BSc Mathematics

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.)

Tuition fees for home students will be £10,050 for the 2027/28 academic year. In England and Wales, tuition fees for home undergraduate students are subject to the Government fee cap, which the Government has announced will be £10,050 for the 2027/28 academic year.

Government policy is to increase the fee cap on an annual basis to reflect inflation (i.e. increasing costs), and the tuition fee home students pay in subsequent academic years will reflect any increases in the Government fee cap (which we will communicate to you as soon as we can in advance of the relevant academic year).

Tuition fees for international students will be £38,100 for the 2027/28 academic year. For subsequent academic years, the University reserves the right to increase the tuition fee by up to 7% each year, including to reflect rising costs associated with delivering our education and wider student experience.

For general information please see the undergraduate finance pages.

All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).

The University of Manchester is committed to attracting and supporting the very best students. We have a focus on nurturing talent and ability and we want to make sure that you have the opportunity to study here, regardless of your financial circumstances.

For information about scholarships and bursaries please visit our undergraduate student finance pages and our Department funding pages