# Università degli Studi di Napoli "Parthenope"  ## Teaching schedule

2018/2019
Belonging course:
Disciplinary sector:
MATHEMATICAL ANALYSIS (MAT/05)
Language:
Italian
Credits:
9
Year of study:
1
Teachers:
Cycle:
First Semester
Hours of front activity:
72

Italian

### Course description

The course aims to provide the basic knowledge of mathematics and the most suitable computing techniques to address the application of mathematics to economics, finance and statistics.

Expected learning outcomes.

Knowledge and understanding
The student will demonstrate knowledge of the mathematical tools to model and solve economic financial and business problems. In particular, the student has to be able to represent the graph of a function of a real variable; to evaluate indefinite, definite and improper integrals; to solve linear systems, to perform operations between vectors and matrices, to compute the rank of an array; to compute the partial derivatives of functions of two variables.

Applying Knowledge and understanding
The student has to be able to apply the mathematical techniques to real economic, business and financial problems. In particular, the student has to be able to solve optimization problems and to evaluate elementary integrals..

Making judgments
The student has to be able to formulate a problem in a mathematical approach to describe a real phenomenon.

Communication
The student has to be able to answer the oral test questions, showing his ability to express and formalize mathematical concepts. He has to be able to explain the techniques learned to solve the questions of the written exam.

Lifelong learning skills
The student has to develop the ability to use mathematical tools to solve application problems.

### Prerequisites

Set theory. Naturals, integers, rational and real numbers. Equations and inequalities of the 1st and 2nd degree. Elements of analytical geometry (equation of a line, parallel and perpendicular lines). Parthenope University of Naples provides beginners with pre-courses of Mathematic (in September).

### Syllabus

1st PART (24 hours)
Functions
Injective, surjective and bijective or invertible functions. Composition of functions.
Global maxima and minima of a function. Supremum and infimum of a function. Monotone functions. Graphic. Domain.

Elementary functions
Linear function. Absolute value function. Power, square root, exponential, logarithm functions. Trigonometric and inverse trigonometric functions. Equations and inequalities with elementary functions.
Limits
Limit of a function. Limits theorems. Computing limits. List of common limits. Indeterminate forms.
Continuity
Continuous function. Classification of discontinuities. Weirstrass theorem (statment). Bolzano theorem (statment).

2st PART (24 hours)
Differential calculus
Derivability of a function on a point and its geometric meaning. Elementary derivatives and derivative rules.

Applications of differential calculus
Test of monotonicity. Local maxima and minima of a function. Convex and concave functions. Test for convexity/concavity. De l’Hopital theorem(statment). Asymptotes of a function. Study of the graphic of a function.

Infinite and infinitesimal functions.

3rd PART (24 hours)
Integral computation
Primitives of a function. Indefinite integral. Immediate integral table. Integration by parts. Integration by substitution. Definite integral. Improper integral.

Linear Algebra
Vectors, operations with vectors and their properties. Linear dipendence and independence of vectors. Matrices, operations with matrices. Determinant of a square matrix . Rank of a matrix. Inverse matrix. Linear systems: Gaussian elimination method. Rouché-Capelli theorem (statment).
Introduction to real functions of two variables
First and second order partial derivatives. Hessian matrix. Maxima and minima of functions of two variables.

Economic applications: maximizing profit, minimizing costs.

### Teaching Methods

The course includes frontal and exercises aimed at the use of mathematical methods studied, with students interaction.

### Textbooks

De Angelis P.L. (2015) Matematica di base. Giappichelli Editore, Torino.

Other references:
- Robert A. Adams (2018), Christopher Essex. Calculus: A Complete Course. Pearson Education Canada.

- Patrick Roger (2013). Analysis and Linear Algebra for Finance, Part I, bookboon.com.

### Learning assessment

The assessment is based on written examination and an oral examination. The written text (duration 90 minutes) is composed of exercises in order to assess the achievement by the student of the learning objectives. It is divided into 4 exercises related to the topics of the program: study of function (12 points), resolution of a linear system of equations (6 points), calculus of an integral (6 points), partial derivatives of a function of two variables and classification of the critical points (6 points). The oral exam focuses on the theoretical topics dealt with during the course and it is designed to evaluate the student's ability to express and formalize mathematical concepts.
Only those who have passed the written test with a minimum vote of 18/30 can hold the oral test. The student must obtain at least 4 point for the first exercise and at least 2 points for the other exercises. The vote of the examination is results of written and oral examination.
During the examination, the use of notes, books and informatics devices (smartphone, tablet, pc, ecc.) is not allowed.