## "Babes-Bolyai" University of Cluj-Napoca Faculty of Mathematics and Computer Science

 Mathematical analysis (1)
 Code Semes-ter Hours: C+S+L Credits Type Section MO020 1 2+2+0 6 compulsory Informatică
 Teaching Staff in Charge
 Assoc.Prof. LUPSA Liana, Ph.D., llupsa@math.ubbcluj.roAssoc.Prof. GOLDNER Gavril, Ph.D., goldner@math.ubbcluj.ro
 Aims Getting to know the topology of the real axis and the differential and integral calculus of functions of one real variable.
 Content 1. Real valued functions of a real variable. The limit and the continuity of a real function. Discontinuity points. Monotone functions. Darboux functions. Uniformly continuous functions, absolutely continuous functions, and Lipschitz functions. Convex functions. 2. Differential calculus on R. The derivative and the differential of a real function. Operations with differentiable functions. Differentiability of the compound function and of the inverse function. Mean value theorems (Fermat, Rolle, Cauchy, Lagrange). The characterization of a monotone function by the sign of its derivative. Side derivatives. L'Hospital rule. Denjoy-Bourbaki theorem. Higher order derivatives. The characterization of convexity by the sign of derivatives. Taylor formula. Local extrem points of a real function and their characterization by the sign of its derivatives. Functions having primitives. The Darboux property of functions having primitives. 3. Integral calculus of a real function. Divions of a compact interval in R. Properties of Riemann integrable functions. Darboux sums. The lower and the upper integral of a bounded function. Their connection with the Rieman integral. Evaluation of the Riemann integral. Newton-Leibniz formula. Integration by parts and by changing of variable. Rieman-Stieltjes integral and its reduction to Riemann integral. Improprious integral. Their convergence criteria. Integrals having parameters and the main operations with them. 4. Series of real functions. Differentiability and integrability propertes of the limit function. Similar properties for series of real functions. Power series. The set and the ray of convergence. Properties of sum function. Expanding of a real function in power series.
 References 1. ANDRICA D., DUCA I.D., PURDEA I., POP I.: Matematica de baza. Cluj-Napoca, Editura Studium, 2000. 2. BALAZS M., KOLUMBAN I.: Analiza matematica. Curs litografiat, Facultatea de Matematica, Univ. "Babes-Bolyai". 3. BRECKNER W. W.: Analiza matematica. Topologia spatiului Rn. Cluj-Napoca, Universitatea, 1985. 4. COBZAS ST.: Analiza matematica (Calcul diferential). Cluj-Napoca, Presa Universitara Clujeana, 1998. 5. COLOJOARA I.: Analiza matematica. Editura Didactica si Pedagogica, Bucuresti, 1983. 6. LUPSA L. si BLAGA L.: Elemente de analiza matematica si teoria campului. Partea I. Bistrita, Editura George Cosbuc, 2001. 7. MARUSCIAC I.: Analiza matematica. I, II. Cluj-Napoca, Universitatea "Babes-Bolyai", 1980. 8. FIHTENHOLT G. M.: Curs de calcul diferential si integral. Vol. I, II. Bucuresti, Editura Tehnica, 1965. 9. NICOLESCU M., DINCULEANU N., MARCUS S.: Manual de analiza matematica. Vol. I. Bucuresti, Editura Didactica si Pedag., 1963.
 Assessment Exam.