Math 4606. Fall 2006.
Solutions to Homework 8
Note: this homework assignment is not collected.
Section 2.1.
Problem 1. Suppose f is dierentiable on an interval I and that f (x) > 0
for all x I except for nitely many points at which f (x) = 0. Show that f
Math 4606. Fall 2006.
Solutions to Homework 7
Section 1.7.
Problem 4. Suppose S1 and S2 are connected subsets in Rn with S1 S2 = .
(a) Show that S1 S2 is notnnected.
(b) Is S1 S2 connected?
Solution. (a) Suppose S = S1 S2 is not connected. Then there is a
Math 4606. Fall 2006.
Solutions to Homework 3
Section 1.3.
Problem 1c. Let
f (x, y ) =
x4 y 4
, for (x, y ) = (0, 0).
(x2 + y 4 )3
Show that lim(x,y)(0,0) f (x, y ) does not exist.
Solution. This problem was solved in class. Briey,
f (0, y ) = 0, for all
Math 4606. Fall 2006.
Solutions to Homework 2
Section 1.1.
Problem 1. Given x, y Rn .
a. Show that |x + y |2 = |x|2 + 2x y + |y |2 .
Solution. Using the denition of the norm in Rn and the commutativity and associativity of the dot product, we have
|x + y
Math 4606. Fall 2006.
Solutions to Exam 2
1. (25 points) Let A and B be two compact subsets of Rn . Dene the distance
between A and B by
d(A, B ) = inf cfw_|x y | : x A, y B .
Show that if A B = then d(A, B ) > 0.
Solution. Note that d(A, B ) 0. Suppose d
Math 4606. Fall 2006.
Solutions to Exam 1
1. (20 points) Let X and Y be two non-empty sets and let f be a
one-to-one function from X to Y . Let A be a subset of X . Show that
f (X \ A) is a subset of Y \ f (A).
Solution. Let y f (X \ A), then y = f (x) fo
MATH 4606. Fall 2006. Advanced Calculus
HOMEWORK 1
Due date: Friday, September 15
PART A. (0 point)
1. Verify the statements, identities, claims, proofs given in lectures.
2. Rewrite some theorems you have learned in the form of mathematical
statements. W
Chapter 1
Setting the stage
1.1
Euclidean spaces and vectors
Let n be a natural number, i.e. n = 1, 2, 3, . . . The n-dimensional Euclidean
space is the set of odered n-tuples of real numbers. We denote this space by
Rn . Then
Rn = cfw_x = (x1 , x2 , . .