Mathematical Analysis

Knowledge and understanding: the student will learn some basic concepts of Mathematical Analysis and will develop both computing ability and the capacity of manipulating  numerical sequences, numerical series, limits and derivatives for functions of real variable.

Applying knowledge and understanding: by means of examples related to applied sciences, the student will learn the central role of Mathematical Analysis within science and not only as an abstract topic. This will expand his cultural horizon.

Making judgements: the student will tackle with rigor some simple meaningful methods of Mathematical Analysis. This will sharpen his logical ability. Many proofs will be exposed in an intuitive and schematic way, to make them more usable also to students that are not committed to Mathematics.

Communication skills: By studying Mathematics and doing guided exercitations, the student will learn to communicate with clarity and rigor both, verbally and in writing. The student will learn that the use of a properly structured language is the key point to clear and effective scientific, and non-scientific, communication.

Learning skills: the students, in particular the more willing, will be stimulated to examine in depth some topics, alone or working in team.

If the teaching is given in mixed or remote mode, the necessary changes may be introduced in order to comply with the program provided and reported in the syllabus

Attendance is compulsory within the minimum limit set by the Laurea Course didactic regulations

1. Sets of numbers. Real numbers. The ordering of real numbers. Completeness of R. Factorials and binomial coefficients. Relations in the plane. Complex numbers. Algebraic operation. Cartesian coordinates. Trigonometric and exponential form. Powers and nth roots. Algebraic equations.
2. Limits. Neighborhoods. Real functions. Limits of functions. Theorems on limits: uniqueness and sign of the limit, comparison theorems, algebra of limits. Indeterminate forms of algebraic and exponential type. Substitution theorem. Limits of monotone functions. Landau symbols and their applications. 
3. Numerical sequences and series. Sequences of Real Numbers. Definitions of convergence and divergence sequences. Some special sequences. The monotone sequences theorem. The number e. Subsequence. Series. Series of nonnegative terms. The harmonic series. The geometric series. The comparison, root and ratio tests. The asymptotic comparison test. The alternating series. The Leibnitz's criterion.
4. Continuity. Continuous functions. Sequential characterization of the continuity. Points of discontinuity. Discontinuities for monotone functions. Properties of continuous functions (zeros theorem, intermediate value theorem, Weierstrass's theorem,). Continuity of the composite and inverse functions. 
5. Differential Calculus. The derivative. Derivatives of the elementary functions. Rules of differentiation. Differentiability and continuity. Extrema and critical points. Theorems of Rolle, Lagrange. Consequences of Lagrange's Theorem. De L'Hôpital Rule. Monotone functions. Higher-order derivatives. Convexity and inflection points. Qualitative study of a function. Recurrences. Numerical methods: Newton method and Secant Method.
6. Combinatorics, Statistics and Probability. Arrangements, permutations, combinations, simple and with repetition. Definition of classical probability and frequentist probability. Conditional probability. Mode, mean and median. Hints to hypothesis tests. Applications. 
7. Integrals. Areas and distances. The definite integral. The Fundamental Theorem of Calculus. Indefinite integrals and the Net Change Theorem. The substitution rule. Integration by parts. Trigonometric integrals. Trigonometric substitution. Integration of rational functions by partial fractions. Strategy for integration. Impropers integrals. Applications of integration.

1. C. Canuto, A. Tabacco – Mathematical Analysis I – Springer-Verlag Italia, Milano, 2015.
2. Lecture notes.

N.B .: The verification of learning can also be carried out electronically, if the conditions require it.

How to give a definition: extremes of a numerical set; various definitions of limit; theorems on limits of functions; continuous functions and their properties; differentiable functions and their properties, indefinite integral and its properties; definite integral and its properties, arrangements, permutations and combinations; probability: definitions, properties and applications.

How to state and prove theorems: fundamental theorems of differential calculus; Taylor's formula and its applications;