27210 - Mathematical Analysis 1 (M-Z)

Learning outcomes

At the end of the course, the student will have basic notions of infinitesimal and integral calculus, simultaneously developing the habit of scientific reasoning and a sensitivity to the analysis of mathematical models, especially through the study of the asymptotic development of functions. Furthermore, he will be able to carry out a detailed study of functions in a variable, of sequences and series, both numerical and of functions.

Course contents

1. Mathematical logic. Hints of logic. Symbols and types of proof.
2. Sets. Definition of number sets N, Z, Q. Axiomatic construction of the reals. Existence of infimum and supremum of a set. Cardinality of a set, countability of the rationals, uncountability of the reals.
3. Elements of line topology. Absolute value and its properties. Open and closed, bounded, compact, connected subsets. Accumulation points.
4. Sequences and series. Converging and diverging successions.
   Algebra of limits. Comparison theorem. Monotone bounded successions are convergent. Definition of Neper's number. Subsequences and Bolzano-Weierstrass theorem. Cauchy sequences and second definition of completeness (a sequence in R is convergent if and only if it is Cauchy). Landau symbols.
   Convergent series. Necessary condition for convergence of a series. Series with nonnegative terms (ratio, root and condensation criteria). Geometric series, harmonic series and generalized harmonic series.
   Absolute convergence. Leibniz criterion (for series with alternate sign).
5. Functions. Definition and operations (composition, invertibility).
   Functions of one real variable. Elementary functions. Limits. Linking theorem.
   Continuous functions (theorem of zeros, intermediate values, Weierstrass) and uniformly continuous (Heine-Cantor theorem).
6. Derivatives. Definition. Algebra of derivatives. Derivative of compound and inverse. Derivability of elementary functions. Fermat's, Rolle's and Lagrange's theorems. Monotonicity tests.
   Convex functions. Higher order derivatives. Tests of convexity. Cauchy and De L'Hopital theorems. Taylor's formula with Peano's remainder.
7. Integrals. Integral according to Riemann. Integrability of continuous functions. Properties. Fundamental theorems of integral calculus. Primitives. Methods of integration: by parts, by substitution, simple fractals. Generalized integrals and convergence criteria.
8. Sequences and series of functions. Pointwise and uniform convergence. Properties of uniform convergence. Absolute and total convergence. Power series.
9. Ordinary differential equations. Linear of the first order. First-order separable variable equations. Second-order linears with homogeneous and non-homogeneous constant coefficients. Cauchy problems. Theorem of contractions. Existence theorem and uniqueness.

Readings/Bibliography

Theory:

• E. Giusti, Analisi Matematica 1, Bollati Borighieri.
• G.C. Barozzi, G. Dore, E. Obrecht, Elementi di Analisi Matematica, Zanichelli.
• P. Marcellini, C. Sbordone, Analisi Matematica uno, Liguori.
• M. Bramanti, C. Pagani, S. Salsa. Analisi matematica 1, Zanichelli.

Exercises:

• M. Bramanti, Esercitazioni di Analisi Matematica 1, Esculapio.
• S. Salsa, A. Squellati, Esercizi di Analisi Matematica 1, Zanichelli.

Teaching methods

Frontal lectures to explain the basic notions, examples, and counterexamples.

Exercises solved by the teacher.

Additional exercise sheets for personal study.

Office hours.

Assessment methods

The final exam for the course consists of a written test and an oral test, both of which are compulsory and to be taken in that order.

The written test aims to test the ability to apply theory to solving exercises of the type of those proposed during the course. Passages should be reported and justified. No books, notes or calculators are allowed; paper and pen only. It lasts 2 hours and 30 minutes.

The written test is passed with a grade greater than or equal to 18/30.

If the written test is passed, one can proceed to the oral test. This aims to test the knowledge and understanding of the theory developed during the course. You will be asked to give definitions and examples of concepts and to give statements and porofs of theorems.

Teaching tools

Tutoring (TBA) and office hours.

Additional material will be made available on the Virtuale page of the class.

Office hours

See the website of Nicola Abatangelo