Math 473
Mid-Term, due Thursday, May 8 at 10 am
1. Suppose g : R R is uniformly continuous and suppose an is a sequence of real
numbers that converge to 0. Let gn : x g(x an ). Show that gn converges uniformly
to g on R.
an is absolutely convergent. Consi
Math 273
Quiz #2, due Friday, April 11
You must use correct notation and show all your relevant work to earn full credit!
y=1
x=0
1. Switch the order of integration and evaluate:
x5 y dx dy.
y=0
s=1
2. Reverse the order of integration and evaluate:
s=0
x=
Math 273 Quiz #5, due Friday, May 9
To earn full credit, you must use correct notation and show your relevant work!
1. Find parametric equations for the line tangent to the curve
r(t) = (e1t ) + cos(t2 ) ln(t)k at t = 1.
2. Find parametric equations for t
Math 473
Mid-Term Solutions
1. Suppose g : R R is uniformly continuous and suppose an is a sequence of real
numbers that converge to 0. Let gn : x g(x an ). Show that gn converges uniformly
to g on R.
Proof. Let > 0 be given. Since g is uniformly continuo
Math 273
Solutions for Exam #1 Practice
1. Calculate
R
x2 y 2 dA if R is the region in the rst quadrant bounded by x = 0 and
the curves y = x2 and y = 2 x2 .
y
2.0
y
2 x2
1.5
1.0
0.5
0.2
Therefore,
x2 y 2 dA =
R
0.4
x=1
x=0
0.6
y=2x2
y=x2
yx
R x2 +y 2
0.8
On Mountain Pass Type Algorithms
James Bisgard
Abstract. We consider constructive proofs of the mountain pass lemma,
the saddle point theorem and a linking type theorem. In each, an initial
path is deformed by pushing it downhill using a (pseudo) gradient
1
Analysis and Linear Algebra
In this section, we want to investigate how we can use our analysis tools to prove some
important and very useful facts from linear algebra. Before we begin, we collect some useful
tools for use in linear algebra. Throughout,
Math 473
Homework Set #4 Solutions
1. (a) Show that if an is a bounded sequence of positive real numbers and bn is a sequence
of positive real numbers that converge to b 0, then lim sup(an bn ) = b lim sup an .
(Hint: recall that
lim sup an = supcfw_a R :
In this section, we assume that youre familiar with R2 as the set of triples of real numbers,
as well as the standard addition and scalar multiplication that you have seen in your linear
algebra class.
1
Analysis on R2
From the previous section, an import
A Sturm Comparison Theorem
Suppose that H C 1 (R2 , R), and H satises:
(H1) H(t + 1, x) = H(t, x) for all (t, x) R2 ,
(H2) H(t, 0) = 0 for all t [0, 1],
(H3) There is > 0 such that Hx (t, 0) < for all t [0, 1]. (Here Hx means
H
x .)
We want to prove the f
Math 273 Exam 2 Practice Solutions
1. Find parametric equations for the line tangent to r(t) = t2 + sin t et k when t = 0.
We have r (t) = 2t + cos t et k. When t = 0, r (0) = k. Geometrically, r (0)
is a vector tangent to the curve at t = 0. To nd a poin
Math 273
Review/Outline for Exam #2
The second exam is Wednesday, May 14. You are allowed a two 8 1 11 sheets with
2
handwritten notes on both sides, as well as a calculator.
Section 16.5 Cylindrical and Spherical Coordinates You should know the relation
Math 273
Review for Exam #1
1
The rst exam is Wednesday, April 23. For the exam, you are allowed a single 8 2 11
sheet with handwritten notes on both sides, as well as a calculator.
Section 16.1 Denition of a Double Integral: You should know the denition
Math 273
Quiz #3, due Friday, April 18
You must correct and proper notation and show all your relevant work to earn full credit!
1. Let D be the region between the surfaces z = 2 (x2 + y 2 ) and z = 3 (x2 + y 2 ).
Calculate the volume of D, leaving your a
Math 473
Homework Set #3 Solutions
1. Show that fn (x) = x
1 2
n
converges uniformly on [a, b]. What is the limit function
f?
The limit function is f (x) = x2 . To see that, let x be xed. Then x
2
since x is continuous on R, we will have x
1
n
0, and
Math 473
Homework #2 Solutions
1. Suppose f, g : [a, b] R are functions such that f g
= 0. Show that f (x) = g(x)
for all x [a, b]. (That is: if d(f, g) = 0, then f and g are the same function.)
Since f g
= 0, we know that supcfw_|f (x) g(x)| : x [a, b] =
Math 473
Homework Set #1 Solutions
1. Suppose that A R, and f : A R is a function. Let an be a sequence of real
numbers such that an 0. Show that fn : x f (x) + an converges uniformly to f on
A.
Notice that for any x A, |fn (x) f (x)| = |an |. Suppose now
Math 473
Homework Set #6
due Thursday, May 29 at 10 am
1. Suppose that f : [a, b] R and g : [a, b] R are bounded functions. Suppose
> 0 is a xed number, and suppose |f (x) g(x)| < for all x [a, b]. Show that
M (f, S) m(f, S) M (g, S) m(g, S) + 2 for any
Introduction to Analysis on R
1
Basic bits of logic
One of the most important developments in mathematics was the notion of proof, and proofs
are one of the reasons that mathematics is not a science. It may be one of the reasons
that math is such a useful
1
Analysis in Innite Dimensions
2
Denition 1.1.
i=1
(N) is the set of all sequences xn such that
Intuitively, it is useful to think of elements of
will typically only write 2 .
Denition 1.2. Suppose
real number a such that
limn n ai .)
i=1
2
x2 converges.
Math 473
Homework Set #1
due Thursday, April 10 at 10 am
1. Suppose that A R, and f : A R is a function. Let an be a sequence of real
numbers such that an 0. Show that fn : x f (x) + an converges uniformly to f on
A.
2. Suppose g : R R is continuous, and
A
Appendix
In this appendix, we prove the various results that we used earlier about the solutions of the
following parabolic partial dierential equation:
(PDE)
ws (s, t) = wtt (s, t) + Vq (t, w(s, t)
w(0, t) = u(t)
In particular, we want to prove
Theorem
Math 473
Homework Set #5
due Thursday, May 22 at 10 am.
Problem #1 will come in handy when we try to show that the integral satises various
useful properties.
1. Suppose that S R is a bounded non-empty set. Do not assume that S consists of
only positive o
Math 473
Homework Set #4
due Thursday, May 1
1. (a) Show that if an is a bounded sequence of positive real numbers and bn is a sequence
of positive real numbers that converge to b 0, then lim sup(an bn ) = b lim sup an .
(Hint: recall that
lim sup an = su
Math 273
Quiz #1, due Friday, April 4
1. Mathography: Please answer the following questions:
(a) What is your hometown and where did you go to high school?
(b) Why are you taking this course? Just saying its required or its a pre-requisite
is insucient. F
Math 273 Quiz #4, due Friday, May 2
To earn full credit, you must use correct notation and show your relevant work!
1. Suppose R is the region inside a sphere of radius 1 and above the xy-plane. If R has
density (x, y, z) = (x2 + y 2 )z, calculate the z-c