16953 - Mathematics 1

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.

Readings/Bibliography

- G. Zwirner "Istituzioni di matematiche. Parte I", CEDAM

- G. Zwirner "Esercizi di Analisi Matematica. Parte I", CEDAM

Teaching methods

Matematica 1 (6 ects) is a first semester course and represents the first part of the integrated course Matematica (12 ects). The second part (Matematica 2, 6 ects) will be held in the second semester.

The part Matematica 1 consists of classroom lectures where, first of all, the theoretical aspects of each topic are illustrated. More precisely, after introducing the basic elements, then the main theorems and results of differential and integral calculus for functions of one real variable are stated and, in some cases, proved. Afterwards, several lectures are devoted to the applications of the notions and techniques previously introduced, and to solving exercises and problems.

Assessment methods

The assessment for the Mathematics 1 module is carried out through a final written exam lasting 3 hours and 30 minutes. In the exam, students will be required to solve exercises and answer theoretical questions. The use of books, notes, calculators, or electronic devices is not allowed.

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

 

Students with learning disorders and\or temporary or permanent disabilities: please, contact the office responsible (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the lecturer, who will assess the appropriateness of the adjustments, taking into account the teaching objectives.

Teaching tools

Teaching resource online platform Virtuale

Office hours

See the website of Emanuele Mingione