Προβολή πηγαίου κώδικα για τη σελίδα Infinitesimal Calculus II (MAY211)

=== General ===

{| class="wikitable"
|-
! School
| School of Science
|-
! Academic Unit
| Department of Mathematics
|-
! Level of Studies
| Undergraduate
|-
! Course Code
| MAY211
|-
! Semester
| 2
|-
! Course Title
| Infinitesimal Calculus II
|-
! Independent Teaching Activities
| Lectures, laboratory exercises (Weekly Teaching Hours: 5, Credits: 7.5)
|-
! Course Type
| General Background
|-
! Prerequisite Courses
| None (from the typical point of view). Without the knowledge earned from the course “Infinitesimal Calculus I” will be nearly impossible to follow this course.
|-
! Language of Instruction and Examinations
| Greek
|-
! Is the Course Offered to Erasmus Students
| Yes (exams in English are provided for foreign students)
|-
! Course Website (URL)
| http://www.math.uoi.gr/GR/studies/undergraduate/courses/may211.htm
|}

=== Learning Outcomes ===

{| class="wikitable"
|-
! Learning outcomes
|
This course is the sequel of the course “Infinitesimal Calculus I”. The student will get in contact with more notions and techniques in the branch of Analysis. In this course the students:
* Are taught the notions of convergence and absolute convergence of series. They learn criteria and theorems concerning these notions as well as they learn how to compute sums of series. They are introduced in the notion of power series and they learn how to calculate the radius of convergence of a power series.
* Are taught the notion of uniform continuity and they learn to distinguish this notion from continuity. 
* Are taught the notion of Riemann integral and various theorems concerning this notion. They also learn various integrating techniques.
* Are taught Taylor’s theorem and they learn to write a given function as a Taylor series.
|-
! General Competences
|
The course provides inductive and analytical thinking, the students evolve their computational skills and they get knowledge necessary for other courses during their undergraduate studies.
|}

=== Syllabus ===

Series, convergence of series and criteria for convergence of series. Dirichlet’s criterion, D’ Alembert’s criterion, Cauchy’s criterion, integral criterion. Series with alternating signs and Leibnitz’s theorem. Absolute convergence and reordering of series, Power series, radius of convergence of power series.
<br/>
Uniform continuity, definition and properties. Characterization of uniform continuity via sequences. Uniform continuity of continuous functions defined on closed intervals.
<br/>
Riemann integral, definition for bounded functions defined on closed intervals.  Riemann’s criterion, integrability of continuous functions. Indefinite integral and the Fundamental theorem of Calculus. Mean Value theorem of integral calculus, integration by parts, integration by substitution.  Integrals of basic functions, integrations of rational functions. Applications of integrals, generalized integrals, relation between generalized integrals and series.
<br/>
Taylor polynomials, Taylor’s Theorem, forms of the Taylor remainder. Taylor series and expansions of some basic functions as Taylor series.  

=== Teaching and Learning Methods - Evaluation ===

{| class="wikitable"
|-
! Delivery
|
Due to the theoretical nature of this course the teaching is exclusively given in the blackboard by the teacher.
|-
! Use of Information and Communications Technology
|
The students may contact their teachers by electronic means, i.e.  by e-mail.
|-
! Teaching Methods
|
{| class="wikitable"
! Activity
! Semester Workload
|-
| Lectures (13x5)
| 65
|-
| Solutions of exercises
| 22.5
|-
| Individual study
| 100
|-
| Course total 
| 187.5
|}
|-
! Student Performance Evaluation
|
* Exams in the end of the semester (mandatory).
* Assignments of exercises during the semester (optional).
|}

=== Attached Bibliography ===

* Γενικά Μαθηματικά - Απειροστικος Λογισμός τόμος Ι, Χ. Αθανασιάδης, Ε. Γιαννακούλιας, Σ. Γιωτόπουλος, Εκδόσεις Συμμετρία.
* Απειροστικός Λογισμός Τόμος ΙΙα, Σ. Νεγρεπόντης, Σ. Γιωτόπουλος, Ε. Γιαννακούλιας, Εκδόσεις Ζήτη. 
* Απειροστικός Λογισμός τομος Β, Σ. Ντούγιας, Leader Books.  
* Thomas, Απειροστικός Λογισμός, R.L. Finney, M.D. Weir, F.R.Giordano, Πανεπιστημιακές Εκδόσεις Κρήτης, (Απόδοση στα ελληνικά: Μ. Αντωνογιαννάκης).  
* Διαφορικός και Ολοκληρωρικός Λογισμός: Μια εισαγωγή στην Ανάλυση, Michael Spivak, Πανεπιστημιακές Εκδόσεις Κρήτης (Μετάφραση στα ελληνικά: Α. Γιαννόπουλος).