# Calculus I

Course ID
1ΚΠ01
Επίπεδο
Είδος
Compulsory
Εξάμηνο
1
Περίοδος
Fall Semester
ECTS
5
Ώρες Θεωρίας
4
Ώρες Εργαστηρίου
-

#### Description

• Sets. Mappings. Real numbers (R). Axiomatic foundations of the real numbers. Rational numbers. Intervals. Distance. Neighborhoods. Classification of points of R. Open and closed sets. • Sequence of real numbers. Limit of a sequence. Operations with limits. The Cauchy criterion. Monotonic sequences. Contraction sequence. Recurrence sequences. Difference equations. Series. Basic tests for convergence of series. • Continuity, derivative of a function. Basic theorems. Leibniz’s rule. Derivative of composite function. Inverse functions. Derivative of an inverse function. Inverse trigonometric functions. Hyperbolic functions. Inverse hyperbolic functions. Derivative of implicit functions. Differential of a function. Derivatives and differentials of higher order. Taylor’s polynomials and Taylor’s series. Power series. • Indefinite integral. Basic methods of computing indefinite integral. The Riemman integral. Properties of the definite integral. The fundamental theorem of calculus. Applications of the definite integral. Fourier series. • Improper integral. Relationship between improper integrals and series. Basic tests for convergence of improper integrals. • Ordinary differential equations. Separable differential equation. Linear differential equations of first order. Linear differential equations of second order with constant coefficients. Euler differential equation.

#### Course objectives

The course aims to teach theorems and rules, to develop critical and analytical thinking so that interdisciplinary problems could be modelled and solved with mathematical accuracy and discipline.

Upon successfully completing this course the student should:

• understand the fundamental concepts of functions of one real variable, such as the limit, the continuity, the derivative, be able to synthesize and apply the properties of these concepts in the study of a real function of one real variable.
• have understood the concepts of the indefinite integral, the definite integral and the improper integral, know methods and techniques of integration so as to apply them when calculating the area of ​​a 2-dimensional region, when calculating volume / surface of ​​a 3-dimensional region by using appropriate integrals, when solving ordinary differential equations or transformation Laplace, Fourier, etc.
• be familiar with distinct concepts, such as sequences and series, know the tests for convergence of the series and how they are applied as regards approaching basic functions.

The course aims at the acquisition of knowledge, ideas and skills that are to be implemented in other courses related to Informatics and Biomedicine.

#### Textbooks/Bibliography

• Μαθηματική Ανάλυση Ι , Ρασσιάς Θ., ΤΣΟΤΡΑΣ ΑΝ ΑΘΑΝΑΣΙΟΣ, 1η έκδ./2014, ΑΘΗΝΑ, 41955063
• Διαφορικός και Ολοκληρωτικός Λογισμός , Spivak Michael, “ΙΔΡΥΜΑ ΤΕΧΝΟΛΟΓΙΑΣ & ΕΡΕΥΝΑΣ-ΠΑΝΕΠΙΣΤΗΜΙΑΚΕΣ ΕΚΔΟΣΕΙΣ ΚΡΗΤΗΣ”, 2H/2010, ΗΡΑΚΛΕΙΟ ΚΡΗΤΗΣ, 213
• Απειροστικός λογισμός, Briggs William, Cochran Lyle, Gillett Bernard, ΕΚΔΟΣΕΙΣ ΚΡΙΤΙΚΗ ΑΕ, 1η έκδ./2018, ΑΘΗΝΑ, 77109719
• Εφαρμοσμένος Απειροστικός Λογισμός, Τσίτσας Λ., Σ.ΑΘΑΝΑΣΟΠΟΥΛΟΣ & ΣΙΑ Ο.Ε., 2η έκδ. /2003, ΑΘΗΝΑ, 45390
• THOMAS ΑΠΕΙΡΟΣΤΙΚΟΣ ΛΟΓΙΣΜΟΣ, [George B. Thomas], Jr., Joel Hass, Christopher Heil, Maurice D. Weir, ΙΔΡΥΜΑ ΤΕΧΝΟΛΟΓΙΑΣ & ΕΡΕΥΝΑΣ-ΠΑΝΕΠΙΣΤΗΜΙΑΚΕΣ ΕΚΔΟΣΕΙΣ ΚΡΗΤΗΣ, 1η/2018, ΗΡΑΚΛΕΙΟ ΚΡΗΤΗΣ, 77107082
• Μαθηματική Ανάλυση, Μαρία Αδάμ-Ιωάννης Χατζάρας-Νικόλαος Ασημάκης, “Ελληνικά Ακαδημαϊκά Ηλεκτρονικά Συγγράμματα και Βοηθήματα – Αποθετήριο “”Κάλλιπος”””, Ηλεκτρονικό Βιβλίο/1η έκδοση2016, 59303650

#### Assessment method

Mandatory written exams at the end of the semester.