### Principal Aims

To give students a more rigorous understanding of the mathematics of real-valued functions. Students will acquire an understanding of basic properties of the field of real numbers, concepts of infinity, limits of functions and methods for calculating them, continuity, differentiation, integration, Taylor series, and differential equations.

### Principal Learning Outcomes

By the end of the module the student should be able to demonstrate an understanding of basic properties of real and complex numbers, functions, and finite and infinite sets; demonstrate an understanding of basic topics in the analysis of real-valued functions, including limits, continuity, differentiation, Taylor-MacLaurin series, and integration; understand formal mathematical definitions and theorems, and apply them to prove statements about real-valued functions.

### Syllabus

The module will typically cover the following topics:

Set theory (notation, basic concepts), Real numbers (basic properties, interval notation), Complex numbers (basic definitions, Cartesian form, polar form, roots of unity, the Fundamental Theorem of Algebra), Functions (injectivity, surjectivity, composition), Counting (cardinality of finite and infinite sets, countability of the rational numbers, uncountability of the real numbers), Limits (basic definitions, the Sandwich Rule, boundedness), Continuity (basic definitions, the Intermediate Value Theorem, numerical methods for solving equations), Differentiation (basic definitions and properties, Rolle’s Theorem, the Mean Value Theorem), L’Hopital’s Rule (techniques and applications), Taylor’s Theorem (generalisation of the Mean Value Theorem, polynomial approximations to functions, convergence criteria), Integration (basic properties, the Newton-Leibniz definition, the Riemann definition, the Fundamental Theorem of Calculus, integration by parts, calculation of improper integrals), Differential equations (first-order separable equations, first- and second-order linear equations)

### Context

- Optional Module
- LM1D (LLD2) - Year 1, L116 - Year 1, V7ML - Year 1, LA99 - Year 1, L100 - Year 1, L1L8 - Year 1, R9L1 - Year 1, R3L4 - Year 1, R4L1 - Year 1, R2L4 - Year 1, R1L4 - Year 1
- Pre or Co-requisites
- A-level Mathematics
- Part-year Availability for Visiting Students
- Available in the Autumn term only (5 x problem sets - 6 CATS)

### Assessment

- Assessment Method
- Coursework (20%) + 2 hour exam (80%)
- Coursework Details
- Five problem sets (worth 4% each)
- Exam Timing
- May/June

### Exam Rubric

Time Allowed: 2 Hours.

Answer THREE questions. Each question is worth 25 marks.

Approved pocket calculators are allowed.

A formula sheet is provided at the end of the exam paper

Read carefully the instructions on the answer book provided and make sure that the particulars required are entered on each answer book. If you answer more questions than are required and do not indicate which answers should be ignored, we will mark the requisite number of answers in the order in which they appear in the answer book(s): answers beyond that number will not be considered.

Previous exam papers can be found in the University’s past papers archive. Please note that previous exam papers may not have operated under the same exam rubric or assessment weightings as those for the current academic year. The content of past papers may also be different.