## 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. Inequalities with fractions. Radicals, inequalities with radicals.

The Mathematic exam is written. Students will have to solve 9 exercises concerning the topics covered in the course (7 exercises on one-variable functions, 2 exercises on n-variable functions). All the will be exclusively mathematical without reference to specific application problems (even if these will be illustrated during the course).

On the Moodle platform several examples of past exam texts are available, with the solution of the exercises.

The score of the written test is in thirty; students will not have to take an oral exam. The maximum score that will be assigned to each exercise will be clearly specified in the exam text. The teacher reserves the right impose, in addition to the written test, also an oral examination to students for whom the autonomy in the written test is for some reason dubious (at the discretion of the teacher). In addition, students with an evaluation of 30/30 in the written test, can ask the teacher (before verbalizing the exam) to also take the oral examination for the possible award of laude (however, in this case, the vote the written exam can also be reduced if the oral examination does not confirm the evaluation of 30/30).

The duration of the exam is one and a half hours. During the test the use of books and notes, calculators, cell phones and other devices that allow internet connection is not allowed.

The aim of the course is to provide 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.

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) Guerraggio A., "Matematica", Pearson, second edition, Milano.

2) Course notes prepared by the teacher -which deal with all the topics covered by the course- are available on the teacher's website in the Moodle platform.

Frontal lessons. Course notes prepared by the teacher -which deal with all the topics covered by the course- are available on the teacher's website in the Moodle platform. On the Moodle platform are also available: a) A file presenting the course (with program, exam mode, list of parts to be studied in the textbook (for non-attending students), facsimile of exam task); b) the exercises (mostly resolved) proposed in the past exams.

Use of computer software (Wolfram Alpha) in the classroom, for drawing the graph of functions and solving maximization and minimization problems.

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

Attending students can take the exam by studying only the lecture notes of the teacher. Non-attending students are also advised to study the textbook of Guerraggio. The parts of the textbook to be studied are indicated in the course presentation file, available on Moodle.

Availability to provide a tutoring service and additional explanations in the classroom, in Italian and English, with special attention for students with disabilities.

Availability of providing teaching material and bibliographic references also in English.

Availability to provide the exams text in English.