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

2021/2022
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

Expected learning outcomes.
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.

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).

### Syllabus

Functions-24 h
Injective, surjective and invertible functions. Composition of functions. Global maxima and minima, 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 their inverses. Equations and inequalities with elementary functions.
Limit of a function. Limits theorems. Computing limits. List of common limits. Indeterminate forms.
Continuous function. Discontinuities points. Weirstrass theorem. Bolzano theorem.
Differential calculus-24 h
Derivative and its geometric meaning. Elementary derivatives and derivative rules. Test of monotonicity. Local maxima and minima of a function. Convex and concave functions. Test for convexity/concavity. De l’Hopital theorem. Asymptotes. Study of the graphic of a function.
Integral computation(12 h)
Primitives of a function. Indefinite integral. Immediate integral table. Integration by parts. Integration by substitution. Definite integral. Improper integral.
Linear Algebra-10 h
Vectors, operations with vectors, 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.
Partial derivatives. Critical points-2h

### Teaching Methods

The course includes frontal and exercises aimed at the use of mathematical methods studied, with students interaction. Students can practice and consolidate their skills in the section dedicated to the course on the e-learning platform of the University "Parthenope", where they can find: slides of the lessons, performed questions / exercises, exam tests.

### Textbooks

- Robert A. Adams (2013), Christopher Essex. Calculus: A Complete Course. Pearson Education Canada.
- Serge Lang (2002), Short calculus. Springer-Verlag, New York.
- Patrick Roger (2013). Analysis and Linear Algebra for Finance, Part I, bookboon.com.
- Slides of the lessons on the platform elearning of Ateneo “Parthenope”.

### 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. In particular, the written test aims to verify the acquisition, by the student, of the mathematical tools necessary to study the behavior of a function, analyze the graph and deduce the properties of the function from this analysis, search for maxima and minima of a function; solve integrals, tools that will be used by the student in subsequent exams (such as statistics); apply the elements of linear algebra and in particular solve linear systems of equations; calculate the partial derivatives of functions of two variables. 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 vote of the examination is results of written and oral examination.

During the course there are two partial tests, one in the middle of the semester and one at the end. These tests will be structured in a similar way to the written test. The final vote will be determined as the average of the votes of the two tests. The achievement of a minimum of 18 points determines admission to the oral exam, without taking the written exam.

During the examination, the use of notes, books and informatics devices (smartphone, tablet, pc, ecc.) is not allowed.
Due to Covid-19, the exams will be held in the presence or at a distance according to the indications of the University.