D1. Knowledge and understanding: At the end of the course, students will acquire the knowledge of fundamental mathematical tools such as differential calculus and integrals. D2. Applying knowledge and understanding: At the end of the course, students will be able to apply the acquired knowledge of differential and integral calculus to solve easy problems and exercises (also drawn from real life with an economic and financial nature). D3. Making judgments: At the end of the course, students will be able to recognize and apply the basic techniques of differential and integral calculus (maximum and minimum of functions, studies of functions) and will also be able to recognize the situations and problems in which these techniques can be useful (simple models of economics and finance). D4. Communication skills: At the end of the course, students will be able to employ appropriate language on topics related to differential and integral calculus with the safety of exposure. D5. Learning skills: At the end of the course, students will be able to read the standard books on differential and integral calculus in one variable.

Basic notions of logic. Basic notions of set theory. Concept of function. Elementary functions. Elementary notions of geometry in the plane.

1. Numerical sets. Natural numbers. Integer numbers and rational numbers. Basic notions of logic. Set theory. Operations on sets. Basic aspects of combinatorics. 2. Axioms of real numbers. The axiom of separation. Upper bound and lower bound. Existence theorem of the upper/lower bound. Topology of real numbers. Neighbourhoods, open and closed sets. Points of accumulation and closure. Cantor's theorem. 3. Relations between sets: functions. The trigonometric functions. Continuous functions. Sum, difference, product, reciprocal, quotient, composition of continuous functions. Continuity of the inverse function. Bolzano’s theorem. Intermediate value theorem. Compactness and continuous functions. Maxima and minima: Weierstrass theorem. 4. Definition of limit, special cases of finite and infinite limits. Properties of limits: uniqueness, permanence of the sign. Comparison theorems and Sandwich theorem. Operations with limits. Change of variable formula. Limits on restrictions: right and left limits. Limit of rational functions. Limit of monotonic functions. Powers with integer and rational exponent. Powers with real exponent. Exponential function and logarithm function. Limits for the exponential, the logarithm, and the trigonometric functions. 5. Differential calculus for real functions of one real variable. The derivative as the limit of the incremental ratio. Higher order derivatives. Rules of derivation: sum, difference, product, reciprocal, quotient, composite functions, inverses. Fermat’s theorem. Rolle’s, Cauchy’s and Lagrange’s theorems. Consequences of Lagrange’s theorem. Rules of de l'Hopital. Taylor's formula. Complete study of a function. 6. Integral calculus for real functions of one real variable. Riemann integrable functions. Indefinite integrals. Integration by substitution and by parts. Definite integrals and area. Mean value theorem. Integral function and fundamental theorem of integral calculus. Torricelli’s theorem.

1) G. Bosi, C. Corsato, M. Zuanon: “Essential Mathematics for Economics” (Apogeo, 2018). UniTs library: https://www.biblio.units.it/SebinaOpac/resource/essential-mathematics-for-economics/TSA2929265 2) Suggested book to review high school topics: S. Lang: “Basic mathematics” (Addison-Wesley Publishing Company, 1971). UniTs library: https://www.biblio.units.it/SebinaOpac/resource/basic-mathematics/TSA03663187

1. Numerical sets. Natural numbers. Integer numbers and rational numbers. Basic notions of logic. Set theory. Operations on sets. Basic aspects of combinatorics. 2. Axioms of real numbers. The axiom of separation. Upper bound and lower bound. Existence theorem of the upper/lower bound. Topology of real numbers. Neighbourhoods, open and closed sets. Points of accumulation and closure. Cantor's theorem. 3. Relations between sets: functions. The trigonometric functions. Continuous functions. Sum, difference, product, reciprocal, quotient, composition of continuous functions. Continuity of the inverse function. Bolzano’s theorem. Intermediate value theorem. Compactness and continuous functions. Maxima and minima: Weierstrass theorem. 4. Definition of limit, special cases of finite and infinite limits. Properties of limits: uniqueness, permanence of the sign. Comparison theorems and Sandwich theorem. Operations with limits. Change of variable formula. Limits on restrictions: right and left limits. Limit of rational functions. Limit of monotonic functions. Powers with integer and rational exponent. Powers with real exponent. Exponential function and logarithm function. Limits for the exponential, the logarithm, and the trigonometric functions. 5. Differential calculus for real functions of one real variable. The derivative as the limit of the incremental ratio. Higher order derivatives. Rules of derivation: sum, difference, product, reciprocal, quotient, composite functions, inverses. Fermat’s theorem. Rolle’s, Cauchy’s and Lagrange’s theorems. Consequences of Lagrange’s theorem. Rules of de l'Hopital. Taylor's formula. Complete study of a function. 6. Integral calculus for real functions of one real variable. Riemann integrable functions. Indefinite integrals. Integration by substitution and by parts. Definite integrals and area. Mean value theorem. Integral function and fundamental theorem of integral calculus. Torricelli’s theorem.

Lectures and traditional classroom tutorials.

The course material will be available on Microsoft Teams. Any changes to the procedures described above will be communicated on the Department website and the Course webpage.

The final exam consists of a written part and an oral part, which will be held in the same exam session. The written test is composed of two parts. The first part contains three simple theoretical questions (for example basic definitions and statements of fundamental theorems) for a total of 3 points. This part is considered passed only if the candidate answers correctly to two questions out of three, which means that the candidate reaches a minimum score of 2 points. The second part instead consists of solving exercises (for example function studies, calculation of limits, derivatives, and integrals) for a total of 30 points. The score obtained in the second part is added to the score obtained in the first part only if the first part is passed, thus obtaining a total of 33 points maximum. Students cannot bring books and course notes into the classroom during the written exam. The use of a non-programmable and non-graphical calculator is allowed. It is worth noting that any type of electronic device is absolutely forbidden during the examination. A minimum grade of 18/33 is required to be admitted to the oral part, which is mandatory. The final grade will be a weighted average between the written and oral parts, with greater weighting given to the oral part. The oral examination has the aim of verifying the theoretical knowledge of the discipline and the ability of the candidate to express concepts and notions using proper mathematical language.

This course explores topics closely related to one or more goals of the United Nations 2030 Agenda for Sustainable Development (SDGs)