MMath Mathematics / Course details
Year of entry: 2021
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Course unit details:
Real Analysis A
|Unit level||Level 2|
|Teaching period(s)||Semester 1|
|Offered by||Department of Mathematics|
|Available as a free choice unit?||No|
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.
|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|
This course unit aims to give a rigorous treatment of Real Analysis (continuity, differentiability and Riemann integration).
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.
- 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. 
- Differentiability. Differentiability of real-valued functions, sums, products and quotients of continuous functions, Rolle's Theorem, the Mean Value Theorem, Taylor's Theorem. 
- Integration. Definition of the Riemann integral, the Fundamental Theorem of Calculus. 
- Coursework; An in-class test in reading week 20%.
- End of semester examination; Weighting within unit 80%.
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.
|Scheduled activity hours|
|Independent study hours|
|Mark Coleman||Unit coordinator|
This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.
Please see Blackboard / course unit related emails for any further updates