37292 - Mathematics

Learning outcomes

The course aims at giving the student a basic knowledge of differential and integral calculus, and linear algebra for the study of economics, financel and statistical analysis. By the end of the course students have the ability to perform basic operations with vectors and matrices, to compute determinants, and to solve linear systems. As far as calculus is concerned, they can apply the methods of differential and integral calculus to plot the graph of functions, to compute the area of plane domains, and to find and classify critical points of functions of two variables.

Course contents

A Crash course covers a number of introductory topics (so-called precalculus), including elementary set theory, sets of real numbers, polynomials, linear and quadratic equations and inequalities, systems of inequalities, absolute value and rational inequalities, exponential and logarithmic equations and inequalities, Cartesian coordinate system, basic analytic geometry.

PLEASE NOTE: It is of FUNDAMENTAL IMPORTANCE that all students have a perfect knowledge of the topics covered in the introductory mathematics course as:

a) they will not be re-explained during the Mathematics lectures;

b) without an excellent command of these topics, the possibility of understanding the contents of Mathematics is nil;

c) a preliminary exercise concerning this topics is present in the first-midterm and in all the full exams. If a student fails this exercise the exam is automatically considered severely insufficient, and the other exercises will not be corrected.

d) students who have attended at least two thirds of the lessons of the crash course can take a test at the end of the same whose outcome gives a bonus from 0 to 6 points to be added to the grade of the first-midterm of Mathematics.

Course content -Mathematics

Linear Algebra

Linear algebra: vector spaces, bases and dimension; matrices and their properties, matrix operations, rank and determinant; linear systems of equations, existence of solutions, cases of one solution and infinitely many solutions, Gaussian elimination, inverse of a matrix and Cramer's rule; eigenvalues and eigenvectors.

Calculus

One-variable functions: basic definitions, graphs and elementary functions (linear, quadratic, polynomial, rational, irrational, power, exponential, logarithmic, absolute value). Odd and even functions. Composite functions. Inverse functions.

Limits and continuity.

Differentiation of one-variable functions: tangents and derivatives, rules of differentiation, chain rule, higher-order derivatives.

Derivatives in use: implicit differentiation and economic examples, differentiation of the inverse function, linear and quadratic approximations, Taylor's formula, elasticities; continuity and differentiability, intermediate-value theorem, De L’Hôpital’s Rule.

Single-variable optimization: local and global extrema, stationary points and first-order condition, simple tests for extreme points, extreme points for concave and convex functions, second-order derivative and convexity, inflection points, study of the graph of a function, asymptotes.

Integration: the Riemann integral and its geometrical interpretation; primitives and indefinite integrals, fundamental theorems of integral calculus. Rules and methods of integration: immediate integrals, integration of rational functions, integration by parts, integration by substitution. Improper integrals.

Multi-variable calculus: partial derivatives with two variables, geometric interpretation;differentials and linear approximations.

Multi-variable optimization; maxima, minima and saddle points; tests based on second derivatives.

Readings/Bibliography

R.A. ADAMS, C. ESSEX. Calculus, a complete course, 9th Edition, Pearson, 2018.

Chapters: preliminaries, 1, 2, 3, 4, 5, 6, 7.9, 9, 10, 12, 13

K. SYDSÆTER, P. HAMMOND, A. STRØM, A. CARVAJAL. Essential Mathematics for Economic Analysis, 5th Edition. Pearson, 2016.

Chapters: 1, 2,3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16.

Lecture notes on Eigenvalues and Eigenvectors will be provided by the Professor.

Teaching methods

Lectures and excercises at the blackboard.

Assessment methods

Partial written exams, according to the academic terms. As an alternative, cumulative written exam.

Written exam: students have to solve different exercises on the course topics. To each exercise a given maximum number of point is associated, and to get it the student has to solve correctly the exercise and all the steps must be justified. The theoretical maximum number of points atteinable in case of a perfect exam is 32.

The test assessment grid will be as follows:

· <18 insufficient

· 18-23 sufficient

· 24-27 average/good

· 28-30 very good

· 30 cum laude excellent/outstanding

If your total is <=30 score, your score corresponds to your mark. If your score is >30, then you get 30 cum laude.

Note: Mistakes concerning properties of arithmetic and all the contents of the preliminary mathematics course will automatically determine the exam failure!

During the exam, students are not allowed to use calculators. Textbooks and other teaching materials are not allowed.

Fraud treatment committed during the examination

If the teacher has doubts about the lawfulness of a student's written test, he can contact the student to verify through an oral test that he actually took the written test without having committed fraud. In case of a positive oral, the mark of the written test will be confirmed.

NOTE: THOSE WHO COMMIT FRAUD DURING THE EXAM WILL NOT BE ABLE TO TAKE THE EXAM FOR A FULL A.A. UNTIL THE CORRESPONDING CALL OF THE FOLLOWING YEAR AND THE TEST AT THAT POINT WILL BECOME BOTH WRITTEN AND ORAL AND WILL INCLUDE ALL THE CHAPTERS OF R.A. ADAMS, C. ESSEX BOOK.

Grade rejection

The only grade that can be rejected without any communication from the student is the one of the first mid-term exam.

Teaching tools

Professor's lecture notes

Office hours

See the website of Gian Luca Tassinari