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16953 - Mathematics 1

Academic Year 2026/2027

                
                        Docente:
                        Emanuele Mingione
                    
                        Credits:
                        6
                    
                        SSD:
                        MATH-04/A
                    
                        Language:
                        Italian
                    
                        Moduli:
                        
                            Emanuele Mingione
                            (Modulo 1)
                        
                            Federica Gerace
                            (Modulo 2)
                        
                        Teaching Mode:
                        
                                    In-person learning (entirely or partially) (Modulo 1); 
                                
                                    In-person learning (entirely or partially) (Modulo 2)
                                
                            Campus:
                            Bologna
                        
                            Corso:
                            First cycle degree programme (L) in
                            Chemistry and Materials Chemistry (cod. 6257)

                                        Course Timetable
                                    
from Sep 23, 2026 to Dec 10, 2026

Learning outcomes

At the end of the module, the student has a basic knowledge of differential and integral calculus for functions of one real variable. In particular, the student is able to: apply differential and integral calculus to functions of one real variable; plot functions on a graph; manipulate trigonometric, polynomial, exponential, and logarithmic functions.

Course contents

Prerequisites: Knowledge of basic mathematics acquired in high school is required.

Syllabus:

Elementary functions: polynomials, logarithms, exponentials, trigonometric functions and their inverses.

Limits and continuity. Definitions and fundamental theorems on limits: uniqueness of the limit, comparison theorem, squeeze theorem, absolute value theorem. Partial limits. Operations on limits. Indeterminate forms. Notable limits. Definition of continuity and points of discontinuity. Theorems on continuous functions on a closed interval (Weierstrass theorem and its applications).

Derivatives and their applications. Definition of the derivative and its geometric meaning. Fundamental theorems of differential calculus: Rolle’s theorem, Lagrange’s mean value theorem, L’Hospital’s rule. Maximum and minimum of a function. Concavity, convexity, and inflection points. Vertical, horizontal, and oblique asymptotes. Graphing a function. Taylor expansion of a function.

Integral calculus and applications. Antiderivative of a function and indefinite integral: definitions and basic properties. Integration by substitution and by parts. Definite integrals and area calculations.

Series od real numbers and power series : Sequence, Series and power series, Convergence tests, Taylor series.

Readings/Bibliography

Note of the course on the online platform Virtuale

Teaching methods

The Mathematics 1 course takes place in the first semester and constitutes the first module (6 ECTS credits) of the integrated Mathematics course (12 ECTS credits). The second module (Mathematics 2, 6 ECTS credits) takes place in the second semester.

The module is structured as classroom lectures, in which the theoretical aspects of the topics covered are presented first. In particular, after introducing the basic notions, the main theorems and results in the field of differential and integral calculus for functions of one real variable are stated, and in some cases proved. Subsequently, ample space is devoted to the applications of the notions and techniques presented, and to solving exercises.

Assessment methods

Assessment for the Mathematics 1 module takes place through a final written exam lasting 3 hours and 30 minutes. The exam will require solving exercises and answering theoretical questions. The use of the following is not permitted: books, notes, calculators, or electronic devices.

The use of AI is prohibited. Any use constitutes a violation of academic integrity.

The grade for the exam of the entire integrated Mathematics course (12 ECTS credits) is calculated as a weighted average, based on the credits, of the grades obtained in the Mathematics 1 exam (6 ECTS credits) and the Mathematics 2 exam (6 ECTS credits).

Students with SLD (Specific Learning Disorders) or temporary or permanent disabilities: it is recommended that they contact the relevant University Office in good time (https://site.unibo.it/studenti-con-disabilita-e-dsa/it). It will be the Office's responsibility to propose any adaptations to interested students, which must in any case be submitted, at least 15 days in advance, for the approval of the instructor, who will assess their appropriateness also in relation to the course's learning objectives.

Teaching tools

Course notes and lecture register updated on the course's virtual platform.

Office hours

See the website of Emanuele Mingione

See the website of Federica Gerace