27991 - Mathematical Analysis T-1

Course Unit Page

Academic Year 2021/2022

Learning outcomes

At the end of the course the student : -knows the basic definitions and main properties of real functions of a real variable (limits of functions, continuity, differential calculus, integral calculus) -knows how to connect these properties -is able to solve suitable exercises on these topics.

Course contents

The field of real numbers, properties of subsets of R, lower bound and upper bound. The principle of induction.

Functions: domain, image, injective, surjective, biunivocal functions; composition of functions; inverse function. Elementary functions of a real variable: power, exponential, logarithm, trigonometric functions and their inverses, hyperbolic functions and their inverses.

Sequences in R; limits of sequences; fundamental theorems about limits; operations on limits. Monotone sequences and their limits. The number e; some noticeable limits of sequences.

Limits of real functions of a real variable; extension of the results established for sequences; limit of compound function. Left and right limit; monotone functions and their limits. Some remarkable limits. Continuity of real functions of a real variable, operations on continuous functions. The theorems of zeros, intermediate values and Weierstrass.

The field of complex numbers; module, argument; powers, nth root, exponential and logarithm in the complex field.

Derivative of a function; derivation rules; derivative of elementary functions. Rolle's and Lagrange's theorems, their consequences; monotony. The de l'Hôpital theorem. Higher order derivatives; Taylor's formula. Local maxima and minima; convexity, inflection points. Asymptotes;  study of functions.

Riemann integral; properties of the integral; mean value theorem for integrals, fundamental theorems of integral calculus; primitive of a function. Integration by parts; integration by substitution; integration of rational functions.

Numerical series, convergence criteria.

Generalized integrals, convergence criteria.



G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 1, Zanichelli.


M. Bramanti: Esercitazioni di Analisi Matematica 1, Esculapio.

Teaching methods

Mixed modality lessons.

Assessment methods

The examination consists of a preliminary written test and an oral one.

The written test is valid to take the oral test in the same appeal or in the immediately following one, if in the same period of exam (Janary.February, June-July, September).

In order to sustain the written test the student must register at least four days before the test through AlmaEsami [https://almaesami.unibo.it/].

The oral test follows the written test; it mainly concerns the theoretical aspects of the course. The student must show to know the concepts explained during the course (in particular definitions and theorems) and how to connect with each other.

Office hours

See the website of Giovanni Dore