28622 - Mathematical Analysis T-A

Learning outcomes

To know the methodological-operational aspects of mathematical analysis, with particular regard to the functions of one real variable,  in order to be able to use this knowledge to interpret and describe engineering problems.

 

Course contents

Functions Domain and range of a function, injective, surjective and bijective functions; composition of functions, inverse function. Elementary functions of a real variable: power, exponential, logarithm, trigonometric and inverse trigonometric functions, hyperbolic and inverse hyperbolic functions.
Real sequences Sequences in R; limit of a sequence; squeeze theorem; algebra of limits. Limits of monotone sequences. Boundedness and bounds of subsets of R. The number e; some remarkable limits of sequences.
Limits and continuity for real functions of a real variable Limits for real functions of a real variable; extensions of the results pertinent to sequences; limit of a composite function. One sided limits, limits of monotone functions. Some remarkable limits of functions. Continuity, algebra of continuity. Intermediate value theorem, Weierstrass theorem.
Differential calculus for functions of one variable Definition and basic properties of the derivative; derivation rules; derivative of elementary functions. Rolle and mean value theorem, derivative and monotonicity. L'Hôpital's rule. Higher order derivatives; Taylor's formula. Local extrema; convexity, inflexion points.
Integral calculus for functions of one variable Definition and basic properties of the integral of a continuous function; mean value theorem for integrals, fundamental theorems of calculus; primitive. Integration by parts, integration by substitution; integration of rational functions.

Readings/Bibliography

Barozzi, Dore, Obrecht, Elementi di Analisi Matematica 1, Zanichelli, Bologna.

M. Bramanti - C.D. Pagani - S. Salsa, Analisi Matematica 1, Zanichelli, Bologna.

Salsa - Squellati, Esercizi di Analisi Matematica 1, Zanichelli, Bologna.

Bramanti, Esercitazioni di Analisi Matematica 1, Esculapio.

 

 

Teaching methods

Frontal lectures and exercises.

Assessment methods

The examination consists of a written test consisting of exercises and theoretical questions related to the topics covered in the course. The student must demonstrate knowledge of the concepts explained in the course (in particular definitions and theorems) and know how to apply them to concrete cases. You need to go to the exam with a university card and an identification document. It is not permitted to keep books, notes, calculators, cell phones or other materials with you. To take the written test you must register on the list, in the indicated time window, via AlmaEsami [http://almaesami.unibo.it/]. For the examination test calendar, always refer to AlmaEsami.

 

The exam is part of the exam of "Analisi Matematica e Geometria e Algebra", together with the module of "GEOMETRIA E ALGEBRA T-A". For more details see the web site of Alessandro Gimigliano.

Office hours

See the website of Francesco Uguzzoni