Credits: 10 | Convenor: Dr. M. Marletta | Semester: 2 |
Prerequisites: | essential: MC144, MC145 | |
Assessment: | Coursework and tests: 20% | One and a half hour examination: 80% |
Lectures: | 18 | Classes: | none |
Tutorials: | 5 | Private Study: | 52 |
Labs: | none | Seminars: | none |
Project: | none | Other: | none |
Total: | 75 |
To be able to understand, reproduce and apply the main results and proofs in this module.
To understand the difference between the real and rational numbers.
To be able to solve routine problems on the continuity of functions, the convergence of sequences, the existence of limits of functions and on the differentiability of functions.
Continuity via ,
language. 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.
Definition of supremum and examples. Proofs of basic properties of suprema. Statement of the completion property of the reals. Proof of the intermediate value theorem. Infima as `dual' ideas to those concerning suprema. Sketch construction of the real numbers (Dedekind cuts).
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 theorems. Applications to computations of limits of sequences defined by rational polynomials and by inductive formulæ. Second sketch construction of the real numbers (equivalence classes of monotone increasing, bounded above sequences).
The limit of a function; continuity via limits. Proof of the equivalence
between
the limit definition of continuity and the ,
version.
Application to reproving 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.
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.
M. Spivak, Calculus, Benjamin Cummings.
T. M. Apostol, Mathematical Analysis, Addison-Wesley.
K. G. Binmore, Mathematical Analysis, Cambridge.
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.