z) = D F 1 (Z c )