Introductory description

Mathematical Analysis is the heart of modern Mathematics. Calculus usually stands for Analysis, focused on calculations rather than proving theorems. This module is the second in a series of modules where the subject of Analysis is developed with a focus on calculations.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• Differentiability
• Taylor's Theorem
• Taylor’s Series
• Riemann Integral
• Methods of integration
• Fundamental Theorem of Calculus
• Improper integrals

Learning outcomes

By the end of the module, students should be able to:

• learn differentiability, including higher derivatives and properties of differentiable functions
• develop the working knowledge of Taylor's series and theorem, ultimately understanding representability of a function by a power series
• develop general understanding of the construction of the Riemann integral
• understand and apply the fundamental properties of the integral such as integrability of continuous functions on bounded intervals or the Fundamental Theorem of Calculus

Indicative reading list

M. Hart, Guide to Analysis, Macmillan.
M. Spivak, Calculus, Benjamin. R.G Bartle and D.R Sherbert, Introduction to Real Analysis (4th Edition), Wiley (2011)
L. Alcock, How to think about Analysis, Oxford University Press (2014)

View reading list on Talis Aspire

Subject specific skills

Calculus gives first-year undergraduates a first excursion in to pure mathematics. The students will gain a new perspective and a deeper understanding of familiar mathematics which they have seen in school (e.g. real numbers, functions and differentiation). In Calculus, these concepts are developed with mathematical rigour, which characterises much of university mathematics to follow.

Transferable skills

Students will acquire key reasoning and problem solving skills, empower them to address new problems with confidence.