Mathematical Analysis I F - O

The aim of the course of Mathematical Analysis I is to give the basic skills on real and complex numbers, differential and integral calculus for real functions of one real variable.

In particular, the learning objectives of the course, according to the Dublin descriptors, are:

• 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, as complex numbers, limits, derivatives and integrals for real functions of one real variable, numerical series.

• Applying knowledge and understanding: The student will be able to apply the acquired knowledge in the basic processes of mathematical modeling of classical problems arising from Engineering.

• Makin​g judgements: The student will be stimulated to autonomously deepen his/her knowledge and to carry out exercises on the topics covered by the course. Constructive discussion between students and constant discussion with the teacher will be strongly recommended so that the student will be able to critically monitor his/her own learning process.

• Communication skills: The attendance of the lessons and the reading of the recommended books will help the student to be familiar with the rigor of the mathematical language. Through constant interaction with the teacher, the student will learn to communicate the acquired knowledge with rigor and clarity, both in oral and written form. At the end of the course the student will have learned that mathematical language is useful for communicating clearly in the scientific field.

• Learning skills: The student will be guided in the process of perfecting his/her study method. In particular, through suitable guided exercises, he/she will be able to independently tackle new topics, recognizing the necessary prerequisites to understand them.

The course Mathematical Analysis I (9 CFU) is divided into two parts carried out in the first and second semester respectively. For the whole course, 49 hours of theory and 30 hours of other activities (typically, these are exercises) are expected. Theory lectures and exercises related to the topics covered will be offered. Theory lectures and exercises will be carried out in frontal mode. Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the program planned and outlined in the Syllabus.

In-depth knowledge of the contents of Arithmetic, Algebra, Analytical Geometry, Trigonometry usually covered in High Schools. All these arguments will be anyway reviewed and recalled during the various “Corsi Zero” taught at the beginning of the academic year. MOOC is strongly suggested as a way to review required mathematical notions.

Lecture attendance is not compulsory, but it is strongly recommended.

The proofs of the topics marked with an asterisk are not required during the exam.

1. Sets of numbers. 

• Natural numbers, integer numbers, rational numbers, real numbers. Basic notions on the set N of natural numbers, on the set Z of integer numbers and on the set Q of rational numbers*. The set R of real numbers. Bounds for numerical sets. Archimedean property*. Density of Q and R-Q in R*. Powers with real exponent*.
• Complex numbers. Basic definitions. Total order. Polar coordinates in the plane. Trigonometric form of a complex number. Product and power of complex numbers in trigonometric form. Exponential form of a complex number. Product and power of complex numbers in exponential form. nth roots of a complex number. Algebraic equations*.

1. Functions and limits. 

• Functions. Basic definitions.
• Limits. Topology in R. Definitions of limits. Algebra of limits. Indeterminate forms. Theorems on limits: uniqueness of the limit, sign permanence, comparison theorems. Theorem on limits of monotone functions*. Theorem on the limit of the composite function*. Sequences: basic definitions, limits, sequential characterization of the limit of a function*, subsequences.

1. Continuous functions and local comparison.
• Continuous functions. Definition of continuous function and basic results. Continuity of elementary functions and operations with continuous functions. Singularity points: removable singularity, singularity of the first kind and singularity of the second kind. Properties of continuous functions: local properties and global properties. Theorem on the existence of zeroes, Intermediate value theorem, Weierstrass Theorem. Injectivity and strict monotonicity for continuous functions. Theorem on the continuity of the inverse function*. Fundamental limits.
• Local comparison of functions. Bachmann-Landau symbols, comparison between infinitesimal and infinite functions. Asymptotes.
• Uniformly continuous functions*.

1. Differential Calculus.
• Definition of differentiable function and definition of derivative. Geometric and kinematic meaning of the first derivative. Relationship between continuity and differentiability. Derivatives of the elementary functions. One-sided derivatives. Point of non-differentiability. Definition of differential. Rules of differentiation. Theorem on the differentiability of the composite function. Theorem on the differentiability of the inverse function.
• Fundamental theorems of Differential Calculus and their consequences. Fermat's Theorem, Rolle's Theorem, Lagrange's Theorem and its consequences (functions with null derivative on an interval, monotonicity test and local extrema, characterization of differentiable strictly monotone funcitons). De L'Hôpital's Theorem*. Theorem on the limit of the derivative. Limit of the derivative and points of non-differentiability. Higher-order derivatives. Taylor formula with Peano’s remainder and Taylor formula with Lagrange’s remainder*. Concave functions, convex functions: definitions of concave function and convex function, inflection points, necessary condition for the inglection points, relationship between the concavity/convexity and the sign of the second derivative, test for the inflection points. Higher-order derivative test for the study of stationary points. Qualitative study of a function.

1. Integral Calculus.
• Indefinite integrals. Antiderivatives of a function on an interval and indefinite integral. Rules of indefinite integration: linearity property, integration by parts, integrations by substitution. Integration of rational functions and applications.
• Definite integrals. Riemann integral: lower sum and upper sum, definition of Riemann integrable function, condition of Riemann integrability. Classes of Riemann integrable functions. Properties of the Riemann integral*. Integral extended to an oriented interval. Definition of integral average and its geometric meaning. Mean value Theorem. Fundamental Theorem of Integral Calculus.
• Improper integrals. Improper integrals on unbounded intervals. Improper integrals on bounded intervals. Convergence criteria: algebra of improper integrals, integrability of nonnegative functions, comparison test, absolute convergence test, asymptotic comparison test.

1. Numerical series.
• Basic definitions. Fundamental series: geometric series, p-series, telescopic series. General convergence criteria: adding, deleting and modifying a finite number of terms in a numerical series*, algebra of numerical series, necessary condition for the convergence of a numerical series.
• Convergence criteria for nonnegative-term series. Comparison test, application of the comparison test to the study of the harmonic series, Asymptotic comparison test, root test. A test for positive-term series: ratio test. Cauchy condensation criterion* and its applications to the study of p-series and Bertrand series. MacLaurin criterion and its applications.
• Convergence criteria for alternating series. Leibniz's alternating series test and its consequences.
• Absolute converge test

Teaching’s contribution to the Goals of the Agenda 2030 for Sustainable Development:

All the topics mentioned in the program can be requested during the exam.

The attendance of the lessons, the study of the recommended texts and the study of the material provided by the teacher (handouts and collections of exercises carried out and proposed) allow the student to have a clear and detailed idea of ​​the questions that may be proposed during the exam.

An adequate exposition of the theory involves the use of the rigorous language characteristic of the discipline, the exposition of simple examples and counterexamples that clarify the exposed concepts (definitions, propositions, theorems, corollaries).

The main types of exercises are:

• Finding the bounds of a numerical set. Finding interior points, boundary points, accumulation points of an assigned numerical set.
• Exercises on complex numbers.
• Calculation of sequences limits.
• Calculation of limits of functions. Study of the continuity, boundedness and invertibility of real functions of a real variable.
• Qualitative study of a function and applications.
• Calculation of indefinite and definite integrals.
• Search the antiderivative of a function satisfying a condition.
• Study of the convergence of improper integrals and calculation of improper integrals.
• Study of the behaviour of a numerical series.