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Μαθηματικός Λογισμός

 Title: Μαθηματικός Λογισμός Lesson Code: 321-1105 Semester: 1 ECTS: 5 Theory Hours: 3 Lab Hours: 2 Faculty: Fotiadis Georgios

Content outline

Mathematical induction. Completeness of the real numbers. Functions. Limits. Continuity, theorems of continuous functions. Uniform continuity. Differentiation, derivative of inverse functions, derivatives of trigonometric functions, differential. Applications of derivatives, extreme values of functions, concavity, curve sketching, Cauchy mean value theorem, L’ Hopital rule, graphical method of solving autonomous differential equations, Newton’s approximation method. Integral, indefinite, definite, techniques of integration. Volume of solids of revolution. Improper integrals. Transcendental functions. Separable, linear differential equations of first order. Taylor’s formula.

Learning outcomes

The purpose of the course is to give a complete and working knowledge of differential and integral calculus. It covers and expands material presented in the last years of high school, including functions, basic calculus, limits, derivatives and integrals. One objective of the course is to provide a solid background to the analysis of functions of a single variable and to expose the mathematical rigor through the proofs of most of the theorems and propositions. For example, one of the goals is to introduce the student to the definitions of the concept of the limit of a function or that of the continuity so that concrete examples of functions can be treated using these definitions. At the same time, it also focuses on direct applications of the material covered to a number of problems from everyday life, from geometry (areas, volumes) or from physics. The student should realize that beyond the terse formalities used in the proofs, there is a very vivid and practical aspect in calculus. Similarly, the definition of the definite integral as summation should be understood, but at the same time a variety of integration techniques should be taught for practically computing complicated integrals. More advanced topics such as improper integrals or solving simple differential equations or a presentation of Taylor theorem should also appear.

Prerequisites

Not required.

Basic Textbooks

1. Finney R.L, Weir M.D, Giordano F.R., Thomas’ Calculus, Vol I, Crete University Press.
2. Instructor’s notes.

1. Calculus, Vol I, S. Negrepontis, S. Giotopoulos, E. Giannakoulias, Symmetria Edt.
2. Calculus, M. Spivak, Publish or Perish, Inc.
3. Answer Book for Calculus, M. Spivak, Publish or Perish, Inc.
4. A first course in Calculus, S. Lang, Springer.

Learning Activities and Teaching Methods

The course evaluation derives from:

• 3 compulsory exercises during the semester: counting 30% in the final grade.
• final exam: counting 70% in the final grade.

Final Grade = (0.3*M.V. Exercises) + (0.7*Final Exam)

One must have: Final Grade >= 5

Lectures: 39 hours

Lab-based exercises: 20 hours

Personal study: 62 hours

Final examination: 3 hours

Total: 124 hours (5 ECTS)

Language of Instruction
Greek, English (for Erasmus students)

Μode of delivery

Face-to-face.

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