1
(Math 360) Free Response Final:
April 28, 2009
2
Rules
Preparation:
In preparing for this part of the nal exam you are allowed to use any books
as well as any source you nd on the internet. You are
MATH 360
University of Pennsylvania
Fall 2015
Midterm 1 Study Guide
by Elaine So
1. General tips.
(1) You can use any fact or theorem provided that it does not make the problem trivial. For
example, i
MATH 360 Homework 8
Due 22 March 2013
1.
i. Give a precise meaning to the statement, Disconnections pull back. Is the statement true?
ii. Show that the image of a connected set under a continuous map
MATH 360 Homework 7
Due 15 March 2013
1. Suppose (V; kk) is a normed space. Show that kk : V 3 R is continuous with respect to the metric
induced by kk and the standard metric on R. (Hint. Use the rev
MATH 360 Homework 6
Due 1 March 2013
Denition. Given two metrics d and on the same set M , we say d is equivalent to if there are positive
real numbers C1 and C2 so that for any x, y M , we have
C1 (x
MATH 360 Homework 6.5
Not due, but do!
Dene
R = (x1 , x2 , . . . , xk , . . .) xi R, all but nitely many xi = 0
Note that under pointwise operations, R is a vector space.
1. For x, y R , show that x,
MATH 360 Homework 5
Due 22 February 2013
Denition. A sequence (xn )n2N in a metric space is called eventually
constant if there is some N so that
! N guarantees x = x .
1. Let A & B be subsets of a me
MATH 360 Homework 4
Due 8 February 2013
Recall that our denition of a closed subset diers from Rudins:
Denition. A subset of a metric space A (M, d) is closed if its complement is open.
1.
i. Show tha
MATH 360 Homework 3
Due 1 February 2013
1. Suppose (an )nNis a sequence of real numbers with an A and A > 0. Consider the sequence
bn = n + an n.
i. Let 0 < c < 1 . Show that there is some N N, depend
MATH 360 Homework 2
Due 25 January 2013
1. (Versions of the Archimedean property) Prove that the following properties are equivalent in an ordered
eld, i.e. any ordered eld F which possesses one of th
MATH 360 Homework 1
Due 18 January 2012
Properties of the Real Numbers
1. Show that the following properties hold in any eld.
i. (Unique identities) If a + x = a for some a, then x = 0. If ax = a for
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MATH 360 Homework 11
Due 23 April 2013
1. For a xed n N, consider the sequence of functions n : [0, 1] R, given by: if x = p is a rational
q
number in reduced form, with q n, then n (x) = q12 ; for ot
MATH 360 Homework 10
Due 12 April 2013
1. (Marsden-Homans problem 4.42) For x > 0, dene L(x) =
this denition of L and theorems about the denite integral.
x1
dt.
1t
Show the following using only
i. L(x
1. A - It is possible to place a linear order on C so that we do NOT get
an ordered eld, and impossible to place a linear order of C so that we DO
get an ordered eld.
2. D - |x + y | = |x| + |y | does
1. Suppose f is twice dierentiable on [a, b], f (a) < 0, f (b) > 0, and
f (x) > 0, and 0 f (x) M for all x [a, b]. Let be the unique
point in (a, b) where f ( ) = 0.
f
a) Pick x1 (, b) and dene xn+1 =
MATH 360
University of Pennsylvania
Fall 2015
Midterm Solutions, 1st Draft
by Elaine So
1.a. (2 pts): Let A, B be any (finite) sets. For set powering, we know that |AB | = |A|B| . For disjoint
unions,
MATH508. ADVANCED CALCULUS
LECTURE 5. DIFFERENTIATION AND INTEGRATION
A. A. KIRILLOV
The goal of this lecture is to recall the basic notions and rules from Calculus and to give them rigorous definitio
MATH 360 (ADVANCED CALCULUS I)
FALL 2015. TIME: TU-TH 10:30-12:00
A. A. KIRILLOV
1. Introduction
1.1. Notations and symbols. .
S1 S2 , S2 S1
the statement S1 implies the statement S2
S1 S2
the stateme
Math 360: Homework 9
Due Friday, November 15
Midterm 2 Redux Question:
1. Prove that for any real constants a0 , a1 , a2 ,
x3 + a2 x2 + a1 x + a0
= .
x
x2 + 1
lim
In class on November 5th, we dened th
Math 360: Homework 10
Due Friday, November 22
1. Fix k 1, and suppose g (t) is real-valued and k -times dierentiable on some neighborhood of the
origin in R (we specically do not require that g (k) is
Exam 1
Math 508
October 14, 2014
Jerry Kazdan
9:00 10:20
Directions This exam has three parts. Part A asks for 8 examples (3 points each, so 24
points), Part B has 4 shorter problems, (8 points each s
THE REAL AND COMPLEX NUMBER SYSTEMS 9
The next theorem could be extracted from this construction with very
little extra effort. However, we prefer to derive it from Theorem 1.19 since this
provides a
MATH 360
University of Pennsylvania
Fall 2015
HW #4 Solutions, 1st Draft
by Elaine So
P
1.a. (4 pts): Recall that a power series n=1 an xn is differentiable term-by-term in its radius of conver
1
1
1
MATH 360
University of Pennsylvania
Fall 2015
HW #5 Solutions, 1st Draft
by Elaine So
1.a. (2 pts): Let X be the set of all sequences in N of finite length. Let pn denote the nth prime number.
Then de
MATH 360
University of Pennsylvania
Fall 2015
HW #3 Solutions, 1st Draft
by Elaine So
1.a. (3 pts): Let xn = n define a sequence cfw_xn Z. Suppose that there is a subsequence cfw_xnk cfw_xn
such th