27991 - Mathematical Analysis T-1

Learning outcomes

Knowing the methodological-operational aspects of the Mathematical analysis and some of its applications, with particular regard to the functions of a variable.

Course contents

Functions: Review of functions: domain, image, injective functions, surjective, two-way functions; composition of functions; inverse function. Elementary functions of real variable: power, exponential, logarithm, trigonometric functions and their inverse, hyperbolic functions and their inverse.

Real sequences: Sequences in R; inheritance limits; theorems of sign permanence and comparison; limit operations. Monotone sequences and their limits; limitations and extremes of subsets of R. The number e; some remarkable limits of successions.

Numerical series: partial sums, definition of the sum of a convergent series. Necessary condition for convergence. Geometric series, generalized harmonic series. Series with positive terms. Absolute convergence and simple convergence. Convergence criteria for series with non-negative terms: comparison, ratio, n-th root.

Limits and continuity for real functions of real variables: Limits of real functions of real variable; extension of the results established for the succession; composite function limit. Right and left boundary; monotone functions and their limits. Some notable limits. Continuity of real functions of real variable, operations on continuous functions. Theorems of zeros, intermediate values and Weierstrass.

Differential and integral calculus for functions of a variable: Derivative of a function; derivation rules; derivative of elementary functions. Rolle and Lagrange theorems, their consequences; growth and decrease. The theorem of de l'Hôpital. Higher order derivatives; Taylor's formula. Relative maxima and minima; convex, flexed functions. Asymptotes; function study. Integral calculation for functions of an integral variable of continuous functions; properties of the integral; the integral mean theorem, fundamental theorems of integral calculus; primitive of a function. Integration by parts; integration by substitution; notes on the integration of rational functions.

Readings/Bibliography

Theory:

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

Exercises:

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

Teaching methods

Frontal lessons

Assessment methods

The exam consists of a preliminary written test and an oral test.

The written test consists of 6 exercises related to the topics covered in the course. To take the written test you must register on the list at least five days before via AlmaEsami [http://almaesami.unibo.it/]. The written test is passed with a minimum score of 15 out of 30; it is valid for taking the exam in the same appeal or in the immediately following one, as long as in the same exam period (January-February, June-July, September).

The oral exam, following the written test, mainly concerns the theoretical aspects of the course. The student must demonstrate to know the concepts explained in the course (in particular definitions and theorems) and to know how to connect them.

Office hours

See the website of Paolo Negrini