Università degli studi di Modena e Reggio Emilia

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Teachers

Alberto ZAFFARONI

Overview

The course aims at providing the students with conceptual tools that should enable them to formalize economic and corporate issues, apply mathematics in the analysis and optimization of economic and business processes, acquire the ability of recognizing the limits and the powerfulness of mathematics.
The course also aims at supplying basic tool of financial calculus, necessary for practical problems encountered in the banking sector as well as in the corporate milieu.

Admission requirements

1. Set theory.
2. Number sets.
3. Powers and roots.
4. Factorization of polynomials and polynomial division.
5. Algebraic equations of first, second and higher degree.
6. Systems of equations.
7. Rational and irrational inequalities, and inequalities involving absolute values.
8. Systems of inequalities.
9. Cartesian plane, line and parabola.
10. Introduction to functions.
11. Exponential and logarithmic functions.
12. Exponential and logarithmic equations and inequalities.
13. Rates, discounts, costs and returns.

These prerequisites are treated in the remedial course in Mathematics, which is held in November.

Course contents

1. Topology of real numbers
2. Functions.
3. Maxima and minima.
4. Sequences.
5. Limits for sequences and functions.
6. Continuity. Weierstrass theorem and intermediate values theorem.
7. Derivative of a function. Derivation rules.
8. Monotonic functions. Monotonicity and derivatives.
9. Concavity. Second derivative. Inflection points.
10. Fermat theorem, Rolle theorem, mean value theorem (Lagrange).
11. Composite functions and inverse function.
12. Order of infinity
13. Graphing functions.
14. Trigonometric functions.
15. Taylor's polynomials.
16. Primitives and integration methods.
17. Definite integral. Integral function. Fundamental theorem of integral calculus.
18. Generalized integral on unbounded intervals. Criteria for integrability.
19. Series. Necessary condition for convergence. Criteria for convergence. Geometric series.
20. Introduction to differential equations. Linear equations, separable equations.
21. Matrices and vectors. Operations. Inverse matrix. Rank and determinant.
22. Systems of linear equations. Methods of Gauss and Gauss-Jordan. Cramer's rule.
23. Topology on R^2 and R^n.
24. Functions of several variables. Linear and quadratic functions.
25. Limits, continuity, partial derivatives.
26. Gradient and Hessian matrix. Taylor's polynomial. Unconstrained Optimization. Optimality conditions. Optimization with Equality Constraints. Lagrangian Function.
27. Capitalization and discounting.
28. Force of interest. Decomposability.
29. Annuity, present and final value.

Teaching methods

Lectures are held in presence and in italian. Each lecture is made of a theoretical part and a practical part (solution of exercises). Classes will be recorded and recordings will be available on MS Teams after the end of the lecture. The web page also contains all informations on the course and further exercises.

Assessment methods

Assessment method: the exam consists of a written part which includes the solution to exercises and theoretical questions. It is followed by an oral part concerning only theory. The written exam last 2 hours. Results are published on Moodle within two weeks. Students whose marks are above 15/30 are admitted to the oral part. During the winter session students can take a partial exam only concerned with matierial of the first semester. Students with marks greater that 15 can take the second part during the summer session. More informations and past exams are available at the Moodle web page of the course.

Learning outcomes

(1) Knowledge and comprehension.
Students learn the basic tools of calculus and of financial mathematics.
(2) Ability to apply knowledge and comprehension.
the student is able to apply the concepts and the acquired skills as tools needed in the following courses of his degree.
(3) Autonomy of judgment.
Students will be able to autonomously interpret economic, business, and financial situations
and model problems in these settings.
(4) Communicative skills.
The course does not aim to develop communicative skills, as it is devoted to improve logical and quantitative skills so that students will be able to make fruitful use of formal models for real-life applications. However, the student develops the ability to present the acquired knowledge in a precise and effective manner, with rigorous terminology.
(5) Learning skills.
Armed with improved logical and quantitative skills, students will be able to tackle their further university courses

Readings

Simon e Blume - Matematica Generale - Egea.
Castagnoli e Peccati - Matematica in Azienda, vol 1 Calcolo Finanziario, Egea.