MATHEMATICS 1
Instructional goals
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.
Intended learning outcomes
Knowledge and understanding: the aim of the course is to provide students with a wide range of mathematical tools in order to understand and solve theoretical and applied problems in the economic, business and financial fields.
Applying knowledge and understanding: students will possess the methodological tools that will allow them to apply the basic quantitative knowledge acquired during the course to the working contexts in which they will operate. They will therefore be able to apply quantitative tools to decision making in the business environment, to the analysis of economic phenomena and to solving problems in the economic and business environment.
Autonomy of judgment: at the end of the class, students will have a working knowledge of quantitative tools useful for understanding economic and managerial phenomena. To this end, standard and non-standard problems will be presented to the students. Students will develop a critical ability in identifying the most suitable solution to the proposed problems.
Learning skills: students will have acquired not only adequate skills and knowledge to pass the exam, but above all ability to update and improve their skills in the field of Mathematics applied to economics and management. Students will have to learn how the general theory can in turn be applied to concrete problems as frequently met in economic studies and how economic reality is a stimulus for the improvement of theoretical concepts.
Course Contents
An introduction to single-variable and two-variable calculus, with applications to economics.
Reference Books
- 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.
- Other freely available online material posted on the course's webpage.
Teaching Methods
Teaching will consist of the following activities:
- classroom teaching;
- TA sessions in the classroom.
The presence of material on the Luiss learn page of the course as well as the tutoring activity and office hours provided by the teacher and their collaborators offers the opportunity to create a stable interaction between the class and the lecturer. This gives the opportunity to check in real time the skills acquired by the students and their ability to apply them to the solution of different problems.
Assessment Method
Written exam, where the students will be asked to solve mathematical problems aimed at evaluating their understanding of the course main topics. There will also be two intermediate tests, called midterms. If a student passes both, they will be exempt from the final exam; otherwise, they will have to take the full written exam.
Thesis assignment criteria
Interview with the teacher after passing the exam.
Week 1
The real line. Space R^2 (rectangles, subsets, circles, distance, vectors). A review of analytical geometry. Some elementary functions. Exponential functions. Inverse functions and logarithms. Domain of function.
New functions from old functions.
Week 2
Functions in two variables, domain and sign. Utility functions (Cobb-Douglas).
TA session on the lectures of the present week.
Week 3
Limit of functions. Computing limits using the laws of limits. Asymptotes and continuity.
TA session on the lectures of the present week.
Week 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.
Week 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.
Week 6
Additional exercises concerning the first part of the course.
First Midterm.
Week 7
Graph of a function.
Integrals: Antiderivatives. The definite integral. The fundamental theorem of calculus.
TA session on the lectures of the present week.
Week 8
Indefinite integrals. Integration by substitution. Integration by parts.
TA session on the lectures of the present week.
Week 9
Partial derivatives. Tangent plane and linear approximation.
TA session on the lectures of the present week.
Week 10
Taylor approximation. Directional derivatives and gradient vector.
TA session on the lectures of the present week.
Week 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.
Week 12
Constrained optimization. Lagrange multipliers.
Second midterm. TA session on the lectures of the present week.