Infinitesimal Calculus II (MAY211)

General

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 (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

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.
Uniform continuity, definition and properties. Characterization of uniform continuity via sequences. Uniform continuity of continuous functions defined on closed intervals.
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.
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

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	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).