Problem 1
Let cfw_xn be a real bounded sequence that is monotonically increasing.
n=1
Dene the set X cfw_x|n where x = xn . Show that limn xn = sup X .
Proof.
The solution follows exactly the logic of the proof of Theorem 3.11. X is
a bounded real set an
Problem 1
A, A Rmn . Show that similarity between matrices is an equivalence relation on
the set Rmn .
Proof.
Note that A A an invertible Q Rmm , P Rnn |A = Q 1 AP
Reexive: Choose Q = Im = Q 1 , P = In . Then Q 1 AP = A = A A.
Symmetric: A A = Q Rmm , P R
Final Solutions
John Rehbeck
October 11, 2013
Point Breakdown
Q1: 10 points (New Solution)
Q2: 10 points (Old solution)
Q3: 5 points (New Solution)
Q4: 15 points (New Solution)
Q5: 10 points (New Solution)
Q6: 10 points (Old Solution)
Q7: 5 points (New So
Problem 1 (part 1)
Let (E , T ) be a toplogical space, and A, B E . Then,
A B = (A B) .
Proof.
First the right inclusion (),
x A B = x A x B
= Oa , Ob T | (x Oa A x Ob B)
= (Oa Ob T ) (x Oa Ob A B)
= x (A B)
To demonstrate left inclusion ()
x (A B) = Oab
Quiz 1 Solutions
John Rehbeck
September 9, 2013
Question 1
Theorem
Let cfw_xn be a real sequence that converges to x. Show that if for all
n=1
n N that xn b, then x b. Is it true that xn > b, then x > b ?
Show why.
Proof.
To prove this, we use proof by c
Economics 205 Quiz 2 Solutions
Joel Watson, Fall 2014
1. Suppose that a function f : R R is integrable and, for any real numbers a and b with a < b,
dene F : [a, b] R by
x
F (x)
f (t)dt.
a
Explain why there must be a number x (a, b) such that F (x ) = F
Outline
Question 1
Question 2
Quiz 2, 2011
Paul Feldman
August 29, 2012
Question 3
Outline
Question 1
Question 2
1
Question 1
Ao B o (A B)o
(A B) A B
Pathological Examples
2
Question 2
Cases we will consider
a R and L R
a R and L =
a = and L R
a = and L
Quiz 2 Solutions
John Rehbeck
September 17, 2013
Question 1
Theorem
Let A R. Show that a A if and only if there exists a sequence cfw_an
in A such that limn an = a.
Proof.
() If a A then we know that for all V nbhd of a that V A = .
1
1
In particular, we
Problem 1
Consider v Rn and the line V Rv. Show that V is a subspace of Rn .
Proof.
0 = 0v = 0 V
Closed under addition.
y1 , y2 V = 1 , 2 R|y1 = 1 v y2 = 2 v
= y1 + y2 = (1 + 2 )v
= (y1 + y2 ) V .
Closed under scalar multiplication.
y1 V = y1 = v
= y1 = (
Econ 205 Quiz Solutions
Quiz 1
1. Give a denition of a weak order (recall that an order is weak if it is complete and
transitive). Consider the following relation on R2 :
v1
v2
f (v1 ) f (v2 ) where f : R2 R.
Show that this is a weak order. Under what add
Econ 205 Quiz Solutions
Quiz 2
1. Let (E, T ) be a topological space, and let A E, B E. Show that
Ao B o (A B)o
and (A B) A B.
Give examples in R to show that each of these inclusions can be strict.
Solution:
First examine Ao B o (A B)o ,
x Ao B o (OA o
Economics 205 Quiz 1
Joel Watson, Fall 2008
1. For both cases below, write expressions for h (x).
(a) h(x) (x + 2)2 [1 xf(x)]
(b) h(x) f(g(x2 + 2)
2. Calculate the following limits. In the case in which one or both of the limits does not
exist, state this
Economics 205 Final Examination
Professors Sobel and Watson, Fall 2008
You have three hours to complete this closed-book examination. You may use scratch paper, but please write your nal answers (including your complete arguments) on these sheets.
Calcula
Economics 205 Quiz 2
Joel Watson, Fall 2008
1. Consider the function f : (0, 00) —> R deﬁned by f(ac) E :52 — 820 + 6lnac.
(a) Calculate the ﬁrst four derivatives of f.
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WOW 17“ 8 “’7‘
PC3300: 12X—
Fm“) : 4926*
(b) Using your observation of the pattern
Economics 205 Quiz 2
Joel Watson, Fall 2008
1. Consider the function f : (0, ) R dened by f (x) x2 8x + 6 ln x.
(a) Calculate the rst four derivatives of f .
(b) Using your observation of the pattern that develops in part (a), write an expression
for f (k
ECON 205 - Past Exam and Quiz Questions
Copyright Ivana Komunjer 2014
These notes are for use of ECON 205 students. They are not to be distributed elsewhere.
1
Quiz I
1. Give a denition of a weak order (recall that an order is weak if it is complete and t
Problem 1
Let E be a set and P(E ) its power set (Note that elements of the power set of E are just subsets
of E ).
a) Show that is a partial order on P(E ).
Proof.
Reexivity: X E , X X .
Anti-symmetry: X , Y E , (X Y ) (Y X ) = X = Y .
Transitivity: X ,
Problem 1
Give the denition of a Weak Order (complete and transitive).
is a Weak Order on set S if it is:
Complete: x, y S|x = y , we have (x y y x).
Transitive: x, y , z S, we have (x y y z) = x
z.
For an arbitrary function f : R2 R. Show the relation on
Problem 1
f (x) = e x . Write nth order Taylor approximation to f (x) around c = 0. Choose n to
e
make En < 24 , x (1, 1).
Note that,
i N, f (i) (x) = e x ,
f (i) (c) = 1.
Then, the Taylor expansion is,
n
f (x) =
i=0
where En =
of En ,
f
(n+1)
(c ) n+1
x
Quiz 2 Solutions
John Rehbeck
September 23, 2013
Question 1
Theorem
Let f : [a, b] R (a < b) be Riemann integrable. Show that the
x
function F : [a, b] R, F (x) = a f (t)dt is continuous on [a, b].
Proof.
Note that f Riemann integrable implies that f (x)