MA203-5-SP-CO:
Real Analysis
PLEASE NOTE: This module is inactive. Visit the Module Directory to view modules and variants offered during the current academic year.
2023/24
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Spring
Undergraduate: Level 5
Inactive
Monday 15 January 2024
Friday 22 March 2024
15
03 January 2024
Requisites for this module
MA101
(none)
(none)
(none)
MA213, MA302
This is an introductory epsilon-delta analysis module. Students will develop their sense of rigour and precision.
The aims of this module are:
- To introduce the idea of epsilon-delta rigorous analysis.
- To enhance students’ ability at understanding and writing proofs of results in real analysis.
- To enhance students’ skills at using results of real analysis.
By the end of the module, students will be expected to:
- Understand basic proofs and proof techniques in relation to real numbers, suprema and infima, limits (sequences, series and functions), continuity and differentiability.
- Be able to give proofs of some simple standard facts in rigorous real analysis.
- Be able to use these techniques on appropriate problems, including working out proofs of simple results related to the module.
Indicative syllabus:
Numbers systems such as the real and rational numbers. Basic properties of real numbers: field structure, order relation, triangle inequality, Archimedes' Axiom. (No formal construction of the real numbers). Suprema and infima. Dedekind's axiom and its use in proving that bounded-above, non-empty sets of reals have suprema.
Sequences and convergence. Sums, differences, scalar multiples, products and quotients of convergent sequences.
Cauchy sequences and the equivalence of the Cauchy property and convergence.
Series. Comparison and ratio tests. Absolute convergence implies convergence.
Power series and the radius of convergence.
Limits of functions, continuous functions of one real variable. Related results such as sums, products, quotients and compositions (chain rule).
Intermediate Value Theorem. Boundedness and attainment of bounds for continuous functions on closed bounded intervals.
Differentiable functions. Examples of differentiable and non-differentiable functions. Differentiable implies continuous.
Theorems related to continuous and differentiable functions, such as Rolle's Theorem and the Mean Value Theorem.
Teaching in the School will be delivered using a range of face-to-face lectures, classes, and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.
This module does not appear to have any essential texts. To see non - essential items, please refer to the module's
reading list.
Assessment items, weightings and deadlines
Coursework / exam |
Description |
Deadline |
Coursework weighting |
Exam |
Main exam: In-Person, Open Book (Restricted), 120 minutes during Summer (Main Period)
|
Exam |
Reassessment Main exam: In-Person, Open Book (Restricted), 120 minutes during September (Reassessment Period)
|
Exam format definitions
- Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
- In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
- In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
- In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary,
for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.
Your department will provide further guidance before your exams.
Overall assessment
Reassessment
Module supervisor and teaching staff
Dr Tao Gao, email: t.gao@essex.ac.uk.
Dr Tao Gao
t.gao@essex.ac.uk
Yes
Yes
No
Prof Stephen Langdon
Brunel University London
Professor
Dr Rachel Quinlan
National University of Ireland, Galway
Senior Lecturer in Mathematics
Available via Moodle
Of 2562 hours, 0 (0%) hours available to students:
2562 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).
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