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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

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

ASISTANT PROFESSOR SEDEF KARAKILIÇ

Offered to

Mathematics (Evening) Mathematics

Course Objective

The aim of the course is to develop rigorously the main concepts and properties of the following topics: Sequences and series of numbers, Continuity, Differentiation, Integration, Uniform convergence of sequences and series of functions.

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 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 in R Limits of sequences. Limit theorems. Cauchy sequences. Monotone Convergence and Bolzano-Weierstrass Theorems. 3 Sequences in R Sequential Compactnes. Compactness. Heine-Borel Theorem. 4 Continuity. Continuity. Uniform continuity. Extreme Value and Intemediate Value Theorems. 5 Continuity. Images and Inverses; Monotone Functions. Limits. 6 Differentiability. The derivative.Algebra of Derivatives. Differentiating Inverses and Compositions. 7 Differentiability. Mean Value Theorems. L Hospital s rule 8 MIDTERM 9 Integrability. Darboux Sums; Upperand Lower Integrals. The Archimedes-Riemann Theorem. 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. Review of sequences and series of numbers. Pointwise and uniform convergence of sequences of functions. 13 Sequences and Series of Functions. Approximation by Taylor Polynomials. The Lagrange Remainder Theorem. Uniform Convergence of Taylor Polynomials. The Cauchy Integral Remainder Theorem. 14 Sequences and Series of Functions. Uniform Convergence of Power series. A Nonanalytic, infinitely differentiable function. A continuous Nowhere Differentiable Function.

Recomended or Required Reading

Textbook(s): Fitzpatrick, P.M., Advanced Calculus, 2. edition, AMS, 2009 Supplementary Book(s):Wade, William R., Introduction to Analysis, 4. edition, Pearson, 2010 William F. T., Introduction to Real Analysis, Pearson, 2003

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 FIN FINAL EXAM 3 FCG FINAL COURSE GRADE MTE * 0.40 + FIN * 0.60 4 RST RESIT 5 FCG FINAL COURSE GRADE MTE * 0.40 + RST * 0.60

Further Notes About Assessment Methods

None

Assessment Criteria

Midterm Final

Language of Instruction

English

Course Policies and Rules

To be announced.

Contact Details for the Lecturer(s)

Room: B219 , sedef.erim@deu.edu.tr

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 13 4 52 Tutorials 13 2 26 Preparations before/after weekly lectures 12 5 60 Preparation for midterm exam 1 34 34 Preparation for final exam 1 36 36 Final 1 3 3 Midterm 1 3 3 TOTAL WORKLOAD (hours) 214

Contribution of Learning Outcomes to Programme Outcomes

PO/LO PO.1 PO.2 PO.3 PO.4 PO.5 PO.6 PO.7 PO.8 PO.9 PO.10 PO.11 PO.12 PO.13 LO.1 5 5 3 3 3 LO.2 5 5 3 3 3 LO.3 5 5 3 3 3 LO.4 5 5 3 3 3 LO.5 5 5 3 3 3 LO.6 5 5 3 3 3