Mathematics 1 & 2  A.Y.2017/2018
Università degli Studi Ca' Foscari of Venice,
A.Y.20172018, Department of Management, curriculum Business
Administration and Management.
Textbook
Knuth Sydsæter, Peter Hammond, Arne Strøm &
Andrès Carvajal, Essential Mathematics for Economic
Analysis, Pearson, 2016 (V edition).
Teaching material
Outline of the topics covered in class
References to the paragraphs of the textbook are given in
parentheses, when possible.

Lecture 1  September 18, 2017: 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).

Lecture 2  September 19, 2017: still 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.

Quadratic functions (4.6).

Power functions (4.8).

Exponentials and logarithms (4.9, 4.10).

Lecture 3  September 20, 2017: still recalls of selected
topics from chapters 1 to 5 of the textbook.
Introduction to limits.

Still exponentials and logarithms.

Properties of logarithms.

Piecewise defined functions. An economic example of
such a function can be found in Pound Drop, on this
web site.

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. A summary of the
introduction to limits, togheter with exercises, can be
found in A brief
introduction to limits.

Lecture 4  September 25, 2017: Limits.

Limits for one variable functions.

Continuity.

Lecture 5  September 26, 2017: limits. Introduction to
the concept of derivative.

Still algebra of limits: indeterminate forms.

Rules of limits.

The strength of infinities.

The slope of a curve: for this subject you can use the
pdf A dash on
derivatives, that you can found on this web
site.

Lecture 6  September 27, 2017: 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).

Composite functions (chain rule) (6.8).

Lecture 7  October 3, 2017: more on differentiation. One
variable optimization. Mockups of first partial.

More on composite functions (chain rule) (6.8).

Compound or piecewise defined functions.

Kinks.

Increasing and decreasing functions (6.3).

Linear approximations (7.4).

Extreme points (8.1). A summary on one variable
optimization can be found in "One variable
optimization", a pdf you can find on this web
site

Local extreme points (8.6).

Mockups of the first partial. All the proposed mockups,
also for the next lectures, can be found on this web
site in the page
of A.Y. 2016/2017.

Lecture 8  October 4, 2017: more on differentiation. One
variable optimization. Mockups of first partial.

Higher order derivatives (6.9).

Convex and concave functions (6.9).

Inflection points (8.7).

Increasing and decreasing functions (6.3).

L'Hôpital's rule (7.12).

Higher order approximations (7.5).

Sample exercises.

Mockups of the first partial. All the proposed mockups
can be found on this web site in the page of A.Y.
2016/2017.

Lecture 9  October 9, 2017: antiderivatives. Mockups of
first partial. See, on this web site, the pdf A dash of integrals for a
summary of integration theory.

Indefinite integrlas (9.1).

Some important integrals.

Some general rules.

Sample exercises.

Mockups of the first partial. All the proposed mockups
can be found on this web site in the page of A.Y.
2016/2017.

Lecture 10  October 10, 2017: more on antiderivaties,
definite integral. Mockups of first partial.

Integration of composite functions.

Integration by parts (9.5).

Area and definite integrals (9.2).

A geometrical interpretation of definite integrals.

Sample exercises.

Mockups of the first partial. All the proposed mockups
can be found on this web site in the page of A.Y.
2016/2017.

Lecture 11  October 11, 2017: conclusions on definite
integrals, improper integrals. Mockups of first partial.

Properties of definite integrals (9.3).

More on areas and definite integrals.

Areas involving piecewise defined functions.

Improper integrals (9.7).

Infinite intervals of integration.

Integration of unbounded functions.

Sample exercises.

Mockups of the first partial. All the proposed mockups
can be found on this web site in the page of A.Y.
2016/2017.

Lecture 12  October 16, 2017: basics of Financial
Mathematics: use the pdf Basics
of Financial Mathematics, that you can find on this web
site. Mockups of first partial.

Further formulas concerning the exponential and
logarithmic functions with a general base.

Final observations about maxima and minima, in
particular concerning the Extreme Value Theorem.

Areas between the graphs of two continuous functions.

The Fundamental Problem of Financial Mathematics.

Interest rate, Present Value, Future Value,
accumulation factor, actualization factor.

Financial regimes and the compound interest.

Subdivision of the period in subperiods and the
effective rate of interest.

Continuous compounding.

Mockups of the first partial.

Lecture 13  October 17, 2017: basics of Financial
Mathematics: use the pdf Basics
of Financial Mathematics, that you can find on this web
site. Mockups of first partial.

Streams of Cash Flow.

Geometric Progressions.

Annuities. Ordinary annuities. Due annuities.

Mockups of the first partial.

Lecture 14  October 18, 2017: Mockups of first partial.

Mockups of the first partial.

Lecture 15  October 23, 2017: Mockups of first partial.

Mockups of the first partial.

Lecture 16  November 6, 2017: Introduction to the functions
of two variables.

Natural domain of a two variables function (11.1).

Plotting the graph of a two variables function.

Partial derivatives (11.2) (skip the formal definition).

Lecture 17  November 7, 2017: local optimization in two
variables.

More on partial derivatives.

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

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 9, 2017: global optimization in two
variables. In addition to the textbook use the notes
Global maximum and minimum for two
variables functions.

Further observations on the theorem of Weierstrass.

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). Exercises and examples.

Lecture 19  November 13, 2017: global optimization in two
variables. In addition to the textbook use the notes
Global maximum and minimum for two
variables functions.

Examples and exercises involving global optimization in
the case where the equation of the border can be used to
reduce a two variables function to a single variable
function.

Discussion on the first partial.

Lecture 20  November 14, 2017: constrained optimization in
two variables. In addition to the textbook use the notes
Lagrangian multipliers
(20172018 edition).

Constraints.

The Lagrangian multiplier method (14.1).
 Exercises and examples.
 Discussion on the first partial.

Lecture 21  November 16, 2017: more on constrained optimization in
two variables.

Bordered Hessian and local constrained maxima and minima.
 The tangent plane to a surface.
 Final exercises and examples on optimization
problems.

Lecture 22  November 23, 2017: 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 27, 2017: 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 in general: minors, complementary
minors, cofactors, expansion by cofactors (16.3, 16.4,
16.5, partly).

Introduction to matrix inversion.

Lecture 24  November 28, 2017: matrix inversion; rank of a
matrix.

The inverse of a matrix (16.6, partly).

The rank of a matrix.

Lecture 25  November 30, 2017: linear systems and matrix
form.

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

Coefficients matrix and augmented matrix.

The RouchéCapelli's theorem. Use the notes:
The theorem of Rouché and Capelli.

Solving linear systems with Cramer's rule.

Solving linear systems using the inverse matrix.

Lecture 26  December 4, 2017: solving general linear
systems, parametric linear systems.

The solution of a linear system in the general case.

Parametric linear systems.

Lecture 27  December 5, 2017: more on parametric linear
systems, linear dependence and
independence.

More on the solution of a parametric linear system.

Introduction to linear dependence and independence of a
set of vectors.

Lecture 28  December 7, 2017: linear dependence and
independence.

Lecture 29  December 11, 2017: tips for the second partial.

A brief summary on the theory of two variables
functions.

Mockups of the second partial.

Lecture 30  December 12, 2017: tips for the second partial.

A brief summary of Linear Algebra.

Mockups of the second partial.
Homeworks.
Old exams.
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.
See the page of A.Y.
2016/2017 for a collection of exercises, mockups and old
exams.
Practice lessons.
You can find the exercises solved or proposed during practice
lessons by prof. Somenzi Damiano Marino on his I.S.A. section of
Unive Platform (http://static.unive.it/isa/index/docente/persona/10594999).
You must use your username and password.
copyright 2017 et seq. luciano battaia