Università degli Studi di Urbino Carlo Bo / Portale Web di Ateneo


A.Y. Credits
2023/2024 9
Lecturer Email Office hours for students
Raffaella Servadei Monday 14-16 or by appointment
Teaching in foreign languages
Course with optional materials in a foreign language English
This course is entirely taught in Italian. Study materials can be provided in the foreign language and the final exam can be taken in the foreign language.

Assigned to the Degree Course

Informatics - Science and Technology (L-31)
Date Time Classroom / Location
Date Time Classroom / Location

Learning Objectives

Aim of the course is to give to the students some basic tools and topics in mathematical analysis, for functions of one variable.


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 Theorem*.

06.13  Mean-Value Theorem* and its applications: monotonicity test and characterization 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  Inflection 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 fundament 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. Complex numbers:

08.01  Algebraic definition of complex number and operations with complex numbers.

08.02  Gauss plane.

08.03  Conjugate and modulus of a complex number.

08.04  Trigonometric representation of a complex number and De Moivre’s Theorem.

08.05  Roots of complex numbers.

08.06  Complex exponential representation and Euler’s formula.

08.07  Complex zeros of algebraic equations.

08.08   The fundamental theorem of algebra.

09. Series:

09.01  Definition and examples: geometric series, harmonic series and generalized harmonic series.

09.02  Necessary condition for the convergence of a series*.

09.03  Convergence tests for positive series: comparison test* and limit comparison test*, root test and ratio test.

09.04  Relation between series and improper integrals. 

09.05  Absolute convergence of a series.

09.06  Alternating series.

09.07  The alternating series test (Leibnitz test).

09.08  Series depending on a parameter.

10. Approximation of functions and Taylor’s formula:

10.01  Differential of a function and linear approximation.

10.02   “small-o” notation.

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

10.04  Taylor  polynomials.

10.05  Taylor’s formula with Peano remainder*.

10.06  Taylor’s formula for elementary functions.

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

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

10.09  Taylor series and Taylor series of elementary functions.

* : this means that the proof is required.

Bridging Courses

There are no mandatory prerequisites. It is recommended to take the exam of Calculus 1 during the first year of the Laurea Degree Program in Applied Computer Science and after Logic, Algebra and Geometry.

Learning Achievements (Dublin Descriptors)

Knowledge and understanding:

The student will learn the basic notions of mathematical analysis for the study of functions of one variable.

Applying knowledge and understanding:

The student will learn the methodologies of mathematical analysis and will be able to apply them to the study of various problems.

Making judgements:

The student will be able to apply the techniques of mathematical analysis in order to solve new problems, also coming from real-world applications.

Communications skills:

The student will have the ability to express the fundamental notions of mathematical analysis using a rigorous terminology.


Learning skills:

The student will learn the ability to study the notions of mathematical analysis, also in order to use it in solving different kind of problems.

Teaching 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

Teaching, Attendance, Course Books and Assessment


Theorical and practical lessons.


Although strongly recommended, course attendance is not mandatory.

Course books

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

Conti - Ferrario - Terracini - Verzini, Analisi matematica, Vol.1, Apogeo

Salsa - Squellati, Esercizi di Analisi matematica 1, Zanichelli


Intermediate exams will be carried out during the period of suspension of teaching activities foreseen by the academic calendar in the second semester.

The exam of Calculus 1 consists of a written exam and an oral one, both of them mandatory.

The written exam, to carry 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 Calculus 1 is the average of the marks of the written exam and the oral one.

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.

Additional Information for Non-Attending Students


As for attending students.


As for attending students.

Course books

As for attending students.


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.

« back Last update: 11/07/2023

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