This module develops the learner’s knowledge of elementary analysis in a rigorous manner and provides a solid foundation in topics in pure mathematics relating to calculus. Sequences and series are analysed. Limits of functions, continuity and properties of continuous functions are investigated. The theory of differentiable functions, Rolle’s theorem, Taylor’s theorem and L’Hopital’s rule are studied. The learner is also introduced to the analysis of integration and the Riemann integral.
Sequences and series
Introduction to sequences, bounds and convergence. Properties of convergent sequences. Cauchy Sequences. Series. Standard limits, subsequences and convergence theorems. Tests for convergence.
Polynomial and rational functions. Limits of functions and continuity. Properties of continuous functions and the intermediate value theorem.
Derivatives and differentiability. Properties of derivatives. Rolle’s theorem, the mean value theorem. Inverse functions. L’hôpital’s rule. Higher derivatives. Taylor’s theorem and Maclaurin and Taylor series. Approximation. The Riemann integral.
Lectures are primarily used to impart module content to the learner. Problem-solving sessions and tutorials support the learner and are designed to encourage learners to work both individually and in groups.
|Module Content & Assessment