The aim of the course is to provide students with basic concepts of the differential calculus for functions with one variable, to make the student able to elaborate main topics critically, to improve reasoning skills.
Lectures and exercises in the classroom.
Elements of set theory. Numerical sets. Topology elements
Cartesian product. Definition of function. Special functions. Succession. Composed function. Injective and surjective functions. Inverse function. Internal, external, border, accumulation, isolated points. The extended line R *.
Real functions of real variable. Positivity and symmetries. Limited functions. Monotone functions. Definition of limit. Theorem of uniqueness of the limit. Right and left limit. Theorem of the permanence of the sign. Operations with function limits. Infinitesimal and infinite. Asymptotes. Limit of a succession. The number "e", some notable limits. Cauchy convergence criterion.
Definition of continuity. Operations on continuous functions. Points of discontinuity. Discontinuity of monotonic functions. Basic properties of continuous functions over a range. Theorem of existence of zeros and intermediate values. First and second Weierstrass theorem. Elementary functions: rational functions; algebraic, exponential and logarithmic functions; hyperbolic functions and their inverse; trigonometric functions and their inverse.
Definition of derivative. Derivability and continuity. Right derivative, left derivative. Operations with derivatives. Differential. Local extremes. Fermat's theorem. Theorem of Lagrange. Consequences of the Lagrange theorem. De L'Hôpital theorems and applications. Taylor's formula. Concave and convex functions. Determination of the nature of stationary points. Determining the graph of a function.
Integrals of functions of one variable
Definition of integral according to Riemann and geometric meaning. Classes of integrable functions. Properties of the integral: additivity; homogeneity; monotony; average theorem; additivity to the integration interval. Integral function. The fundamental theorem of integral calculus. Undefined integral. Rules for integration by decomposition, by parts and by substitution. Improper integrals.
Definition of numerical series and first properties. Cauchy criterion. Series with non-negative terms. Convergence criteria. Convergence and absolute convergence. Leibniz criterion. Operations on the series.
1) C.D. Pagani, S. Salsa - Analisi Matematica I - Zanichelli
2) G. Zwirner, Esercizi di Analisi Matematica I, CEDAM
3) G. Sbordone-P. Marcellini, Esercitazione di Matematica, primo volume, Liguori Editore