MATHEMATICS 1
Obiettivi formativi
The course provides students with the basic mathematical tools needed in economics, management, finance, and insurance. Students will also get acquainted with setting up and solving mathematical problems, with the aid of many examples. By the end of the course, students will be able to tackle and solve non trivial mathematical exercises, together with a clear understanding of the most important theoretical results discussed in the class.
Risultati di apprendimento attesi
Basic algebra, first- and second-degree equations and inequalities, rational equations and inequalities, first and second degree systems of equations and inequalities, equations and inequalities involving logarithms and exponentials, set theory. Students must attend the online preliminary courses available at learn.luiss.it.
Contenuti Del Corso
An introduction to single-variable and two variables calculus, with applications to economics.
Testi Di Riferimento
- Lecture Notes of the course will be provided during the course. - Calculus Early Trascendentals, James Stewart, Saleem Watson, Daniel K. Clegg. - Calculus, Laurence D. Hoffmann, Gerald L. Bradley.
-Principi di matematica per economia (Principles of Mathematics for Economics)
-Other freely available online material posted on the course's webpage.
Metodologie Didattiche
Teaching will consist of the following activities: - classroom and online teaching; - classroom and online TA sessions. The presence of multimedia content available on the Moodle platform (videos, slides, tutorials) as well as the tutoring activity and office hours provided by the teachers and their collaborators provides the opportunity to create a stable interaction between class and teacher giving the opportunity to check in real time skills acquired by students and their ability to apply them to the solution of different problems and their capability to argue their choices.
Modalità di verifica dell'apprendimento
The exam will take place in several steps: Written exam: The exam will be divided into two parts. The first part consists of multiple choice or numerical questions. The answers must be given directly on your laptop. The second consists of open-ended questions that must be solved out on sheets to be delivered by the end of the test. Phones must be turned off during the exam. The written test can also take place as two distinct tests: - The first partial test, called the midterm test, takes place in the middle of the semester (the date will be announced in the first weeks of the course). - The second partial test, called the final test, takes place during the last week of the course. The grade will be a weighted average between the first part and the second part. If the student fails any of the two partial tests, she/he will only be able to retake the complete written test (on the entire syllabus) in the exam session. At the beginning of the session, students have access to their graded papers and can ask for clarifications about the grading.
Criteri per l’assegnazione dell’elaborato finale
Interview with the teacher after passing the exam.
Settimana 1
The real line. Space R^2 (rectangles, subsets distance, vectors). A review of analytical geometry. Some elementary functions. Exponential functions. Inverse functions and logarithms. Domain of function. New functions from old functions.
Settimana 2
Functions in 2 variables, domain and sign. Utility functions (Cobb-Douglas). TA session on the lectures of the present week.
Settimana 3
Function limits. Computing limits using the laws of limits. Asymptotes and continuity. TA session on the lectures of the present week.
Settimana 4
Derivatives and rates of change. The derivative as a function. Rules of differentiation: derivatives of polynomials and exponential functions. Product and quotient rules. TA session on the lectures of the present week.
Settimana 5
Derivatives of composite functions. Derivatives of logarithmic functions. Applications of the derivative: Taylor expansion. Maximum and minimum values. TA session on the lectures of the present week.
Settimana 6
Additional exercises concerning the first part of the course. First Midterm.
Settimana 7
Graph of a function. Integrals: Antiderivatives. The definite integral. The fundamental theorem of calculus. TA session on the lectures of the present week.
Settimana 8
Indefinite integrals. Integration by substitution. Integration by parts. TA session on the lectures of the present week.
Settimana 9
Partial derivatives. Tangent plane and linear approximation. TA session on the lectures of the present week.
Settimana 10
Taylor approximation. Directional derivatives and gradient vector. Implicit differentiation. TA session on the lectures of the present week.
Settimana 11
Chain rule and implicit function theorem. Stationary points of function in two variables. Unconstrained optimization. TA session on the lectures of the present week.
Settimana 12
Constrained optimization. Lagrange multipliers. TA session on the lectures of the present week.