| Module Code: MAT1032 |
Module Title: REAL ANALYSIS 1 |
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Module Provider: Mathematics
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Short Name: MAT1032
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Level: HE1
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Module Co-ordinator:
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Number of credits: 15
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Number of ECTS credits: 7.5
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| Module Availability |
| Semester 1 |
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| Assessment Pattern |
Assessment Pattern
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Unit(s) of Assessment
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Weighting Towards Module Mark( %)
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One class test:
Exam:
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40
60
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Qualifying Condition(s)
An aggregate mark of 40% is required to pass this module.
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| Module Overview |
| The objective of this module is to provide an introduction to analysis, which is the branch of mathematics that rigorously studies functions, continuity and limit processes, such as differentiation and integration. The module intends to lead to a deeper understanding of what it means when a sequence or series is said to converge. Historic motivation and the rigorous use of definitions and logic play a central role. Tools such as convergence tests are presented and their validity proved. |
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| Prerequisites/Co-requisites |
| None. |
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| Module Aims |
| The objective of this module is to provide an introduction to analysis, which is the branch of mathematics that rigorously studies functions, continuity and limit processes, such as differentiation and integration. The module intends to lead to a deeper understanding of what it means when a sequence or series is said to converge. Historic motivation and the rigorous use of definitions and logic play a central role. Tools such as convergence tests are presented and their validity proved. |
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| Learning Outcomes |
At the end of the module a student should be able to:
• demonstrate understanding of the real numbers, their axioms and the role of completeness in the existence of limits and solutions to equations.
• calculate limits of sequences and (power) series, and prove/disprove converge using the definitions.
• properly interpret and apply quantifiers in mathematical statement.
• quote and apply basic theorems in analysis, notably the Theorem of Bolzano-Weierstrass and convergence tests.
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| Module Content |
• The axioms of real numbers. Denseness of rational and irrational numbers. Maximum, minimum, supremum and infimum of sets, sequences and functions. The triangle inequality.
• Natural induction, set notation, cardinalities of sets (in particular the rationals and reals)
• Axiom of Completeness, and its consequence to existence of limits. Role of quantifiers in stating and verifying mathematical definitions.
• Sequences and convergence, and their properties. Boundedness, Cauchy sequences, subsequences and the Theorem of Bolzano-Weierstrass.
• Infinite series, convergence and absolute convergence. Convergence tests. Power series, radius and region of convergence. Rearrangement of series.
• Historical motivation of analysis in mathematics.
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| Methods of Teaching/Learning |
Teaching is by lectures, tutorials and tests. Learning takes place through lectures, tutorials, tests, exercises and background reading.
3 lecture/tutorial hours per week for 11 weeks.
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| Selected Texts/Journals |
Essential
J. M. Howie, Real Analysis, Springer (2001) 2nd edition, Available in paperback from Springer or Amazon.co.uk
Recommended
R.P. Burn, Numbers and Functions, Steps into Analysis, Second Edition, Cambridge University Press (2000).
Available in paperback from CUP or Amazon.co.uk
Supplementary Reference Texts (all available in the library)
P.E. Kopp, Analysis, Arnold Publishers, (1990).
K.E. Hirst, Numbers Sequences and Series, Arnold Publishers, (1995).
C. McGregor, J. Nimmo and W. Stothers, Fundamentals of University Mathematics, Albion Publishers, (1994).
M. Spivak, Calculus W.A. Benjamin (1967).
S. Lang, Analysis I Addison-Wesley (1968).
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| Last Updated |
21 April 2011 |
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