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# BSc Mathematics with Financial Mathematics / Course details

Year of entry: 2021

## Course unit details:Real Analysis A

Unit code MATH20101 10 Level 2 Semester 1 Department of Mathematics No

### Overview

This course explains how the basic ideas of the calculus of real functions of a real variable (continuity, differentiation and integration) can be made precise and how the basic properties can be deduced from the definitions. It builds on the treatment of sequences and series in MATH10242. Important results are the Mean Value Theorems, leading to the representation of some functions as power series (the Taylor series), and the Fundamental Theorem of Calculus which establishes the relationship between differentiation and integration.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Foundations of Pure Mathematics A MATH10101 Pre-Requisite Compulsory
Foundations of Pure Mathematics B MATH10111 Pre-Requisite Compulsory
Calculus and Vectors A MATH10121 Pre-Requisite Compulsory
Calculus and Vectors B MATH10131 Pre-Requisite Compulsory
Sequences and Series MATH10242 Pre-Requisite Compulsory

### Aims

This course unit aims to give a rigorous treatment of Real Analysis (continuity, differentiability and Riemann integration).

### Learning outcomes

On completion of this unit successful students will be able to:

• state the definition of the limit of a function;  calculate  the  limit  for  simple  functions; prove and apply the Rules for Limits to calculations for more complicated functions,
• state the definition of continuity; prove that simple functions are continuous at given points; prove and apply the Rules for Continuous functions to more complicated functions,
• state the definition of differentiable; prove that simple functions are differentiable and calculate their derivatives at given points; prove and apply the Rules for Derivatives to more complicated functions,
• prove and apply the Intermediate Value Theorem; Inverse function Theorem; various results on the composition of functions; various mean value theorems,
• calculate Taylor polynomials; state Taylor's Theorem with an error term; derive bounds on the error terms; state criteria when a Taylor series for a function converge to that function,
• state the definition of the Riemann integral; calculate the Riemann integral for various functions.

### Syllabus

• Limits. Limits of real-valued functions, sums, products and quotients of limits. [7 lectures]
• Continuity. Continuity of real-valued functions, sums, products and quotients of continuous functions, the composition of continuous functions. Boundedness of continuous functions on a closed interval. The Intermediate Value Theorem. The Inverse Function Theorem. [7]
• Differentiability. Differentiability of real-valued functions, sums, products and quotients of continuous functions, Rolle's Theorem, the Mean Value Theorem, Taylor's Theorem. [5]
• Integration. Definition of the Riemann integral, the Fundamental Theorem of Calculus. [3]

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Coursework; An in-class test in reading week 20%.
• End of semester examination; Weighting within unit 80%.

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• Mary Hart, Guide to Analysis, Macmillan Mathematical Guides, Palgrave Macmillan; second edition 2001.
• Rod Haggerty, Fundamentals of Mathematical Analysis, Addison-Wesley, second edition 1993.

### Study hours

Scheduled activity hours
Lectures 22
Tutorials 11
Independent study hours
Independent study 67

### Teaching staff

Staff member Role
Mark Coleman Unit coordinator