![]() | Department of Mathematics | |||
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Credits: 10 | Convenor: Dr. F. Neumann | Semester: 2 (weeks 15 to 26) |
Prerequisites: | essential: MA1101, MA1102 | |
Assessment: | Coursework and tests: 20% | One and a half hour examination: 80% |
Lectures: | 18 | Problem Classes: | 5 |
Tutorials: | none | Private Study: | 52 |
Labs: | none | Seminars: | none |
Project: | none | Other: | none |
Surgeries: | none | Total: | 75 |
Students should be able to understand, reproduce and apply the main results and proofs in this module.
Students should understand the difference between the real and rational numbers and be able to solve routine problems on the convergence of sequences, the existence of limits of functions and on the continuity and differentiability of functions.
Questions of the nature of the real numbers and how they differ from the rationals. Irrational numbers. The axioms of the real numbers. Definition of supremum and examples. Proofs of basic properties of suprema. Statement of the completion property of the reals. Infima as `dual' ideas to those concerning suprema. Construction of the real numbers.
Concept of a sequence and the notion of convergence to a limit. Examples. Proof of results on sums, products and quotients of convergent sequences. Proof of monotone convergence theorem. Applications to computations of limits of sequences defined by rational polynomials and by inductive formulæ.
The limit of a function; continuity via limits. Application to proving results on sums and products of continuous functions from the work on sequences. Examples (with proofs) of functions having or not having limits at certain points.
Continuity via ,
language. Proof of the equivalence between the limit definition of continuity and the
,
version. Examples, with full proof, of continuous and of non-continuous functions. Proof of continuity of sums, products and
composites of continuous functions; continuity of polynomial functions. Proof of the intermediate value theorem.
Application of the idea of limit to rigorous definition of differentiablity. Proof of differentiablity of polynomial functions, via that of sums, products etc. Examples (with proofs) of functions differentiable or not differentiable at certain points. Proof that differentiable implies continuous.
K. G. Binmore, Mathematical Analysis, Cambridge.
R. Hoggarty, Fundamentals of Mathematical Analysis, Addison-Wesley.
S. Abbott, Understanding Analysis, Springer.
R. G. Bartle, D. R. Sherbert, Introduction to Real Analysis, Jon Wiley & Sons..
J. Lewin, An Interactive Introduction to Mathematical Analysis, Cambridge.
R. A. Adams, Calculus, Addison-Wesley.
J. C. Burkill, A First Course in Mathematical Analysis, Cambridge.
M. Hart, An Guide to Analysis, Macmillan.
J. B. Read, An Introduction to Mathematical Analysis, Oxford.
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Author: C. D. Coman, tel: +44 (0)116 252 3902
Last updated: 2004-02-21
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This document has been approved by the Head of Department.
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