The course provides the student with the main notions, results and methodologies of basic mathematics, real functions of a single variable, linear algebra and matrix theory, with particular attention to logical deduction and analysis of the arguments of the course.

01. Numbers:

01.01  Natural numbers, integers, rational numbers and real numbers.

01.02  Summations, factorials, binomial coefficients and Newton’s binomial formula.

01.03  Algebraic properties and geometric representation of rational numbers.

01.04   From rational numbers to real ones.

01.05  Absolute value and distance in the real line.

01.06  Intervals in the real line. Bounded and unbounded sets in the real line. Maximum and minimum of a subset of the real line. Supremum and infimum of a subset of the real line.

01.07  Mathematical induction and its applications.

02. Functions of one variable:

02.01  What is a function.

02.02  Real functions of one real variable: preliminaries, bounded functions, even and odd functions, monotone functions, periodic functions.

02.03  Elementary functions.

02.04  Scaling and shifting a graph.

02.05 Piecewise-defined functions.

02.06  Composition of functions.

02.07  Inverse function.

02.08  Inverse trigonometric functions.

03. Limits of functions:

03.01  Finite limits at a point.

03.02  Uniqueness theorem for limits*.

03.03  Finite limits at infinity.

03.04  Horizontal asymptotes.

03.05  Infinite limits at infinity.

03.06  Oblique asymptotes. Infinite limits at a point.

03.07  One-sided limits.

03.08  Vertical asymptotes.

03.09  Non-existence of a limit.

03.10  Algebra of limits and indeterminate forms.

03.11  Sign-stability theorem*.

03.12  The squeeze theorem*.

03.13  Change of variable in the limits.

03.14  Asymptotic functions.

03.15  Notable special limits. 

03.16  Comparison between infinite functions.

04. Sequences:

04.01  Definition of sequence.

04.02  Convergent, divergent and indefinite sequences.

04.03  Monotone sequences.

05. Continuity:

05.01  Continuous functions.

05.02  Algebra of continuous functions.

05.03  Continuity of elementary functions.

05.04  Continuity of the composition of functions.

05.05  Limits for polynomial functions.

05.06  Limits for rational functions.

05.07  Notable special limits.

05.08  Points of discontinuity.

05.09  Continuous functions on an interval: Bolzano Theorem (bisection method)*, Weierstrass Theorem and Intermediate-Value Theorem*.

05.10  Continuity of the inverse function.

06. Differential calculus for functions of one variable:

06.01  Derivative of a function.

06.02  Geometric interpretation of the derivative.

06.03  Tangent line to the graph of a function.

06.04  Derivatives of elementary functions.

06.05  Relations between differentiability and continuity*.

06.06  Algebra of derivatives*.

06.07  Differentiation rules for product and quotient*.

06.08  The chain rule*.

06.09  Derivative of the inverse function*.

06.10  One-sided derivative and non-differentiability points.

06.11  Critical points, local and global maxima and minima.

06.12 Fermat's Theorem*.

06.13  Mean-Value Theorem* and its applications: monotonicity test and characterisation of functions with zero derivative in an interval.

06.14  Finding maxima and minima of a function.

06.15  De L’Hospital Rule.

06.16  Second-order derivatives.

06.17  Concavity and convexity of a function.

06.18 Inflexion points.

06.19  Sketching the graph of a function.

07. Integral calculus for functions of one variable:

07.01  Antiderivative and indefinite integral of a function.

07.02  Antiderivatives of elementary functions.

07.03  Areas of plane regions.

07.04  Definition of definite integral.

07.05  Integrable functions.

07.06  Properties of definite integral.

07.07  A Mean-Value Theorem for integrals*.

07.08  The fundamental theorem of integral calculus*.

07.09  Techniques of integration: scomposition and substitution.

07.10  Integrals of rational functions.

07.11  Integration by parts*.

07.12  Integrals of trigonometric functions.

07.13  Integration of irrational functions.

07.14  Improper integrals.

07.15  Integrability criteria: comparison and limit comparison*.

08. Approximation of functions and Taylor’s formula:

08.01  Differential of a function and linear approximation.

08.02   “small-o” notation.

08.03  Approximating values of functions and application to the calculus of limits.

08.04  Taylor's polynomials.

08.05  Taylor’s formula with Peano remainder*.

08.06  Taylor’s formula for elementary functions.

08.07  Taylor’s formula with Lagrange remainder and integral remainder.

08.08  Applications: approximations of functions, error estimates and calculus of limits.

08.09  Taylor series and Taylor series of elementary functions.

09. Linear algebra:

09.01 Vectors and operations on them: sum, difference, scalar multiplication, scalar product and vector product.

09.02 Vector spaces: definition and properties*.

09.03 Vector subspaces. Necessary and sufficient condition for a subset to be a subspace*.

09.04 Linear combination of vectors. Subspaces spanned by a set of vectors.

09.05 Linear dependence and linear independence of vectors.

09.06 Basis and dimension of a vector space. Uniqueness of the representation of a vector as a linear combination of the elements of a basis*.

09.07 Operations on subspaces: sum and intersection. Grassmann Theorem.

09.08 Direct sum of vector subspaces. Necessary and sufficient condition for a vector space to be the direct sum of its subspaces.

09.09 Preliminaries and operations with matrices: sum, scalar multiplication, product and their properties. Vector spaces of matrices (m,n).

09.10 The determinant of a square matrix and its properties.

09.11 Sarrus rule and Laplace Theorem for the determinant.

09.12 Invertible matrices.

09.13 Binet Theorem and its consequences.

09.14 Rank of a matrix. Kronecker Theorem.

09.15 Eigenvalues and eigenfunctions of a matrix. Characteristic polynomial and characteristic equation.

09.16 Linear systems: preliminaries and methods of solution.

09.17 Cramer Theorem.

09.18 Rouché-Capelli Theorem.

09.19 Linear systems depending on parameters.

09.20 Linear maps. Necessary and sufficient condition for a map to be linear*. Image of the zero vector by a linear map*.

09.21 Kernel and image of a linear map and their properties*.

09.22 Rank-nullity theorem.

09.23 Matrix associated with a linear map.

*: This means that the proof is required.

There are no mandatory prerequisites. It is recommended to take the exam in Mathematics I during the first academic year.

Knowledge and understanding: The student will achieve a deep understanding of the structure of mathematical reasoning, both in a general context and in the context of the arguments of the course. The student will achieve mastery of the computational methods of basic mathematics and of the other arguments of the course.

Applying knowledge and understanding: The student will learn how to apply the knowledge: to analyse and understand results and methods regarding both the arguments of the course and arguments similar to those of the course; to use a clear and correct mathematical formulation of problems pertaining to or similar to those of the course.

Making judgements: The student will be able to construct and develop logical reasoning pertaining to the arguments of the course with a clear understanding of hypotheses, theses and rationale of the reasoning.

Communication skills: The student will achieve: the mastery of the lexicon of basic mathematics and of the arguments of the course; the skill of working on these arguments autonomously or in a work group context; the skill of easily fit in a team working on the arguments of the course; the skill of expose problems, ideas and solutions about the arguments of the course both to a expert and non-expert audience, both in written or oral form.

Learning skills: The student will be able to autonomously deepen the arguments of the course and of other but similar mathematical and scientific theories; to easily refer to literature and other material about the arguments of the course and similar theories, and to add knowledge by a correct use of the bibliographic material.

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

The teaching material and specific communications from the lecturer can be found, together with other supporting activities, inside the Moodle platform › blended.uniurb.it

Teaching

Theorical and practical lessons.

Innovative teaching methods

The classroom lectures will be integrated with exercises individually and in groups.

Attendance

Although strongly recommended, course attendance is not mandatory. 

Course books

Abate - De Fabritiis, Geometria analitica con elementi di algebra lineare, McGraw Hill
Abate - De Fabritiis, Esercizi di geometria, Mc Graw Hill
Adams, Calcolo Differenziale 1, Casa Editrice Ambrosiana
Adams - Essex, Calculus: a complete course, Pearson Canada
Bramanti - Pagani - Salsa, Analisi matematica 1, Zanichelli
Bramanti - Pagani - Salsa, Analisi matematica 1 con elementi di geometria e algebra lineare, Zanichelli
Salsa - Squellati, Esercizi di Analisi matematica 1, Zanichelli

Assessment

The exam for Mathematical Analysis 1 consists of a written exam and an oral exam, both of which are mandatory.

The written exam, to be carried out in two hours, consists of exercises related to the topics of the course. The written exam is passed if the mark is at least 15/30. During the written exam, it is not allowed to use textbooks, workbooks or notes. Moreover, it is not allowed to use scientific calculators and mobile phones, under penalty of disqualification.

The oral exam consists of a discussion related to the topics of the course.  The oral exam can be taken only if the written one has been passed. If so, the oral exam can be taken only in the same call in which the written exam has been passed or in the other calls of the same session.

The final mark of Mathematical Analysis 1 is the average of the marks of the written exam and the oral exam.

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

As for attending students.

Attendance

As for attending students.

Course books

As for attending students.

Assessment

As for attending students.

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.