Mathematics
A.Y. 2024/2025
Learning objectives
The course aims to lead the student to master the basic concepts of mathematical analysis (relative to the study of real functions of real variable), as well as to recognise the appropriate mathematical tools for solving problems related to the physical and biological world.
Expected learning outcomes
By the end of the course, the student should be able to (1) recognize which mathematical representations are the most appropriate for representing and studying the observables of the physical and biological world that he or she encounters in the course of his or her studies, and (2) find, independently and creatively, the way to solve seemingly complicated mathematical problems. It is also hoped that the student will, by the end of the course, come to be able to gather and compare the information necessary to achieve the objectives from different sources and as independently as possible, always being able to rely on the possibility of confrontation with the teacher.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Lesson period
First semester
Course syllabus
The course begins with a consideration of number sets and measurable quantities in biology: from natural numbers, through generalization of elementary operations, we gradually arrive at studying the real number field. The next leap is the complex numbers: a bridge between algebra and geometry, they allow solving any algebraic equations (fundamental theorem of algebra).
After offering a detailed introduction of the concept of a function, we continue with a rigorous but intuitive study of the notions of a circle and a limit. Limits of elementary functions and techniques for calculating limits of any functions are then discussed. Finally, the concepts of continuity and derivability, as well as extremal points and local/global minimum and maximum of a real variable real function are addressed.
The course continues with the study of the Riemann integral: from the calculation of the primitives of a function (indefinite integral), we move on to the notion of measuring the area subtended by the graph of a real function with real variable (definite integral) and its relation to the primitives of that function (fundamental theorem of calculus). Improper integrals and the concept of convergence are then addressed. The course closes with an introduction to ordinary differential equations, with a focus on solving techniques for first-order ones (autonomous, separable variable, homogeneous, Bernoulli, linear with constant coefficients).
After offering a detailed introduction of the concept of a function, we continue with a rigorous but intuitive study of the notions of a circle and a limit. Limits of elementary functions and techniques for calculating limits of any functions are then discussed. Finally, the concepts of continuity and derivability, as well as extremal points and local/global minimum and maximum of a real variable real function are addressed.
The course continues with the study of the Riemann integral: from the calculation of the primitives of a function (indefinite integral), we move on to the notion of measuring the area subtended by the graph of a real function with real variable (definite integral) and its relation to the primitives of that function (fundamental theorem of calculus). Improper integrals and the concept of convergence are then addressed. The course closes with an introduction to ordinary differential equations, with a focus on solving techniques for first-order ones (autonomous, separable variable, homogeneous, Bernoulli, linear with constant coefficients).
Prerequisites for admission
Basic skills acquired in high school are necessary for successful use of the course: solving linear, quadratic and fractional equations and inequalities, elementary equations and inequalities with radicals, exponentials, logarithms and trigonometric functions. The ability to graph elementary functions is also very important.
Teaching methods
The course (lasting 48 hours) is organized in lectures, all of which include theory and exercises. During the lectures wide attention is paid to examples, often drawn from physical or biological problems. Attendance is compulsory, and although the student can enter the exam having attended 75 % of the lectures, attending all lectures is highly recommended.
Teaching Resources
The reference text is Mathematics for Science, A. Guerraggio, Pearson.
Slides projected in class are provided on the days following each lecture. These slides, which do not exhaustively cover everything explained or done in class, are to be considered a mere mnemonic aid, and are in no case a substitute for notes and textbook. It is strongly recommended to take notes.
Finally, we would like to point out the availability, on the University's Ariel portal, of the excellent and very useful "Assisted Mathematics" handouts (theory + exercises with solutions).
Slides projected in class are provided on the days following each lecture. These slides, which do not exhaustively cover everything explained or done in class, are to be considered a mere mnemonic aid, and are in no case a substitute for notes and textbook. It is strongly recommended to take notes.
Finally, we would like to point out the availability, on the University's Ariel portal, of the excellent and very useful "Assisted Mathematics" handouts (theory + exercises with solutions).
Assessment methods and Criteria
The exam for the Mathematics course is only written and will also include a theory part. During the test it is not permitted to consult notes, books or anything else, nor to use electronic equipment. For attending students there are two partial tests whose cumulative result is considered equivalent to the written test.
MAT/03 - GEOMETRY - University credits: 6
Lessons: 48 hours