Fundamentals on the calculus of functions of one real variable. Sequences and series. Convergence tests. Integration techniques. Improper integrals. Applications on the evaluation of area, arclength and volume
- Teacher: Ιωάννης Γάσπαρης
ECTS : 5
Language : el
Learning Outcomes : Knowledge: The course is an introduction to Mathematical Analysis I, whose main subject is the study of real functions of one variable. Mathematical Analysis is a valuable tool for many scientific fields, including mathematics and physical sciences, engineering, economics and management sciences, as well as computer science. At the same time, it has great value for anyone who wishes to develop the ability to think abstractly, analyze mathematical situations, and extend ideas to a new context. The course introduces students to the basic concepts of Analysis, which are convergence (i.e., the existence of the limit of a sequence or function), the derivative, and the integral of a function. Initially, the method of Mathematical Induction is presented, which is a very powerful and flexible method for proving propositions concerning natural numbers. Then, a complete study of sequences and series of real numbers and their convergence is made, with emphasis on notable limits of sequences and basic criteria for the convergence of series. Also, the concept of the limit and continuity of a real function is given, and the close relationship that connects them with the concept of the limit of a sequence (e.g., the principle of transfer). In addition, the fundamental properties of continuous functions defined on intervals are presented. The concept of the derivative, as well as the basic related theorems (e.g., Mean Value Theorem), are presented immediately after. Particular emphasis is given to Taylor s Theorem and its applications (e.g., approximate calculations, proof of functional inequalities, finding local extrema of a function, etc.). Next, power series are studied, which are the simplest and most useful forms of series of functions (radius and interval of convergence, differentiation and integration of power series), and the power series expansions of basic functions (e.g., exponential, trigonometric functions, logarithm) are proven. Then, the concept of the antiderivative, the indefinite integral, and some basic integration techniques are presented. The definite Riemann integral of a real function has its foundations in Archimedes method of exhaustion and plays a dominant role in Analysis and its applications (e.g., calculation of areas and volumes). The course presents the definition, examples, and basic properties of the definite integral. The main focus is on the definite integral of continuous or piecewise continuous functions. The Fundamental Theorem of Calculus is also presented, which allows us to analytically calculate a plethora of definite integrals. Finally, generalized integrals of the first and second kind are studied, and the basic convergence criteria are given. Skills: Upon successful completion of the course, the student will mainly be able to: • Understand the concepts of the limit of a sequence or function, continuity, derivative, and definite integral of a function. • Study sequences, series, and generalized integrals for convergence. • Calculate definite and indefinite integrals with ease. • Apply Taylor s Theorem for approximate calculations, proving inequalities, and locating local extrema of a function. • Expand a function whose formula involves some of the basic real functions into a power series.