# ANALYSIS OF REAL NUMBERS AND FUNCTIONS

MAT2ANA

2015

Credit points: 15

## Subject outline

The limits of sequences and limits of functions are studied in this subject. Initially we study them in one-dimensional space and then in higher dimensions. We also study series and various tests are derived to determine the convergence or otherwise of these series. We then extend the basic idea of limit to include sequences of functions and sequences of sets in metric spaces. A powerful theorem called The Contraction Mapping Theorem will be derived. This theorem plays a fundamental role in analysis and its applications. We will use it to establish the existence and uniqueness of solutions to certain differential equations.

SchoolSchool Engineering&Mathematical Sciences

Credit points15

Subject Co-ordinatorYuri Nikolayevsky

Subject year levelYear Level 2 - UG

Exchange StudentsYes

## Subject particulars

### Subject rules

Prerequisites MAT1CLA or (MAT1NLA and MAT1CDE)

Co-requisitesN/A

Incompatible subjectsN/A

Equivalent subjectsN/A

Special conditionsN/A

## Learning resources

Resource TypeTitleResource RequirementAuthor and YearPublisher
ReadingsPrinted subject text available from University BookshopPrescribedN/AN/A

## Graduate capabilities & intended learning outcomes

01. Calculate limits of certain sequences and functions and justify these calculations.

Activities:
Discussed and demonstrated in lectures. Related problems solved by students in practice classes. Assignment questions, with feedback.
Creative Problem-solving(Creative Problem-solving)
Critical Thinking(Critical Thinking)
Writing(Writing)
Quantitative Literacy/ Numeracy(Quantitative Literacy/ Numeracy)
Discipline-specific GCs(Discipline-specific GCs)

02. Prove the convergence or otherwise of certain series by applying appropriate tests.

Activities:
Discussed and demonstrated in lectures. Related problems solved by students in practice classes. Assignment questions, with feedback.
Inquiry/ Research(Inquiry/ Research)
Critical Thinking(Critical Thinking)
Discipline-specific GCs(Discipline-specific GCs)
Creative Problem-solving(Creative Problem-solving)
Quantitative Literacy/ Numeracy(Quantitative Literacy/ Numeracy)
Writing(Writing)

03. Manipulate bounds and least upper bounds

Activities:
Discussed and demonstrated in lectures. Related problems solved by students in practice classes. Assignment questions, with feedback.
Creative Problem-solving(Creative Problem-solving)
Discipline-specific GCs(Discipline-specific GCs)
Critical Thinking(Critical Thinking)
Writing(Writing)
Quantitative Literacy/ Numeracy(Quantitative Literacy/ Numeracy)

04. Perform calculations involving function and metric spaces.

Activities:
Discussed and demonstrated in lectures. Related problems solved by students in practice classes. Assignment questions, with feedback.
Critical Thinking(Critical Thinking)
Quantitative Literacy/ Numeracy(Quantitative Literacy/ Numeracy)
Writing(Writing)
Creative Problem-solving(Creative Problem-solving)
Discipline-specific GCs(Discipline-specific GCs)

05. Apply the contraction map theorem in various situations.

Activities:
Discussed and demonstrated in lectures. Related problems solved by students in practice classes. Assignment questions, with feedback.
Discipline-specific GCs(Discipline-specific GCs)
Quantitative Literacy/ Numeracy(Quantitative Literacy/ Numeracy)
Creative Problem-solving(Creative Problem-solving)
Writing(Writing)
Inquiry/ Research(Inquiry/ Research)
Critical Thinking(Critical Thinking)

06. Communicate your understanding of analysis using both words and precise mathematical symbolism.

Activities:
Discussed and demonstrated in lectures. Related problems solved by students in practice classes. Assignment questions, with feedback.
Discipline-specific GCs(Discipline-specific GCs)
Writing(Writing)

## Subject options

Select to view your study options…

Start date between: and    Key dates

## Melbourne, 2015, Semester 1, Day

### Overview

Online enrolmentYes

Maximum enrolment sizeN/A

Enrolment information

Subject Instance Co-ordinatorYuri Nikolayevsky

### Class requirements

LectureWeek: 10 - 22
Two 1.0 hours lecture per week on weekdays during the day from week 10 to week 22 and delivered via face-to-face.

PracticalWeek: 10 - 22
Two 1.0 hours practical per week on weekdays during the day from week 10 to week 22 and delivered via face-to-face.