27991 - Mathematical Analysis T-1

Learning outcomes

At the end of the year, after a positive result at the final exam, the student should be able to know and to use the fundamental concepts and techniques of calculus concerning the functions of one real variable: limits, continuous functions, differential calculus, integral calculus.

Course contents

LIMITS AND CONTINUOUS FUNCTIONS. Properties of the real numbers. Definition of a convergent and of a divergent sequence of real numbers. The theorems about limits of sequences: uniqueness of the limit, comparison theorems. The algebra of limits. Monotone sequences and their limits. The number e. Recall about the functions: composition of functions, invertible functions and inverse functions. Basic concerning the real functions with one real variable. Definition of a continuous function of one real variable. The Weierstrass theorem and the intermediate value theorem. Definition of the limit of a real function of one real variable; extension of the results established for the sequences. Composition of two or more functions, inverse function of an invertible function. Continuity of the composition of two continuous functions and the theorem on changing variable in a limit. One-sided limits. Monotone functions and their limits. Asymptotes. The inverse circular functions. The hyperbolic functions and their inverse. DIFFERENTIAL CALCULUS. Definition of a differentiable function and of derivative of a function. The algebra of derivatives. The mean value theorems and their application to study the monotonicity of a function. Higher order derivatives. Taylor's formula. Relative maxima and minima of a function: definitions, necessary conditions, sufficient conditions. Convex functions. INTEGRAL CALCULUS. Definition of the Riemann integral. Properties of the integral: linearity, additivity, monotonicity, the mean value theorem. The fundamental theorems of the integral calculus. The theorems of integration by substitution and of integration by parts. Improper integrals. COMPLEX NUMBERS. Definiton of the field of the complex numbers. Agebraic form. Modulus and argument of a complex number. Exponential form of a complex number. De Moivre's formula. Complex roots of a complex number. Algebraic equations in C. The exponential function in C. NUMERICAL SERIES.Real and complex numerical series. Convergence and absolute convergence of a series. Criteria of convergence. DIFFERENTIAL EQUATIONS. Linear differential equations. The general solutions of homogeneous and nonhomogeneous linear differntial equations. The Cauchy problem. Solution of linear differential equations (of order one and of order n) with constant coefficients.

Readings/Bibliography

Marco Bramanti,Carlo Domenico, PaganiSandro Salsa, Analisi matematica 1. Ed. Zanichelli, 2008.

S. Salsa & A. Squellati:Esercizi di Matematica, Vol. I, Ed. Zanichelli.

Giulio Cesare Barozzi, Giovanni Dore, Enrico Obrecht, Elementi di Analisi Matematica. Vol 1.

Ed. Zanichelli 2009

Teaching methods

The lessons will be taught at the bleackboard. During such lessons the fundamental theory of the differential and integral calculus of one real variable will be introduced. In order to clarify these arguments and to help the student to understand the mathematical methods explained some lessons will be dedicated to the solutions of exercises.

Assessment methods

A preliminar written exam is planned before the oral exam. The preliminar written exam can be substituted by two partial written exams during the period of the lessons.

Teaching tools

Exercises of the exams of previous years will be available in theunofficial web page.

Links to further information

http://www.unibo.it/SitoWebDocente/default.htm?UPN=fausto.ferrari@unibo.it

Office hours

See the website of Fausto Ferrari