27991 - Mathematical Analysis T-1

Learning outcomes

To provide a solid methodological and operational mastery of the fundamental aspects of differential and integral calculus for functions of a single variable.

Course contents

Module 1 Contents

Introduction. Natural, integer, rational, and real numbers. Maximum and minimum values. Upper and lower bounds. Supremum and infimum. Examples and exercises.

Sequences. Bounded sequences. Definition of limit. Convergent, divergent, and indeterminate sequences. Subsequences. Uniqueness of the limit. Algebra of limits. Sign permanence theorem. Squeeze theorem. Bolzano-Weierstrass theorem. Bounded sequences. Properties and theorems. Infinity and infinitesimals. Asymptotic equivalence and little-o notation. Monotone sequences. Existence of limits for monotone sequences. Bolzano-Weierstrass theorem. Examples and exercises.

Limits of functions of a real variable. General concepts: definition of function, domain, and codomain; injective and bijective functions. Upper-bounded, lower-bounded, bounded, periodic, even, and odd functions. Monotonicity. Examples I. Composite and inverse functions. Theorem on the invertibility of a monotonic function (without proof). Examples II. Definition of limit. Uniqueness of the limit. Right-hand and left-hand limits. Equivalence between definitions of limit using neighborhoods and sequences. Algebra of limits. Sign permanence theorems. Squeeze theorem. Variable substitution theorem for limits. Bounded and monotonic functions. Limits for monotonic functions. Examples and exercises.

Module 2 Contents

Continuous functions. Definitions and examples. Basic properties of continuous functions. Composition of continuous functions. Weierstrass theorem. Uniformly continuous functions. Basic properties of functions. Examples and exercises.

Differential calculus. Definition of a differentiable function and derivative of a function. Characterization of differentiable functions. Relationship between differentiability and continuity. Derivatives of sums, products, and quotients of differentiable functions. Derivative of a composite and inverse function. Derivatives of elementary functions. Local extrema. Critical points. Fermat's theorem. Rolle's theorem. Lagrange's theorem. Cauchy's theorem. Differentiable functions on intervals and monotonicity. Darboux's theorem. L'Hôpital's rule. Higher-order derivatives. Determination of extrema or inflection points using successive derivatives. Study of function graphs. Polynomial functions and properties. Taylor's theorem with Peano's remainder. Taylor's theorem with Lagrange's remainder. Taylor polynomials of elementary functions. Convex functions. Theorems and characterizations. Examples and exercises.

Integral calculus. Partitions. Lower and upper sums. Definitions and properties. Lower and upper integrals. Riemann integrability. Riemann's theorem. Properties of integrable functions. Integrability of monotonic and continuous functions. Mean value theorem for integrals. Integral function and antiderivatives. Fundamental theorem of calculus. Integration by parts. Integration by substitution. Antiderivatives of rational functions. Improper integrals. Definitions and properties. Comparison tests. Absolute convergence. Convergence tests for functions with varying signs. Examples and exercises.

Series. Partial sums. Definitions and properties. Telescoping series. Geometric series. Comparison tests. Cauchy's theorem. Root test. Ratio test. Cauchy's condensation test. Integral test. Absolute convergence. Abel-Dirichlet test. Leibniz's criterion. Examples and exercises.

Complex numbers. Imaginary unit and algebraic definition of a complex number. Addition and multiplication of complex numbers. Additive inverse. Conjugate, modulus, geometric interpretation of the modulus. Multiplicative inverse. Triangle inequality. Trigonometric form of a complex number. De Moivre's formulas. Euler's function. Theorem on the roots of a complex number. Fundamental theorem of algebra. Examples and exercises.

Readings/Bibliography

Boooks for the Theoretical part

Main textbook

• G.C.Barozzi, G.Dore, E.Obrecht, Elementi di Analisi Matematica 1, Zanichelli.

Other recommended textbooks

• P.Marcellini, C.Sbordone. Elementi di Analisi Matematica Uno,
  Liguori Editore.
• M.Bramanti, C.Pagani, S.Salsa. Matematica. Calcolo infinitesimale e algebra lineare, Zanichelli Editore.
• E.Giusti. Analisi matematica 1, Bollati Boringhieri Editore.
• E.Lanconelli, Lezioni di Analisi Matematica 1, Pitagora

Recommended Exercise books

• P.Marcellini, C.Sbordone. Esercitazioni di Matematica - I vol.,
  Liguori Editore.
• S.Salsa, A.Squellati. Esercizi di Analisi matematica Vol. 1,
  Zanichelli Editore.
• E.Giusti. Esercizi e complementi di analisi matematica,
  Bollati Boringhieri Editore.

In addition to the aformentioned Exercise books, students can use any Exercis book covering the arguments of the course.

Teaching methods

Lectures and Exercises conducted by the Course Instructors.

Assessment methods

The assessment method consists of a written exam followed by an oral exam. Admission to the oral exam is contingent upon passing the written exam.

Teaching tools

Teaching resourses on the online poortal Virtuale.

Office hours

See the website of Serena Federico

See the website of Giovanna Citti