Basic mathematics and analysis

[2 CFU] Real numbers and functions. Elements of set theory. The axioms of real numbers. Some consequences of the axioms of real numbers. Natural numbers, integers, rational numbers; the set of rational numbers does not satisfy the completeness axiom. Functions and Cartesian representation; injective, surjective, invertible functions. Image and preimage of sets through a function. Monotone functions; linear functions; absolute value function; maximum, minimum, infimum, supremum; bounded and unbounded sets. Existence of the supremum and infimum. Archimedean property. Density of Q in R. Power, root, exponential, logarithmic functions; trigonometric and inverse trigonometric functions. The principle of mathematical induction. Bernoulli’s inequality.

[1 CFU] Limits of sequences. Definitions; theorem of the uniqueness of the limit; bounded sequences; operations with limits; indeterminate forms. Comparison theorems: permanence of sign theorem, Carabinieri’s theorem. Bounded sequences and convergent sequences. Infinitesimal sequences; theorem on the limit of the product of a bounded sequence by an infinitesimal sequence. Subsequences, Bolzano-Weierstrass theorem (s.d.). Cauchy sequences. Monotone sequences; regularity theorem of monotone sequences. The number eee. Recursively defined sequences. Increasing order infinities. Notable limits.

[1 CFU] Limits of functions. Continuous functions. Accumulation points. Definition of limit. Connection between limits of functions and limits of sequences: the bridging theorem. Examples and properties of limits of functions. Operations with limits of functions. Limits of composite functions (s.d.). Continuous functions. Classification of discontinuities. Permanence of sign theorem. Zero existence theorem. Intermediate value theorem; Weierstrass’s theorem (s.d.). Notable limits. Infinites and infinitesimals. Principle of substitution of infinitesimals and infinites. Comparison between infinitesimals, comparison between infinites, and applications.

[1.5 CFU] Derivatives. Definition of derivative. Differentiability and continuity. Operations with derivatives. Derivatives of composite functions. Derivatives of inverse functions (s.d.). Derivatives of elementary functions; geometric meaning of the derivative.

[1.5 CFU] Applications of derivatives. Relative maxima and minima. Fermat’s theorem. Rolle’s and Lagrange’s theorems. Monotonicity criterion. Characterization of constant functions in an interval. Convex and concave functions; convexity and concavity criterion (s.d.). Theorem on left and right derivatives (s.d.). Finding absolute extrema of continuous functions on closed and bounded intervals. De L’Hospital’s theorems. Horizontal, oblique, vertical asymptotes. Graph analysis of a function. Taylor’s formula with Peano’s remainder. McLaurin series of elementary functions. Using Taylor’s formula in limit calculation. Some consequences of Taylor’s formula: sufficient conditions for relative extrema, sufficient conditions for inflection points.

[1.5 CFU] Riemann integration for single-variable functions. Definitions and notations; Riemann integrability criterion (s.d.). Properties of definite integrals. Uniform continuity, Cantor’s theorem (s.d.). Integrability of continuous functions. Mean value theorems for integrals. Area of a rectangular solid.

[1.5 CFU] Indefinite integrals. The fundamental theorem of calculus. Antiderivatives. Fundamental theorem of calculus. The indefinite integral. Indefinite integrals of elementary functions. Immediate integrals. Integration by decomposing into sums. Integration of rational functions. Integration by parts. Integration by substitution.

[1.5 CFU] Numerical series. Definitions; necessary condition for the convergence of a series. Cauchy condition. Series with non-negative terms. Geometric series. Harmonic series. Generalized harmonic series (s.d.). Series remainder and properties. Convergence criteria: comparison criterion, infinitesimal criterion, ratio criterion. Root criterion (s.d.). Absolutely convergent series. Alternating series. Alternating harmonic series. Leibniz’s convergence criterion for alternating series (s.d.).

[0.5 CFU] The field of complex numbers. Definitions. Operations in the complex field. Algebraic and trigonometric forms of a complex number. Powers of a complex number. Roots of a complex number.

Examples and exercises completed during lessons are an integral part of the program.

(s.d.) = without proof