## MATH201 Real Analysis

First Semester |

MATH 201 is an introduction to the basic techniques of real analysis in the familiar context of single-variable calculus.

### Paper details

Analysis is, broadly, the part of mathematics which deals with limiting processes. The main examples students have met in school and first-year university are from calculus, where the derivative and integral are defined using quite different limiting processes. Real analysis is about real-valued functions of a real variable — in fact, exactly the kind of functions which are studied in calculus. The methods of analysis have been developed over the past two centuries to give mathematicians rigorous methods for deciding whether a formal calculation is correct or not. This paper discusses the basic ideas of analysis, and uses them to explain how calculus works. At the end of the semester, students should have a broader overview of calculus and a grounding in the methods of analysis which will prove invaluable in later years.

### Potential students

MATH 201 is compulsory for the Mathematics major and is of particular relevance also for students majoring in Statistics, Physics or any discipline requiring a quantitative analysis of systems and how they change with space and time. It is a prerequisite for MATH 301 (Hilbert spaces) and MATH 302 (Complex Analysis).

### Prerequisites

MATH 170

### Course Outline

The paper will cover the following topics:

- A review of the real number system
- The completeness axiom
- Limits of sequences and the algebra of limits
- Limits of functions and the algebra of limits
- Continuous functions and their algebraic properties
- The intermediate value theorem
- Differentiable functions and the algebra of differentiation
- The mean value theorem and Taylor's theorem
- The Riemann integral
- The fundamental theorems of calculus

### Lecturer

Dr. Lisa Orloff Clark, room 220, ext 7769, lclark@maths.otago.ac.nz

### Lectures

3 per week: Mondays at 12 noon, Wednesdays at 12 noon and Fridays at 9 a.m. (Lecture rooms TBA)

### Tutorial

2 per week: (Times TBA) Tutorials start in the second week of semester.

### Assessment

The internal assessment is made up of 50% from 4 assignments and 50% from a class test.

### Exam format

A combination of true/false, multiple choice, short-answer and long-answer questions; all questions to be answered.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = max(E, (4E + A + T)/6)**

where:

- E is the Exam mark
- A is the Assignments mark
- T is the Test mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic endeavours at the University of Otago are built upon an essential commitment to academic integrity.**

The two most common forms of academic misconduct are *plagiarism* and *unauthorised collaboration*.

#### Academic misconduct: **Plagiarism**

Plagiarism is defined as:

- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

#### Academic misconduct: **Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.