### Course Catalog 2014-2015 Basic

Basic Pori International Postgraduate Open University Search

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# MAT-60206 Mathematical Analysis, 5 cr

Janne Kauhanen

#### Lessons

 Study type P1 P2 P3 P4 Summer Implementations Lecture times and places Lectures Excercises 3 h/week  3 h/week +3 h/week +3 h/week MAT-60206 2014-01 Monday 12 - 14 , S2 Wednesday 13 - 14 , SE203

#### Requirements

Two mid-course exams or final exam. The grade of the course may be improved by bonus points collected from the weekly exercises.
Completion parts must belong to the same implementation

#### Learning Outcomes

This course introduces students to the fundamentals of mathematical analysis at an adequate level of rigor. Upon successful completion of the course, student will be able to: - Read mathematical texts and proofs, - Use the definitions and apply the basic results that are introduced during this course and - Produce rigorous proofs of results that arise in this course using direct and indirect proof, induction and epsilon-delta technique.

#### Content

 Content Core content Complementary knowledge Specialist knowledge 1. The real numbers: the structure and properties of the real numbers, least upper bound, open and closed sets. Limit points and the Bolzano-Weierstrass Theorem. 2. The limit and continuity of functions defined on subsets of the real line: arithmetic operations and the Intermediate Value theorem. Uniform continuity. 3. Differentiability of functions defined on subsets of the real line: linear approximation, arithmetic operations, the Chain Rule, the Mean Value Theorem and l'Hospitals Rule. Taylor's Theorem. 4. The Riemann integral: the definition using Riemann sums and upper and lower sums, arithmetic operations, existence of the integral using upper and lower sums, integrability of continuous functions, the Mean Value Theorem for Integrals and the Fundamental Theorem of Calculus. Testing the convergence of improper integrals and conditional convergence. 5. Sequences: convergence, monotonic sequences, pointwise and uniform convergence of sequences of functions. Caychy sequence. The effect of uniform convergence on the limit function with respect to continuity, differentiability and integrability.

#### Instructions for students on how to achieve the learning outcomes

If the student is mastering the concepts, results and short proofs in concrete examples the evaluation is 3. For the grades 4 and 5 the student should in addition to the previous level be able to independently apply theory to deduce new results.

#### Assessment scale:

Numerical evaluation scale (1-5) will be used on the course

#### Partial passing:

Completion parts must belong to the same implementation

#### Study material

 Type Name Author ISBN URL Edition, availability, ... Examination material Language Book Introduction to real analysis (Edition 2.04) William Trench Chapters 1-4 No English Summary of lectures Matemaattinen analyysi Janne Kauhanen Ladattavissa Moodlesta No Suomi

#### Prerequisites

 Course Mandatory/Advisable Description MAT-01160 Matematiikka 1 Mandatory MAT-01260 Matematiikka 2 Mandatory MAT-01360 Matematiikka 3 Mandatory MAT-01460 Matematiikka 4 Mandatory

Vaihtoehtoisesti Laajan matematiikan opintojaksot tai hyvin suoritetut Insinöörimatematiikan opintojaksot (18-19 op).

Prerequisite relations (Requires logging in to POP)

#### Correspondence of content

 Course Corresponds course Description MAT-60206 Mathematical Analysis, 5 cr MAT-43650 Mathematical Analysis, 6 cr MAT-60206 Mathematical Analysis, 5 cr MAT-60200 Mathematical Analysis, 5 cr

#### More precise information per implementation

 Implementation Description Methods of instruction Implementation MAT-60206 2014-01 The course is lectured in english.