BSc Actuarial Science and Mathematics / Course details

We apply the fundamental properties of real numbers, sequences and limits, introduced in Mathematical Foundations and Analysis, to study series, including power series. This allows us to rigorously investigate exponential and trigonometric functions. We continue with an in-depth study of differentiation and integration and derive the important results such as the Fundamental Theorem of Calculus.

The unit aims to give a rigorous treatment of convergence, differentiation and integration. This provides a theoretical foundation for basic calculus techniques, already familiar to the students, and allows these techniques to be extended to much more general situations later in the Mathematics degree course. The material in this unit serves as a basis for many of the applications of mathematics in other disciplines.

On the successful completion of the course, students will be able to:

• Establish convergence or divergence of series and calculate the radius of convergence of a power series in a wide class of examples.
• State the definition of differentiable; prove that simple functions are differentiable and calculate their derivatives; construct proofs for the rules for differentiation and apply the rules to more complicated functions.
• Construct proofs of the Chain Rule; Rolles Theorem; the Mean Value Theorem and Cauchy's Mean Value Theorem, with applications, e.g. L’Hôpital’s rule. State the Inverse Rule for monotonic differentiable functions and apply to examples, e.g. the logarithm and powers.
• Calculate Taylor polynomials and series; state and apply Taylor's Theorem with an error term, and state criteria for a Taylor series for a function to converge to that function.
• Construct proofs about Riemann sums and Upper and Lower integrals; state the definition of the Riemann integral and calculate the Riemann integral for various functions; construct a proof of the Fundamental Theorem of Calculus.

There are supervisions in alternate weeks which provide an opportunity for students; work to be marked and discussed and to provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Burkill, J.C. . A first course in Mathematical Analysis. Cambridge University Press, 1962.

Hart, F.M. Guide to Analysis. Palgrave, London, 1988.

Tao, T. Analysis I. 3rd edition, Springer, 2016. (Available freely online to the University via SpringerLink.)