Mathematics 1 & 2  A.Y.2016/2017
Università degli Studi Ca' Foscari of Venice,
A.Y.20162017, Department of Economics, curriculum Economics,
Markets and Finance.
***** Ca' Foscari  Harvard Summer School *****
On Monday November 28 prof. Paolo Pellizzari, during our class,
introduced the Ca' Foscari Harvard Summer School, in
particular the Course Math 1b, of which you can download the Syllabus. Other info on the web
page www.unive.it/pag/10033/.
All parts of the prerequisites have been studied in our course,
except for trigonometry, so no problem if you want to take this
great and important opportunity. If any student wants to enroll
in this course (as I hope!!) and
has the need to review trigonometry, please tell me: I can help
and, among other things, I can write a brief summary of the
needed concepts of trigonometry.
Textbook
Knuth Sydsæter, Peter Hammond, Arne Strøm &
Andrès Carvajal, Essential Mathematics for Economic
Analysis, Pearson, 2016 (V edition).
Office hours
Mathematics 1: Tuesday and Wednesday
3.15pm to 3.45pm and 5.15pm to 5.45pm, room D0004, complex D, only during the period of
the lectures.
Mathematics 2: Tuesday and Thursday 3.15pm to 3.45pm
and 5.15pm to 5.45pm, room
D0004, complex D, only during the period of the lectures.
Teaching material

Approaching the
course.

Saying maths: hints for
reading numbers and mathematical formulas in english and
american english (this material is found on this web site).

Precalculus,
by Luciano Battaia, Giacomo Bormetti and Giulia Livieri: A
prelude to calculus with exercises (this material is found on
this web site). Notes for a crashcourse in mathematics. This
booklet can help in revising basic maths concepts, and
contains mainly the same topics as in the first five chapters
of the textbook.

A brief introduction to limits.
These notes supplement the concept of limit, treated in
paragraphs 6.5 and 7.9 of the textbook.

Limits and
continuity: just the gist: a concise introduction
to limits and continuity, that can be used to deepen the same
concepts treated in "A brief introduction to
limits" (this material is found on this web site).

A dash on derivatives. These
notes are to be considered as a summary of some concepts
included in chapters 6 and 7 of the textbook.

One variable optimization. These
notes are to be considered as a summary of chapter 8 of the
textbook.

A dash of integrals. These notes
are to be considered as a summary of chapter 9 of the
textbook.

Global maximum and minimum for
two variables functions. These notes are to be considered as
a supplement of paragraph 13.5 of the textbook.

Lagrangian multipliers.
These notes are to be considered as a supplement of paragraph
14.1 of the textbook.

Constrained
Optimization in two variables. Notes, in italian, used
for another course at Ca' Foscari: you can find here
additional info and solved exercises.

The theorem of
Rouché and Capelli. These notes are a supplement
of the textbook.

Linear dependence or
independence. These notes are a supplement of the
textbook.
Outline of the topics covered in class
References to the paragraphs of the textbook are given in
parentheses, when possible.

Lecture 1  September 19, 2016: recalls of selected topics
from chapters 1 to 5 of the textbook. It is important to
remember that all the topics contained in these chapters
are considered known.

Approaching the course.

Informations about the software packages that can help
during the course, in particular Wolfram Alpha and
Geogebra. As far as Geogebra is concerned the software
can be used directly on line in a browser's window
or downloaded and installed to your personal computer.
There is also a version suitable for use on portable
media without leaving setting files on the host
computer: see the web site geogebra.org.

Basic definitions about functions (4.2, 4.3).

Linear functions (4.4).

Quadratic functions (4.6).

Power functions (4.8).

Lecture 2  September 20, 2016: still recalls of selected
topics from chapters 1 to 5 of the textbook.

Exponentials and logarithms (4.9, 4.10).

Some very simple exponential and logarithmic
inequalities.

Piecewise defined functions (or compound functions)
(5.4).

New functions from old: sums, products and quotients
(5.2, partly)

Graphs of equations, in particular circles: differences
compared to the graphs of functions (5.4).

Lecture 3  September 21, 2016: conclusion of recalls of
selected topics from chapters 1 to 5 of the textbook.
Introduction to limits.

Injective functions.

Surjective functions.

Bijective or onetoone functions; properties of graphs
of such functions.

Composite functions (5.2, partly).

A graphical approach to limits (6.5, partly). The
limits will be treated in a somewhat different way from
the textbook. Geogebra has been used to illustrate in a
simple way what a limit is. You can download the files
used: FirstExampleOnLimits, SecondExampleOnLimits.

Lecture 4  September 26, 2016: limits and continuity.
Introduction to derivatives.

Definition of limit, continuity, horizontal and
vertical asymptotes, always using a graphical approach,
as discussed in "A brief introduction to
limits".

First exercises on limits and continuity.

A graphical approach to the concept of derivative:
tangents to curves. For this topic you can use the
geogebra file SecantTangent. Directions for
using this software: using the mouse move the point P
on the given curve, the point Q moves accordingly as
the difference between x(P) and x(Q) is given by the
number a. Changing the value of a on the slider (click
on the slider and then use the left and right arrows on
the keyboard), Q moves accordingly. When a is different
from 0, you can see the secant line (in black) through
P and Q. When a reaches 0, in what case Q is over P,
the secant disappears and you can see a new line in
red: this line is exactly the tangent to the curve. By
tiping a new function in the input line you can
experiment with different curves. Pay attention
while typing a new function: the abscissa of the point
P must be in domain of the new function, otherwise the
point P itself (and consequently Q) becomes undefined.
Functions to test (type exactly the written formulas in
the input line, or copy and paste them):

f(x)=ln(x)

f(x)=exp(x)

f(x)=sqrt(x)

f(x)=1/x

f(x)=x^32x^2+1/2

Lecture 5  September 27, 2016: differentiation. A concise
summary on differentiation can be found in "A dash on
derivatives".

Slopes of curves (6.1).

Tangents and derivatives (6.2).

Simple rules for differentiation (6.6).

Sums, products and quotients (6.7).

Exponential functions (6.10, partly).

Logarithmic functions (6.11, partly).

Lecture 6  September 28, 2016: more on differentiation.

Composite functions (chain rule) (6.8).

Compound or piecewise defined functions.

Kinks: to understand the graphical meaning of
"kink" you can use the geogebra file Kink.

Increasing and decreasing functions (6.3).

Sample exercises.

Lecture 7  October 3, 2016: limits in practice.

Rules for limits (6.5).

Indefinite forms and simple resolution techniques.

Sample exercises.

Lecture 8  October 4, 2016: more on derivatives.

Linear approximations (7.4)

Higher order derivatives (6.9).

Convex and concave functions (6.9).

Inflection points (8.7).

L'Hôpital's rule (7.12).

Sample exercises.

Lecture 9  October 5, 2016: onevariable optimization.

Extreme points (8.1).

Tests for extreme points (8.2).

The extreme value theorem (8.4).

Local extreme points (8.6).

Sample exercises.

Lecture 10  October 10, 2016: antiderivatives.

Indefinite integrlas (9.1).

Some important integrals.

Some general rules.

Integration of composite functions.

Sample exercises.

Lecture 11  October 11, 2016: more on antiderivaties,
definite integral.

Integration by parts (9.5).

Area and definite integrals (9.2).

A geometrical interpretation of definite integrals.

Area of the region between two curves.

Sample exercises.

Lecture 12  October 12, 2016: conclusions on definite
integrals, improper integrals.

Properties of definite integrals (9.3).

Improper integrals (9.7).

Infinite intervals of integration.

Integration of unbounded functions.

Sample exercises.

Lecture 13  October 17, 2016: polynomial approximations,
review exercises.

Polynomial approximations (7.5).

Exam type exercises.

Lecture 14  October 18 2016: review exercises.

Lecture 15  October 19, 2016: review exercises.

Lecture 16  November 7, 2016: functions of two variables.

Functions of two variables (11.1).

Partial derivatives with two variables ( 11.2) (skip
the formal definition).

Geometric representation, graphs, level curves,
geometric interpretation of the partial derivatives
(11.3).

Lecture 17  November 8, 2016: local optimization in two
variables.

The Young's theorem (11.6, partly).

Two variables optimization: necessary conditions (13.1).

Two variables optimization: sufficient conditions
(13.2).

Local extreme points (13.3).

Lecture 18  November 10, 2016: global optimization in two
variables. In addition to the textbook use the notes Global maximum and minimum for two
variables functions.

The equation of the tangent plane to a surface (12.8,
partly).

Bounded, open and closed subsets of the plane and the
extrem value theorem (13.5).

Finding global maximum and minimum using the equations
of the boundary (13.5 partly).

Lecture 19  November 14, 2016: constrained optimization in
two variables. In addition to the textbook use the notes Lagrangian multipliers.

Global maximum and minimum for a two variables function
in unbounded sets in simple cases.

The Lagrangian multiplier method (14.1).

Exercises.

Lecture 20  November 15, 2016: constrained optimization in
two variables.

Bordered Hessian and second order sufficient conditions
for local constrained maxima and minima (14.5 partly
and 16.2 partly).

Exercises.

Lecture 21  November 17, 2016: multivariable optimization.

Use of level curves for problems of two variables
optimization, both constrained and not.

Exercises.

Lecture 22  November 22, 2016: matrices and operations
between matrices.

Systems of linear equations (15.1).

Matrices and matrix operations (15.2).

Matrix multiplication (15.3).

Rules for matrix multiplication (15.4).

Lecture 23  November 24, 2016: matrices and determinants.

The transpose of a matrix (15.5).

Symmetric matrices (15.5).

Vectors and operations on vectors (15.7).

The inner product (15.7).

Determinants of order two (16.1) (execpt geometric
interpretation).

Determinants of order three (16.2) (except geometric
interpretation).

Determinants in general: minors, complementary
minors, cofactors, expansion by cofactors (16.3, 16.4,
16.5, partly).

Introduction to matrix inversion.

Lecture 24  November 28, 2016: matrix inversion; rank of a
matrix.

The inverse of a matrix (16.6, partly).

The rank of a matrix.

Lecture 25  November 29, 2016: linear systems and matrix
form.

The matrix form of a linear system (15.3, partly).

Coefficients matrix and augmented matrix.

The RouchéCapelli's theorem.

Solving linear systems with Cramer's rule.

Solving linear systems using the inverse matrix.

Lecture 26  December 1, 2016: solving general linear
systems.

Lecture 27  December 5, 2016: parametric linear
systems.

The solution of a parametric linear system.

Lecture 28  December 6, 2016: linear dependence and
independence.

Lecture 29  December 12, 2016: mockups of second partial.

Solutions of sample exercises for the second partial:
real functions of two variables.

Lecture 30  December 13, 2016: mockups of second partial.

Solutions of sample exercises for the second partial:
linear algebra.
Homeworks
Selected exercises from the texbook.

Chapter 6: 6.6.3; 6.7.3, 6.7.4; 6.8.3; 6.10.1, 6.10.4;
6.11.3,6.11.6,6.11.7; Review exercises n. 3, 5, 14, 15.

Chapter 7: 7.12.1, 7.12.2, 7.12.3, 7.12.4, 7.12.5.

Chapter 8: 8.2.1, 8.2.5, 8.2.6, 8.2.7; 8.4.2; 8.6.2, 8.6.3;
8.7.2, 8.7.3; Review exercises n. 1, 6, 7.

Chapter 9: 9.1.1, 9.1.2, 9.1.4, 9.1.5, 9.1.6, 9.19,
9.1.10,9.1.11, 9.1.12; 9.2.3, 9.2.4, 9.2.5; 9.3.1, 9.3.2,
9.3.3; 9.5.1, 9.5.2; 9.7.1, 9.7.9; Review exercises n. 1, 2,
3, 4.

Chapter 11: 11.1.6, 11.1.7; 11.2.4, 11.2.5; Review exercises
n. 6, 7, 9, 10, 12.

Chapter 13: 13.3.1, 13.3.2, 13.3.3; 13.5.1, 13.5.2; Review
exercises: 4.

Chapter 14: 14.1.1, 14.1.3.c, 14.1.4.
Exercises sheets used during practice sessions.
Mockups of partial examinations.
Old Exams.
Tips.
Notebooks and pdfs used during the lectures.
The following teaching materials have been used during the
lectures. They contain no explanation, so they can't be used
without the support of the classroom lesson. There are two kinds
of materials: the Notebooks are dinamical pages that
must be downloaded and then opened with the program
Mathematica of Wolfram (commercial and copyrighted
software!) or with CDF Player, a viewer that can be
downloaded (about 700 Mb) from https://www.wolfram.com/cdfplayer/
and installed on your computer; the pdfs are normal pdf
files that reproduce the same content of the notebooks, but
without interactivity. Many files have explanations in italian,
as they have been used during other courses at Ca' Foscari
University.
copyright 2016 et seq. luciano battaia
first published on 2016/09/18  last updated on 2017/02/06