28622 - Mathematical Analysis T-A

Learning outcomes

The student knows the basic concepts and the main properties of real functions of a real variable (limits of sequences and functions, continuity, differential calculus, integral calculus) and is able to solve simple exercises on these topics.

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

G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 1, Zanichelli (2009).

M. Bramanti: Esercitazioni di Analisi Matematica 1, Esculapio (2011).

Teaching methods

Lectures and exercises in the classroom.

Assessment methods

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

The written test consists of 6 exercises related to the arguments of the course. In order to sustain the written test the student must register at least four days before the test through AlmaEsami. The written test is passed with a score of 12/30; it is valid to take the oral exam in the same appeal or in the immediately following one, if in the same period of exam (Janary.February, June-July, September).

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.

Links to further information

http://www.dm.unibo.it/~dore/Analisi_T-A/index.html

Office hours

See the website of Giovanni Dore