## MATHS 2100 - Real Analysis II

### North Terrace Campus - Semester 2 - 2018

Much of mathematics relies on our ability to be able to solve equations, if not in explicit exact forms, then at least in being able to establish the existence of solutions. To do this requires a knowledge of so-called "analysis", which in many respects is just Calculus in very general settings. The foundations for this work are commenced in Real Analysis, a course that develops this basic material in a systematic and rigorous manner in the context of real-valued functions of a real variable. Topics covered are: Basic set theory. The real numbers and their basic properties. Sequences: convergence, subsequences, Cauchy sequences. Open, closed, and compact sets of real numbers. Continuous functions and uniform continuity. The Riemann integral. Differentiation and Mean Value theorems. The Fundamental Theorem of Calculus. Series. Power series and Taylor series. Convergence of sequences and series of functions.

• General Course Information
##### Course Details
Course Code MATHS 2100 Real Analysis II Mathematical Sciences Semester 2 Undergraduate North Terrace Campus 3 Up to 3.5 hours per week Y MATHS 1012 ongoing assessment 30%, exam 70%
##### Course Staff

Course Coordinator: Dr Raymond Vozzo

##### Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
##### Course Learning Outcomes
On successful completion of this course, students will be able to

1. describe fundamental properties of the real numbers that lead to the formal development of real analysis;

2. comprehend rigorous arguments developing the theory underpinning real analysis;

3. demonstrate an understanding of limits and how they are used in sequences, series, differentiation and integration;

4. construct rigorous mathematical proofs of basic results in real analysis;

5. appreciate how abstract ideas and rigorous methods in mathematical analysis can be applied to important practical problems.

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)
Deep discipline knowledge
• informed and infused by cutting edge research, scaffolded throughout their program of studies
• acquired from personal interaction with research active educators, from year 1
• accredited or validated against national or international standards (for relevant programs)
1,2,3,4,5
Critical thinking and problem solving
• steeped in research methods and rigor
• based on empirical evidence and the scientific approach to knowledge development
• demonstrated through appropriate and relevant assessment
1,2,3,4,5
Teamwork and communication skills
• developed from, with, and via the SGDE
• honed through assessment and practice throughout the program of studies
• encouraged and valued in all aspects of learning
1,2,3,4,5
Self-awareness and emotional intelligence
• a capacity for self-reflection and a willingness to engage in self-appraisal
• open to objective and constructive feedback from supervisors and peers
• able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
1,2,3,4,5
• Learning Resources
None.
##### Online Learning
Assignments, tutorial exercises, handouts, and course announcements will be posted on MyUni.
• Learning & Teaching Activities
##### Learning & Teaching Modes
This course relies on lectures as the primary delivery mechanism for the material. Tutorials supplement the lectures by providing exercises and example problems to enhance the understanding obtained through lectures. A sequence of written assignments provides assessment opportunities for students to gauge their progress and understanding.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

 Activity Quantity Workload hours Lecture 36 90 Tutorials & Quizzes 6 18 Assignments 6 42 Test 1 8 TOTAL 158
##### Learning Activities Summary
Lecture Outline

Basic logic and set theory. The real numbers and their defining properties. (Lectures 1-4)
Sequences: convergence, properties of limits, subsequences. (Lectures 5-8)
The topology of the real numbers. (Lectures 9-11)
Continuity. The key properties of continuous real-valued functions of a real variable. (Lectures 12-14)
The Riemann integral. (Lectures 16-19)
Differentiation. Mean Value theorems, l'Hôpital's rules. (Lectures 20-22)
The inverse function theorem. The fundamental theorem of calculus. (Lectures 23-25)
Series. Tests for convergence. (Lectures 26-29)
Power series. Taylor's theorem. Sequences and series of functions. (Lectures 30-33)
Convergence of sequences and series of functions. (Lectures 34-36)

Tutorials

Tutorials will be held in every odd week, covering material from the preceding two weeks of lectures. The first tutorial, in Week 1, will be a review of relevant ideas from Mathematics IA and IB Calculus.
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

##### Assessment Summary
 Component Task type Due Weighting Objective Assessed Class Exercises Formative and summative Even weeks 10% All Tutorial participation Formative and summative Odd weeks 3% All Quizzes Formative and summative Ongoing 5% All Test Summative Lecture 15 12% 1,2,3,5 Exam Summative Exam period 70% All
##### Assessment Related Requirements
An aggregate score of at least 50% is required to pass the course.
##### Assessment Detail
Assignments will be distributed in the middle of each odd week and due at the end of each even week. Each will mainly cover material from the previous weeks' lectures, but may include material from the next one or two lectures.  In all, assignments count 10% towards the final mark for the course.

The midsemester test will be held during the 15th lecture period. It will cover the material covered in lectures up to the end of Lecture 12. It will be worth 12% of the final mark.

Quizzes, based on lectures and tutorials, will be administered throughout the semester on my-uni.  These will be worth 5% of the final mark.  Attendance and participation at tutorials is expected and will be worth 3% of the final mark subject to satisfactory performance.  Satisfactory performance is defined to be attendance and active participation in a minimum of 5 out of the 6 tutorials.

##### Submission

All written assignments are to be submitted to the designated hand-in boxes in the School of Mathematical Sciences.

Late assignments will not be accepted.

Assignments will have a two week turn-around time for feedback to students.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
FNS   Fail No Submission
F 1-49 Fail
P 50-64 Pass
C 65-74 Credit
D 75-84 Distinction
HD 85-100 High Distinction
CN   Continuing
NFE   No Formal Examination
RP   Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
• Fraud Awareness

Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's studentâ€™s disciplinary procedures.

The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.

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