27991 - Mathematical Analysis T-1

Learning outcomes

At the end of the course, after passing the final examination, the student should possess basic knowledge relating to Mathematical Analysis and in particular relating the functions of a real variable: interpretation of graphs, limits, derivatives, integrals and their meaning.

Course contents

Introduction: Properties of the real numbers and of N, Z, Q.  Real-valued functions of one real variable; injectivity, surjectivity, invertibility, inverse function, composition of function, monotone functions. Elementary functions: recalls.

Complex numbers: Definiton of the field of the complex numbers. Algebraic 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.

Limits: Accumulation point, definitions of limit; one/two-sided limits. Elementary properties of limits: unicity, locality. Algebraic properties of the limit, comparison theorems, limits of monotone functions. Indeterminacy. Landau symbols

Continuity: Definition of a continuous function of one real variable. The Weierstrass theorem, the Bolzano theorem and the intermediate value theorem. Continuity of the composition of two continuous functions. Continuity of the inverse function.

Differential calculus and applications: Definition of a differentiable function and of the 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. Hospital's Rule. Taylor's formula. Relative maxima and minima of a function: definitions, necessary conditions, sufficient conditions. Convex functions.

Integration: 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.

Differential equations. Linear differential equations. The general solutions of homogeneous and nonhomogeneous linear differential equations. The Cauchy problem. Solution of linear differential equations (of order one , of order n with constant coefficients). The method of Lagrange. Differential equations solvable by separation.

Readings/Bibliography

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

AmsCampus.

S. Salsa, A. Squellati: Esercizi di Analisi Matematica 1, Zanichelli Editore (Bologna), 2011

Teaching methods

Class lessons and exercises. Theoretical lectures; examples and exercises done in class; possible exercises left to the student. Exercises regularly published in the institutional AMS campus site.

Assessment methods

Written and oral examinations. A detailed program for the oral part will be published in the institutional AMScampus site. The written part of the examination will check the knowledge of ALL the topics presented in the exercises, regularly published on the AMScampus site; in the written part, some questions may be posed on theoretical topics as well. During the oral examination, the student will be asked at least three theorems/proofs/examples/definitions, presented during the lectures.

Dates:

3 exams in January/February

1 in June: 1 in July, 1 in September

Teaching tools

Regularly, exercises will be published on-line in the AMS-Campus site. As for the preparation for the oral examination, at the end of the semester, a very-detailed list of the questions for the oral part will be published in the AMS-Campus site.

Office hours

See the website of Andrea Bonfiglioli