## GENERAL MATHEMATICS

Literal equations, use of brackets. Polynomials. The difference of squares. Expressions with fractions. Identity and equations, notion of solution. First and second-degree algebraic equations. First and second-degree algebraic inequalities. Rational inequalities. Radicals and radical inequalities.

For more detailed and updated information about the exam, please refer to the webpage of the course on the Moodle platform at https://edisea.uniss.it

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The Mathematics exam consists of a written test. Students will have to solve 9 exercises. On the Moodle platform several examples from past exams are available, including solutions. The maximum score for each exercise will be clearly specified in the exam text and the marks are out of 30.

The teacher reserves the right to require the student to take an additional oral examination whenever the autonomy in the written test is dubious (at the discretion of the teacher). In addition, students with a score of 30/30 in the written test are given the option to also take the oral examination for possible award of honors. In this case, oral examination must be required before registering the result. Please note that the initial score from the written exam will be reduced if the oral examination does not confirm the score 30/30.

The duration of the exam is 90 minutes. The use of books, notes, calculators, cell phones ore other devices during the exam is strictly forbidden.

Distance exam modality - COVID-19 period: The exam consists of a written test with up to 10 problems to solve.

Each problem is assigned a maximum score from a minimum of 1 to a maximum of 5, depending on the complexity of the problem.

Maximum scores are clearly specified in the exam text and the marks are out of 30.

The duration of the exam is 60 minutes. The use of books, notes, calculators, cell phones or other devices is strictly forbidden.

For remote examinations, students are asked to make themselves visible accessing a video conference on Google Meet.

The course provides the mathematical tools needed to study economic and financial models.

At the end of the course the student will have acquired:

1) (knowledge and understanding) the ability to solve maximization and minimization problems of functions of one or more variables;

2) (applying knowledge and understanding) the ability to develop and analyze simple mathematical models aimed at describing economic phenomena;

3) (making judgements) the ability to evaluate economic choices;

4) (communication skills) the ability to represent relationships of dependence between economic quantities via graphs;

5) (learning skills) the basic notions of differential calculus required to understand the contents of the first and second year courses in microeconomics and macroeconomics.

For a more detailed and updated syllabus, please refer to the webpage of the course on the Moodle platform at https://edisea.uniss.it

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ONE-VARIABLE FUNCTIONS

1) Definition of function. Numerical sequences. Domain and image of a function. Injective, surjective and bijective functions.

2) Invertible functions. Economic example: inverse demand function of a consumer.

3) Composite functions. Economic example: reduction of the number of variables in a mathematical model.

3) Maximum and minimum of a function. Economic example: rofit function and cost function of a firm.

4) Elementary functions. Linear functions; power functions; exponential functions; logarithms. Economic phenomena represented by elementary functions.

LIMITS AND CONTINUITY OF ONE-VARIABLE FUNCTIONS

5) Definition of limit of a function. Limit of a sequence.

6) Definition of continuous function in a point and in a whole interval.

7) Theorems on continuous functions (Zeros Theorem, intermediate values Theorem, Weierstrass Theorem).

8) Applications to the economic models of theorems on continuous functions: existence of solutions to problems of maximization and minimization, existence of equilibrium prices in a competitive market.

DERIVATIVES OF ONE-VARIABLE FUNCTIONS

9) Average rate of change of a function. Economic examples: average increase in the costs of a company, average increase in consumer utility.

10) Definition of derivative of one-variable functions. Economic examples: marginal cost, marginal utility.

11) Rules of derivation.

12) Theorems on derivable functions: Fermat's Theorem and Lagrange's Theorem. Economic applications: first-order conditions relating to the problems of maximization and minimization in economic models. Economic examples: maximization of consumer's utility, maximization of firm's profits.

13) Taylor's formula of the second order. Necessary and sufficient second order conditions related to maximization and minimization problems in economic models.

14) Concave and convex functions. Procedure for identifying solutions in economic models in which concave or convex functions are involved.

VECTORS AND n-VARIABLE FUNCTIONS

15) Vectors. Scalar product between vectors, norm of a vector, distance between two vectors. Internal points of a set, border points. Open sets and closed sets. Economic application: analysis of the structure of the constraint set in the consumer's problem of utility maximization.

16) Functions of n variables. Economic applications: utility function, production function.

17) Lines of level of a function. Economic example: indifference curves.

18) Continuous functions. Weierstrass's Theorem for functions of n variables. Economic application: existence of the solution in maximization and minimization economic problems.

19) Partial derivatives, gradient.

20) Methods for solving problems of maximization and minimization of functions of n variables with and without constraints. Lagrange multiplier method for maximization and minimization problems with equality constraints. Maximization and minimization problems with inequality constraints. Economic applications: minimizing the costs of a firm, maximizing the utility of a consumer.

1) A. Guerraggio, "MATEMATICA", Pearson, second edition, Milano.

2) Course notes available on the teacher's website through the Moodle platform.

Academic lectures. Course notes will be available on the teacher's website in the Moodle platform. Students can also download from the platform: a) A presentation file for the course (with program, exam mode, list of parts to be studied in the textbook, example exam text); b) the exercises (mostly solved) proposed in the past exams.

Possible use of mathematical software in the classroom, especially for drawing the graph of functions and solving optimization problems.

Tutoring service for specific topics covered during the lessons and for solving exercises.

1) Teaching material with notes, examples and solved problems will be available on the course's website through the portal https://edisea.uniss.it

2) A tutoring service (office hours and additional explanations in the classroom), in Italian or English, is provided.

3) International students are given the option to take the exam in English.

Office hours are posted on www.edisea.uniss.it

Contacts:

- Danilo Delpini email: ddelpini@uniss.it phone: 079/213019