The course aims to provide students with a theoretical and applied basis in the fundamentals of Mathematics. In particular, the fundamental notions of Algebra and Mathematical Analysis will be explored in depth, with the aim of providing students with the necessary tools to undertake subsequent scientific and technical studies. The course involves the introduction and study of real numbers, the notions of limit, continuity and differentiability for real functions of a real variable, presenting the methods for drawing the graph of a function after analyzing its behavior. The notions of integral and primitive of a function with real variables will also be defined and studied and differential equations will be introduced, with a focus on equations with separable variables. At the end of the course, students will be able to understand and use mathematical language, applying the methods acquired to the resolution of theoretical and practical problems.

Numerical systems. Familiarization with mathematical formalism, essential elements of logic and combinatorics. Major and minor of a set. Upper bound and lower bound. Natural, integer and rational numbers. Real numbers. Complex numbers. Algebraic form of complex numbers. Trigonometric form of complex numbers. Operations with complex numbers and roots in the complex field nth root. Exponentials and Logarithms, related properties.

Limits of real functions of a real variable and continuity.  Definition of function, real functions of real variable. Definition of around. Limit definitions for a real variable function. Accumulation points and isolated points. Limit uniqueness theorem*. Sign permanence theorem*. Comparison theorem*. Operations on limits. Determinate forms and indeterminate forms. Cauchy theorem. Operations between functions. Inverse function and composite function. Limits of monotonic functions. Definition of infinitesimals and infinities. Definition of asymptote (vertical, horizontal, oblique). Notable limitations and application. Notes on numerical sequences and series. Definition of continuity. Continuity of elementary functions, composite functions and inverse functions. Points of discontinuity of a function. Existence theorem of zeros. Weierstrass theorem, Bolzano theorem. Darboux theorem on intermediate values.

Differential calculus.  Definition of differentiability and derivative: its geometric meaning. Angular points and cusps. Relationship between differentiability and continuity*. Derivatives of elementary functions. Algebra of derivatives. Derivatives of composite functions and inverse functions. Differential of a function. Higher order derivatives. Absolute and relative extrema of a function. Fermat's theorem. Rolle's theorems* and its geometric meaning, Lagrange's theorem* and its geometric meaning. Cauchy theorem. Characterization of monotonicity for differentiable functions*. Functions with zero derivative in an interval. Higher order derivatives. De L'Hôpital theorems. Taylor's formula. Convex function theorem. Qualitative study of the graph of a function.

Integration according to Riemann.  Integrability and integral according to Riemann. Definitions, properties and geometric meaning. Integrability of continuous functions. Integrability of monotone functions. Integrability of generally continuous and limited functions. Example of a non-integrable function. Properties of integrals. Integral mean theorem. Primitives. Integral function of a continuous function. Fundamental theorem of integral calculus. Indefinite integral. Integration by parts and by replacement. Integration of rational functions. Integration of irrational and transcendent functions. Improper integrals (outlines). First order differential equations and Cauchy problem. Differential equations with separable variables. Linear equations. Second order differential equations with constant coefficients and the Cauchy problem. Mathematical models applied to reality.

Elements of linear algebra Vector spaces and subspaces. Linearly independent vectors. Bases and dimensions. Matrices and operations between matrices. Inverse matrix. Rank of a matrix. Eigenvalues ​​and eigenvectors of a matrix. Linear algebraic systems. Cramer's rule. Rouché-Capelli theorem. Homogeneous and non-homogeneous systems.

* with demonstration.

Knowledge and understanding: At the end of the course the student must have acquired good mastery, calculation and manipulation skills of the basic Algebra and Mathematical Analysis topics covered in the course.

Applying knowledge and understanding: At the end of the course, the student must have acquired a good ability to use basic mathematical tools. In particular, they must be able to apply theoretical knowledge to practical problems, demonstrating an in-depth understanding of the concepts learned; independently solve exercises and problems of a mathematical nature, even in contexts slightly different from those addressed during the lessons; use the skills acquired by demonstrating flexibility and adaptability. During the lessons and exercises, practical examples will be proposed aimed at consolidating theoretical knowledge and stimulating the development of skills.

Making judgments. At the end of the course the student must have acquired a good ability to analyze general mathematical themes and problems. In particular, they must be able to evaluate in depth and independently the validity and effectiveness of different proposed solutions, correctly interpreting the mathematical results with arguments, even in slightly different contexts from those addressed during the lessons. To encourage the acquisition of these skills, independent in-depth study of the topics covered will be encouraged through carrying out exercises and individual study activities. We will also promote constant constructive discussion between students and with the teacher, in order to encourage critical reflection on one's learning path and consolidate the knowledge acquired.

Communication skills. At the end of the course the student must have acquired a good ability to communicate mathematical concepts and reasoning clearly and rigorously. Through active participation in the lessons and in-depth study of the recommended texts, the student will become familiar with the mathematical language and learn its peculiarities. Thanks to constant interaction with the teacher, they will be able to express their knowledge precisely and clearly, both in oral and written form. The course aims to provide the student with the necessary tools to communicate effectively and efficiently in the scientific field, enhancing the fundamental role of mathematical language.

Ability to learn (learning skills). At the end of the course the student must have acquired a good self-learning ability, demonstrating autonomy in reading and interpreting formulas, texts and data relating to the topics studied; knowing how to search for information relevant to deepening the concepts covered in class. The course will provide the student with the methodological tools necessary to perfect their study strategies. In particular, through classroom exercises, they will be able to independently tackle new topics, identifying the essential prerequisites for their understanding and evaluating the results. In this way, the student will be able to continue their studies independently, developing a critical and constructive approach towards mathematical knowledge.

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Teaching

Frontal lessons

Attendance

Not compulsory, though strongly recommended

Course books

Erin N. Bodine, Suzanne Lenhart and Louis J. Gross, Mathematics for the Life Sciences, Princeton Uniersity Press, 2014.

Assessment

The examination consists of a written and an oral part. 

The written examination is 120 minutes long.

Students are admitted to the oral examination if they have passed the written one with the minimum mark of 15/30.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.

Teaching

None

Course books

Erin N. Bodine, Suzanne Lenhart and Louis J. Gross, Mathematics for the Life Sciences, Princeton Uniersity Press, 2014.

Assessment

The examination consists of a written and an oral part. 

The written examination is 120 minutes long.

Students are admitted to the oral examination if they have passed the written one with the minimum mark of 15/30.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.