The aim of the course of Analisi Matematica I is to give the basic skills real and complex numbers, differential and integral calculus for real functions of one real variable.
In particular, the learning objectives of the course, according to the Dublin descriptors, are:
The course is divided into two parts. The first partdeals with the construction and the properties of the field of real numbers, complex numbers, basic topology notions, sequences and functions, differential calculus and its applications. The second part deals with series and integrals. The lessons are complemented by exercises related to the topic of the course and both the lessons and the exercises will be carried out in frontal mode and/or online, according to rules determined by the department. Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the program planned and outlined in the Syllabus.
Sets of numbers. Real numbers. The ordering of real numbers. Completeness of R. Factorials and binomial coefficients. Relations in the plane. Complex numbers. Algebraic operation. Cartesian coordinates. Trigonometric and exponential form. Powers anndth roots. Algebraic equations.
Limits. Neighbourhoods. Real functions. Limits of functions. Theorems on limits: uniqueness and
sign of the limit, comparison theorems, algebra of limits. Indeterminate forms of algebraic and exponential type. Substitution theorem. Limits of monotone functions. Sequences. Limit of a sequence. Sequential characterization of a limit. Cauchy's criterion for convergent sequences. Infinitesimal and infinite functions. Local comparison of functions. Landau symbols and thei applications.
Continuity. Continuous functions. Sequential characterization of the continuity. Points of discontinuity. Discontinuities for monotone functions. Properties of continuous functions (Weierstrass's theorem, Intermediate value theorem). Continuity of the composition and the inverse functions.
Differential Calculus. The derivative. Derivatives of the elementary functions. Rules differentiation. Differentiability and continuity. Extrema and critical points. Theorems of Rolle, Lagrange and Cauchy. Consequences of Lagrange's Theorem. De L'Hôpital Rule. Monotone functions. Higher-order derivatives. Convexity and inflection points. Qualitative study of a function. Recurrences.
Integrals. Areas and distances. The definite integral. The Fundamental Theorem of Calculus. Indefinite integrals and the Net Change Theorem. The substitution rule. Integration by parts Trigonometric integrals. Trigonometric substitution. Integration of rational functions by partial fractions. Strategy for integration. Impropers integrals. Applications of integration.
Numerical series. Round-up on sequences. Numerical series. Series with positive terms. Alternating series. The algebra of series. Absolute and Conditional Convergence. The Integral Test and Estimates of Sums.
C. Canuto, A. Tabacco – Mathematical Analysis I – Springer-Verlag Italia, Milano, 2015.
J. Stewart, D. Clegg, S. Watson – Calculus. Early Transcendentals – Ninth Edition, Cengage Learning, Boston, USA, 2021.
H. D. Junghenn – A Course in Real Analysis – CRC Press, Boca Raton, FL, 2015.