Exercises 1 Structures, introduction to functions
Multiple-choice questions
1) Consider the two intervals A = [ 0,3] , B = ( 3, 7 ) ; then A B
[ a ] is the union of two intervals, therefore it is not an interval
[b] is an interval, but it is neither open
Chapter 16
Problems of optimum
16.1
Generalities
The problems of optimum are fundamental in Economics, whose study is based on
the analysis of the problems of optimum of the economic agents, such as individuals
(consumers, producers and investors), famili
UNIVERSIT BOCCONI - A.Y. 2012/2013
30062 GENERAL MATHEMATICS - HOMEWORK _
Instructions
Multiple-choice questions and open-answer questions will be corrected in a classroom by the
course tutor, according to the schedule prepared by each tutor. As for the t
General Mathematics (30062)
Midterm Test - Simulation
Surname
Bachelor program
Name
Exam code*
ID code
BIEMF
Class
* 30062 for I-II-III year students; otherwise 6230, 6007, 5131, 5015
Rules of conduct
I hereby undertake to respect the regulations describe
Mathematics (6007), Midterm Test, November 7th 2006
I Shift
Group: A
1 The graph of the function f (x) = 1
log0:5 (x) is:
.
.
5
f( x)
5
0
2
f( x)
4
5
0
5
5
a)
b)
x
x
.
.
5
5
f( x)
f( x)
5
0
2
4
5
0
5
5
5
c)
d)
x
2 The natural domain of the function f (x)
MATEMATICA GENERALE - A.A. 2014/2015
(codice 30062, CLEAM - CLES - CLEF - BIEM - BIEF)
PROGRAMMA DETTAGLIATO
Prima parte
Strutture ( 1.1-5, 2.1-3, 4.1, 5.1-4, 5.6, 8.6, 13.1)
Insiemi, operazioni con gli insiemi, insiemi numerici. Linsieme R dei numeri rea
American Economic Association
Population, Technology, and Growth: From Malthusian Stagnation to the Demographic
Transition and beyond
Author(s): Oded Galor and David N. Weil
Source: The American Economic Review, Vol. 90, No. 4 (Sep., 2000), pp. 806-828
Pu
Malthus Was Right after All: Poor Relief and Birth Rates in Southeastern England
Author(s): George R. Boyer
Source: Journal of Political Economy, Vol. 97, No. 1 (Feb., 1989), pp. 93-114
Published by: The University of Chicago Press
Stable URL: http:/www.j
Economics 2099 Market Design
Scott Duke Kominers
Logistics
Time. Tuesdays, 16:0018:45 (beginning September 8, 2015).
Location. Littauer M-17.
Oce Hours.
By appointment Please email kominers@fas.harvard.edu to schedule, using the subject line 2099
Oce Hou
Communication Complexity
Alexander A. Razborov
Abstract. When I was asked to write a contribution for this book about
something related to my research, I immediately thought of Communication
Complexity. This relatively simple but extremely beautiful and i
Stat 310A/Math 230A Theory of Probability
Homework 2
Andrea Montanari
Due on 10/7/2015
Solutions should be complete and concisely written. Please, use a separate sheet (or set of sheets) per
each problem. Staple sheets referring to the same problem, and w
Stat 310A/Math 230A Theory of Probability
Homework 1
Andrea Montanari
Due on September 30, 2015
Solutions should be complete and concisely written. Please, use a separate sheet (or set of sheets) for
each problem. Staple sheets referring to the same prob
Homework 5 Limits of functions and continuity
Multiple-choice questions
x4
x
1) Let f ( x)
. Then lim f ( x)
x 4
x4
3
a 3
c does not exist
b 4
2) The function f ( x)
a
x0
ln( x 1)
x
d none of the preceding
has the following straight line as asymptote
UNIVERSIT BOCCONI - A.Y. 2012/2013
30062 GENERAL MATHEMATICS - HOMEWORK _
Instructions
Multiple-choice questions and open-answer questions will be corrected in a classroom by the
course tutor, according to the schedule prepared by each tutor. As for the t
Chapter 14
Concave functions
14.1
Convex sets
In Economics it is important to be able to combine among them the dierent alternatives among which the decision makers must choose. For example, if x and y are
bundles of goods or vectors of input, we want to
Exercises 1 Structures, introduction to functions
Multiple-choice questions
1) Consider the two intervals A 0,3 , B 3, 7 ; then A B
a is the union of two intervals, therefore it is not an interval
b is an interval, but it is neither open nor closed
c
Exercises 3 Limits of sequences
Multiple-choice questions
1) For which values of n is the sequence an = n 2 12n positive and strictly increasing?
[a]
n 0
[b]
[c]
n 12
n 13
[d ] none of the preceding
a0 = 2
2) The term a3 of the recursively defined sequenc
Exercises 3 Limits of sequences
Multiple-choice questions
1) For which values of n is the sequence an = n 2 12n positive and strictly increasing?
[a] n 0
[b] n 12
[c] n 13
[d ] none of the preceding
a0 = 2
2) The term a3 of the recursively defined sequenc
Exercises 2 One-variable and n-variable functions
Multiple-choice questions
1) Let f ( g ( x) = e7 x 6 . Then:
[a] f ( x) = 7 x 6, g ( x) = e x
[c] f ( x) = e7 x , g ( x) = e6
[b]
[d ]
f ( x) = e x , g ( x) = 7 x 6
f ( x) = e7 x , g ( x) = x 6
3
2) Let f
Exercises 2 One-variable and n-variable functions
Multiple-choice questions
1) Let f ( g ( x) = e7 x 6 . Then:
[a] f ( x) = 7 x 6, g ( x) = e x
[c]
f ( x) = e7 x , g ( x) = e 6
b f ( x) = e x , g ( x) = 7 x 6
[d ] f ( x) = e7 x , g ( x) = x 6
3
2) Let
Exercises 4 Number series
Multiple-choice questions
4 ( 1)
1) Consider the series
n
n =0
[a] 1
. Let sn be the sequence of its partial sums. Then s4 =
[b ] 4
3
22
2) The series
n =0
[b ]
n=0
[b ]
2<k <4
3
2
[ d ] none of the preceding
[c]
2
3) For which
Exercises 4 Number series
Multiple-choice questions
4 ( 1)
1) Consider the series
n
n =0
[a] 1
. Let sn be the sequence of its partial sums. Then s4 =
[c] 0
b 4
3
22
2) The series
n =0
[b ]
n=0
[b ]
2<k <4
[ d ] none of the preceding
[c]
2
3) For whic
Exercises 5 Limits of functions and continuity
Multiple-choice questions
1) If the graph of f is
1
=
x 0 f ( x )
b
then lim
a
sin( x ) 3
2
2) lim
a
x
x
ln( x 1)
3) Let f ( x) 1
e x
a
b
0
ln(1 x )
2
4) lim
x 0
a
x 0
x sin 3 x
ln 1 x x
0
c
x0
x0
d it doe
Exercises 5 Limits of functions and continuity
Multiple-choice questions
1) If the graph of f is
1
=
x 0 f ( x )
a
b
then lim
sin( x ) 3
2
2) lim
x
a
x
ln( x 1)
3) Let f ( x) 1
e x
a
b
0
ln(1 x )
2
4) lim
x 0
a
x 0
0
x sin 3 x
d none of the precedi
UNIVERSIT BOCCONI - A.Y. 2012/2013
30062 GENERAL MATHEMATICS - HOMEWORK _
Instructions
Multiple-choice questions and open-answer questions will be corrected in a classroom by the
course tutor, according to the schedule prepared by each tutor. As for the t
UNIVERSIT BOCCONI - A.Y. 2012/2013
30062 GENERAL MATHEMATICS - HOMEWORK _
Instructions
Multiple-choice questions and open-answer questions will be corrected in a classroom by the
course tutor, according to the schedule prepared by each tutor. As for the t
Chapter 22
Complements
22.1
Study of functions
22.1.1
Inection points
Denition 601 Let f : A R R and x0 A A . The function f is said to be
(i) concave at the point x0 if there exists a neighbourhood of this point (eventually
only a right-neighbourhood or
UNIVERSIT BOCCONI - A.Y. 2012/2013
30062 GENERAL MATHEMATICS - HOMEWORK _
Instructions
Multiple-choice questions and open-answer questions will be corrected in a classroom by the
course tutor, according to the schedule prepared by each tutor. As for the t