MATHEMATICAL ANALYSIS I A - L

Knowing how to solve simple equations in the complex domain, determine the extrema of a numerical set, calculate the limits of sequences and functions, as well as studying the character of numerical series. Knowing how to draw the graph of real functions of one real variable, identifying the main properties, calculate indefinite, definite, improper and generalized integrals. Knowing how to solve simple differential equations of the first order and of the second order, linear and with constant coefficients.

Elements of set theory. Symbols and fundamental set operations. Functions: definition; composition; injective; surjective; bijective; inverse.

Numerical sets. The set of natural numbers. Induction principle. Bernoulli's inequality. Relative integers. Rational numbers. Existence of irrational numbers. The set of real numbers: algebraic structure, sorting. absolute value, powers and roots of a real number. Logarithms. The set of rational numbers is dense in the set of real numbers. Bounded sets of real numbers. Extrema of a numerical set and its properties. The extended straight. Intervals. The set of complex numbers. algebraic form, trigonometric form, powers and roots of a complex number.

Elements of topology in R. Neighbourhoods of a point. Interior, exterior, and boundary points. Interior and boundary of a set. Accumulation points. Derivative of a set. open sets, closed sets. Theorem of Bolzano-Weierstrass.

Real functions of one real variable. Definitions. Geometric representation. Extrema of a function. Definition of limit. Some examples. The unique limit theorem. Theorems of the comparison. Theorem of sign permanence. Left and right limits. Operations on the limits and indeterminate forms. Monotone functions and their limits. Infinitesimal and infinite. Asymptotes vertical, oblique or horizontal. Sequences and their limits. Characterization of the limit of a function by means of the limits of appropriate sequences. Monotone sequences. The number of Neper. Some known limits. Subsequences. Cauchy's convergence test. Sequentially compact sets and their characterization. Averages of the terms of a sequence.

Continuous functions. Definition of continuity in a point and in a set. Discontinuities. Discontinuities of monotone functions. Operations on continuous functions. Fundamental properties of continuous functions: theorem of the existence of zeros and of the existence of intermediate values, Weierstrass theorem. Continuity of composite functions and inverse functions. The functions arcsin x, arccos, arctan x. Uniform continuity. Cantor's theorem. Lipschitz continuity.

Differential calculus for real functions of one real variable. Derivative and its kinematic and geometric meanings. Differentiability and continuity. Derivatives of elementary functions. Rules of derivation. Derivatives of composite functions and inverse functions. Higher order derivatives. Maxima and minima, Fermat's theorem. The theorems of Rolle, Cauchy, and Lagrange. Some consequences of the Lagrange theorem: functions having zero derivative; characterization of the monotonicity for differentiable functions in an interval; functions with bounded derivative. Search of the maximum and minimum points of a function. Theorems of de l'Hospital and indeterminate forms. Convex functions in an interval. Property. Taylor's formula and applications. Study of a function's graph. Continuity of the derivative function.

Numerical series. Definitions and first properties. Cauchy convergence criterion. Geometric, Mengoli's, and harmonic series. Series with non-negative terms; convergence and divergence criteria: comparison, root, Raabe, and condensation criteria. Generalized harmonic series, criteria of infinitesimals. Absolute convergence. Series with terms having alternating sign, criterion of Leibniz. Operations on the series: sum, product by a constant. commutative property.

Integrals of real functions of one real variable. Integrability and integral of Riemann for limited functions in a closed bounded interval. A characteristic condition for the integrability and geometrical meaning. Example of function not Riemann integrable. Classes of integrable functions: continuous functions, monotonic functions, functions discontinuous at a finite set. Properties: distributivity, positivity, additivity, integration of the absolute value. The theorems of average. Definite integrals. Integral function and the fundamental theorem of calculus. Primitive and indefinite integrals. Elementary methods of indefinite integration: sum decomposition, by parts, by substitution. Integrals of rational functions. Integration for rationalization. Calculation of areas and volumes. Generalized integrals and improper integrals. Absolute integration, convergence criteria.

Notes on differential equations of the 1st and 2nd order. Model of Malthus. Damped harmonic oscillator. Differential equations of the 1st order with separable variables, homogeneous, linear and of Bernoulli's type. 2nd order differential equations, linear and with constant coefficients: structure of the space of solutions, method of variation of arbitrary constants.

1) G. DI FAZIO – P. ZAMBONI, Analisi Matematica Uno, Monduzzi Editore, Bologna, 2007.

2) P. MARCELLINI – C. SBORDONE, Analisi Matematica uno, Liguori Editore, Napoli, 1998.

3) G. DI FAZIO – P. ZAMBONI, Analisi Matematica Uno (Eserciziari per l’Ingegneria), EdiSES, Napoli, 2013.

4) P. MARCELLINI – C. SBORDONE, Esercitazioni di Matematica, Volume I, Liguori Editore, Napoli, 1988.