This module is not currently running in 2024 to 2025.
This module will consider many concepts you know from Calculus and put them on a more rigorous basis. The concept of a limit is basic to Calculus and, unless this concept is defined precisely, uncertainties and paradoxes will creep into the subject. Based on the foundation of the real number system, this module develops the theory of convergence of sequences and series and the study of continuity and differentiability of functions. The notion of Riemann integration is also explored. The syllabus includes the following: Sequences and their convergence. The convergence of bounded increasing sequences. Series and their convergence: the comparison test, the ratio test, absolute and conditional convergence, the alternating series test. Continuous functions: the boundedness theorem, the Intermediate Value Theorem. Differentiable functions: The Mean Value Theorem with applications, power series, Taylor expansions. Construction and properties of the Riemann integral.
48 hours, Teaching methods involve a mix of lecture and example class activity.
80% Examination, 20% Coursework
FM Hart Guide to ANALYSIS. (Palgrave Macmillan, 2001).
D.A. Brannan A First Course in Mathematical Analysis. (C.U.P. 2006)
See the library reading list for this module (Canterbury)
The intended subject specific learning outcomes. On successful completion of the module students will:
a. have acquired a competence with the basic techniques of Analysis so that when these are needed as tools for the development and exploration of topics encountered in subsequent parts of their programme these same students can engage with confidence and some facility
b. have revisited and, as a remedial activity, practised the fundamental manipulative skills of elementary algebra and Calculus
c. understand the limitations, both in rigour and in scope, of the methods of Calculus they have previously met
d. begin to appreciate the effectiveness of topological ideas when it is necessary to work beyond the range of algorithmic methods
e. have attained a competence with limiting arguments and processes, in the contexts of convergence of real sequences and series, continuity and differentiability of real functions, and integrals
f. know the power of the Intermediate Value Theorem and the Mean Value Theorem as tools both for establishing results about functions in general, and also for analysing individual functions they are likely thereafter to encounter
g. have attained a competence with Taylor expansions, in the variety of examples and their ranges of validity, and with power series more generally.
The intended generic learning outcomes. On successful completion of the module, students will :
a. have matured in their problem formulating and solving skills, by a shift from the uncritical formal approach often adequate at earlier levels, towards a preoccupation with the sense and meaning conveyed by the symbols of their mathematical language
b. have enhanced their capacity to communicate mathematical statements and conclusions, both symbolically and literally
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