MATHEMATICS 1

The objectives of the course are the following:

Knowledge and understanding: the student will learn some basic concepts of Mathematical Analysis and will develop both computing ability and the capacity of manipulating some common mathematical structures among which 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 rigour 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 rigour 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.

Traditional (teacher up front) lessons.

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. Neighbourhoods. 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. Sequences. Limit of a sequence. Sequential characterization of a limit. Cauchy's criterion for convergent sequences*. Infinitesimal and infinite functions. Local comparison of functions. Landau symbols and their applications.
3. Continuity. Continuous functions. Sequential characterization of the continuity. Points of discontinuity. Discontinuities for monotone functions. Properties of continuous functions (Weierstrass's theorem, Intermediate value theorem). Continuity of the composition and the inverse functions.
4. Differential Calculus. The derivative. Derivatives of the elementary functions. Rules of differentiation. Differentiability and continuity. Extrema and critical points. Theorems of Rolle, Lagrange and Cauchy. 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.
5. 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.

C. Canuto, A. Tabacco – Mathematical Analysis I – Springer (2015).