Mathematics

LUIGI ORSINA

GENERAL OBJECTIVES: to obtain a general knowledge of the basic techniques of Differential and Integral Calculus and of the standard applications to problems of maxima-minima of functions of a real variable, to the study of their graph, to the convergence of numerical series and to the calculus of definite and indefinite integrals.

SPECIFIC OBJECTIVES:

Knowledge and understanding: at the end of the course, students will master the basic notions of Differential and Integral Calculus, in particular the notions of function, limit, continuity, numerical series, derivatives and definite integrals.

Applying knowledge and understanding: students will be able to solve typical problems from Differential and Integral Calculus, such as the explicit calculation of derivatives, of maxima and minima of a function, to plot an approximate graph of functions of a real variable, to determine the convergence of a numerical series and to compute a definite integrals.

Critical and judgment skills: students will be able to use a graph as a tool to analyse concrete phenomena which admit a mathematical description. They will also acquire the tools that have historically led to the solution of classic problems and the basic tools needed in other courses of mathematical analysis, numerical analysis and mathematical phisics.

Communication skills: ability to display the contents in the oral part of the verification and in any theoretical questions present in the written test.

Learning skills: the notions and techinques learned will give the student access to more advanced notions, either in a further course or in the form of self-study, concerning further aspects of Differential and Integral Calculus.

General objectives: to acquire basic notions of differential and integral calculus, as well as the ability to apply these notions to both theoretical and practical problems.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the main notions relative to function, continuous functions, derivative, integral.

Applying knowledge and understanding: at the end of the course the student will be able to solve medium to high level problems in differential and integral calculus, will have acquired familiarity with the notions of calculus, and will be able to apply these techniques to the solution of various concrete problems.

Critical and judgmental skills: the student will have the basis for understanding when calculus techniques can be useful as tools for solving problems in various areas of analysis and its applications.

Communication skills: ability to expound content in the oral part of the test and answer theoretical questions.

Learning ability: the knowledge acquired will allow a study, individually or in a course, of more advanced aspects of analysis, and of more specific application topics.

Arithmetic and algebra: operations on numbers, use of powers, roots and logarithms, literal calculus, polynomials (operations, decomposition into factors). Algebraic equations and inequalities of first and second degree or reducible to them. Systems of first degree equations. Fratte rational equations and inequalities.

Geometry: measurement and properties of segments and angles. Straight lines, planes, properties of the main plane and solid figures and their lengths, areas, volumes and surface areas.

Analytical geometry: Cartesian coordinates, the intuitive concept of function, equations of lines, parabolas, circumferences, ellipses. Graphs and properties of elementary functions.

Elements of trigonometry: graphs and geometric meaning of sine, cosine and tangent. Main trigonometric formulae (addition, subtraction, duplication, bisection). Trigonometric equations and
trigonometric equations and inequalities. Relations between elements of a triangle.

Numbers: Natural numbers, relative and rational integers, operations and ordering. Introduction to real numbers. Upper and lower extremes. Induction and combinatorics.

Real functions of real variable: Definition of function, examples. Operations between functions, composition. Limited functions, monotonous functions, symmetries and periodicity. Injectivity, suriectivity, invertible functions. Basic functions: powers and roots, exponentials and logarithms, trigonometric functions.

Limits: Finite to finite limits. Tools for calculating limits. Infinite and/or infinite limits, indeterminate forms. Asymptotes. Comparison between infinitesimals and comparison between infinities, principles of substitution.

Successions and series: Limits of successions. Bolzano-Weierstrass theorem. Cauchy Criterion. Sequential characterisation of limits. Monotone successions, Nepero's number. Successions defined by recurrence. Numerical series. Series with non-negative terms: regularity, Comparison Criterion, Asymptotic Comparison Criterion, Root Criterion and Ratio Criterion. Simple and absolute convergence, Leibniz Criterion.

Continuity: Definition and first properties. Classification of discontinuities. Weiestrass Theorem, Zero Theorem, Intermediate Value Theorem.

Differential Calculus: Definition of derivative and calculation tools. Fermat's Theorem, Rolle's Theorem, Lagrange's Theorem and its direct consequences, Darboux's property of derivatives, Cauchy and L'Hopital's Theorem. Subsequent derivatives and Taylor's Formula, convexity. Study of functions.

Integration: Primitives, integration by parts and by substitution. Integration of rational functions. Other methods of integration. Defined integrals. Definition and general properties. Classes of integrable functions. Integral Mean Theorem. Fundamental Theorem of Integral Calculus. Some applications of Integral Calculus. Improper integrals.

All course topics are contained in a series of notes prepared by the lecturer and available to students from the beginning of the course.
The material provided is sufficient for comprehensive preparation

Any book on first level analysis

Lessons in class.

Attendance is not mandatory (but strongly recommended)

Passing the course involves a written and an oral test, both of which are compulsory. During the course of the course there will be in itinere tests, the passing of which will exempt from the written test

Written test: exercises similar to those carried out in class
Oral examination: discussion of some results included in the course programme and their application

- Give the definition of limit of a sequence, with examples.
- Give the definition of continuity, with examples.
- Prove one theorem on continuous functions.
- Give the definition of derivative, with examples, and prove a theorem on differentiable functions.
- Give the definition of Riemann integrable function, and prove a theorem on integrals.

• Il corso è diviso in varie parti: 1) numeri reali e topologia della retta reale; 2) successioni e serie a termini reali; 3) limiti e continuità; 4) derivate e formula di Taylor; 5) teoria dell'integrazione secondo Riemann.