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
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
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(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.
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
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 de
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 s
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 li
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
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
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)
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
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 Denit
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
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
s
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
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.
Notic
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]. S
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.
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
r
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 unif
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)
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-empt
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 b
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 say
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