COURSE UNIT TITLE

: ANALYSIS I

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 2043 ANALYSIS I COMPULSORY 4 2 0 9

Offered By

Mathematics (English)

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

ASSISTANT PROFESSOR DIDEM COŞKAN ÖZALP

Offered to

Mathematics (English)
Mathematics (Evening)

Course Objective

The aim of the course is to develop rigorously the main concepts and properties of the sequences and series of numbers, continuity, differentiation, integration, sequences and series of functions together with the exact and formal proofs.

Learning Outcomes of the Course Unit

1   will be able to distinguish the Completeness axiom by understanding its consequences such as Monotone Convergence, Bolzano-Weierstrass and Heine-Borel Theorems.
2   will be able to use the definitions of continouos and uniform continouos functions and /or their sequential characterizations to prove their properties.
3   will be able to use the definition and the properties of a differentiable function.
4   will be able to understand the Riemann integrability of a bounded function on a bounded interval by means of Darboux sums and the Fundamental Theorems.
5   will be able to distinguish between pointwise and uniform convergence of the sequences and series of functions.
6   will be able to write complete and formal proofs for the problems related with the above topics.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 The Real Number System: Ordered field axioms. Well-ordering principle. Completeness axiom.
2 Sequences of Real Numbers: Limits of sequences. Limit theorems. Cauchy sequences. Monotone Convergence and Bolzano-Weierstrass Theorems.
3 Sequences of Real Numbers: Sequential Compactnes. Compactness. Heine-Borel Theorem.
4 Continuity: Limits. Images and Inverses. Monotone Functions.
5 Continuity: Uniform continuity. Extreme Value and Intemediate Value Theorems.
6 Differentiability: The derivative. Properties of Derivative.
7 Differentiability: Derivatives of inverse and composite functions.
8 Differentiability: Mean Value Theorem.
9 Integrability: Darboux Sums. Upper and Lower Integrals. Additivity, Monotonicity and Linearity.
10 Integrability: Continuiuty and Integrability. The Fundamental Theorems.
11 Integrability: Convergence of Darboux and Riemann Sums.
12 Sequences and Series of Functions: Pointwise and uniform convergence of sequences of functions.
13 Sequences and Series of Functions: Approximation by Taylor Polynomials. Uniform Convergence of Taylor Polynomials.
14 Sequences and Series of Functions: Uniform Convergence of Power series.

Recomended or Required Reading

Textbook(s): Fitzpatrick, Patrick M., Advanced Calculus, AMS, 2nd ed., 2009
Supplementary Book(s): Wade, William R., Introduction to Analysis, 4th ed., Pearson, 2010
Trench, William F., Introduction to Real Analysis, Pearson, 2003
Apostol, Tom M., Calculus and Linear Algebra, Vol. I, John Wiley & Sons, 1967
Apostol, Tom M., Calculus and Linear Algebra, Vol. II, John Wiley & Sons, 1969
Apostol, Tom M., Mathematical Analysis, Addison-Wesley Publishing, 1974
Rudin, W., Principles of Mathematical Analysis, McGraw-Hill, 1976
Kaplan, W., Advanced Calculus, Addison-Wesley Publishing, 1991
Taylor, A. E. and Mann, W.R., Advanced Calculus, John Wiley & Sons, 1983
Joel Hass, Maurice D. Weir, and George B. Thomas Jr., University Calculus, Early Transcendentals, 2nd ed.,International Edition Pearson, 2012

Planned Learning Activities and Teaching Methods

Lecture Notes
Text Book(s)
Solving Problems

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 MTE MIDTERM EXAM
2 QUZ QUIZ
3 FIN FINAL EXAM
4 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.40 + QUZ * 0.10 + FIN * 0.50
5 RST RESIT
6 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.40 + QUZ * 0.10 + RST * 0.50


Further Notes About Assessment Methods

None

Assessment Criteria

The weighted average of the student's midterm, quiz and final grades will be taken and the letter grade will be given according to the relative scoring method. If the student's letter grade is FD or FF, or a student except given exemption from attendance does not
satisfy the requirement of attendance, she/he will be assumed to be unseccessfull.

Language of Instruction

English

Course Policies and Rules

Any unethical behavior that occurs either in lessons or in exams will be dealt with as outlined in school policy. You can find the document "Fen Fakültesi Öğretim ve Sınav Uygulama Esasları" at https://fen.deu.edu.tr/tr/belge-ve-formlar/ and the document
"Önlisans ve Lisans Öğretim ve Sınav Yönetmeliği" at https://ogrenci.deu.edu.tr/regulations-and-directives/

Contact Details for the Lecturer(s)

Asst.Prof.Dr. Didem COŞKAN ÖZALP
E-mail: didem.coskan@deu.edu.tr
Office: B 351-2 (Faculty of Science, Block B, Third Floor)
Phone: +90 232 301 86 07

Office Hours

Will be announced later.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 14 4 56
Tutorials 14 2 28
Preparations before/after weekly lectures 14 5 70
Preparation for midterm exam 1 20 20
Preparation for quiz etc. 1 5 5
Preparation for final exam 1 30 30
Midterm 1 2 2
Quiz etc. 1 1 1
Final 1 2 2
TOTAL WORKLOAD (hours) 214

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11PO.12PO.13
LO.155333
LO.255333
LO.355333
LO.455333
LO.555333
LO.655333

CONTACT INFORMATION
Dokuz Eylül Üniversitesi Cumhuriyet Bulvarı No: 144 35210 Alsancak / İZMİR
Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

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