The aim of this course is to learn basic calculus and some theorems of real analysis (differential and integral calculus, sequences and series of real numbers and real functions). A further aim is to apply analytical techniques in other scientific disciplines.
Learning outcomes (declined compared with the Dublin descriptors)
-Knowledge and understanding. Knowledge of the differential and integral calculus for functions of one real variable. The student will be able to state and prove basic theorems of Mathematical Analysis.
-Applying knowledge and understanding. The ability to understand the problems proposed during the course, the ability to correctly apply the theoretical knowledge. The student will be able to study of the graphs of elementary functions, to solve integration problems of elementary character, to discuss the nature of numerical sequences and series.
-Making judgments. Develop the ability to critically evaluate the problems and propose the most appropriate approach
-Communication skills. Ability to report and present the results with a ogical-deductive and synthetic exposition
-Ability to learn.Ability to develop, outline, summarize the contents
Algebra of polynomials. Elements of analytic geometry. Elements of goniometry and trigonometry.
Elementary equations and inequalities.
Real numbers and numeric functions (2 CFU-16 hours). Review of elementary set theory. The real number system. The concept of function.; elementary functions.
Sequences of real numbers and series (2 CFU-16 hours). Sequences: limit definition; and related theorems; operations with indefinite limits and forms; monotone sequences.
Series: definitions and first properties, geometric series, harmonic series and generalized harmonic series; series of non-negative terms and convergence criteria; series with alternate signs: Leibnitz's criterion; absolute convergence and property.
Differential Calculus for Function of one Variable (3 CFU-24 Hours) Numeric Functions:
limit of a function and its properties; continuous functions; considerable limits; monotone functions; Weierstrass theorem. Derivative, Definition and Its Geometric meaning; derivation of elementary functions and derivation rules; maximum and minimum relative; Rolle theorem, Lagrange theorem and consequences; de l'Hopital theorem; Taylor's formula; concave and convex functions, asymptotes.
Integral Calculus for Function of one Variable (2 CFU-16 Hours) Primitive of a Function, Indefinite Integral; integration rules; Riemann integral; integrability of continuous functions; mean value theorem, fundamental theorem of integral calculus.
The real number system. The concept of function. Sequences of real numbers. Series.
Continuous functions and related theorems. Differential calculus for functions of one variable. Integral calculus for functions of one variable.
Robert A. Adams, Christopher Essex, Calculus: A Complete Course, Pearson Canada.
The objective of the exam is to check the level of achievement of the above-mentioned training objectives.
The exam is divided into two parts:
- A written test that aims to evaluate the ability to correctly use the theoretical knowledge acquired during the course to solve mathematical problems. The student who does not show sufficient mastery of the arguments is not admitted to the next test. The expected time is 2 hours.
- an oral test in which the ability to link and compare different aspects of the course will be evaluated.
The final vote takes into account the evaluation of both tests.
Lectures are in Italian. The professor is fluent in English and is available to interact with students in English, also during the examination.