# EC119: Mathematical Analysis

### Principal Aims

This module provides students with a strong background in pure mathematics, particularly the theory of sets and functions, real and complex number systems, logic and proof, analysis of real-valued functions, and differential equations. This allows the students to develop a fluency with abstract mathematical reasoning, and gives a deeper understanding of techniques used in mathematical economics and econometrics.

### Principal Learning Outcomes

Subject knowledge and understanding: … demonstrate an understanding of basic properties of real and complex numbers, functions, and finite and infinite sets. The teaching and learning methods that enable students to achieve this learning outcome are: Lectures, tutorials, problem sheets and independent study. The assessment methods that measure the achievement of this learning outcome are: Problem sheets and unseen examination.

Subject knowledge and understanding: … demonstrate an understanding of basic topics in the analysis of real-valued functions, including limits, continuity, differentiation, Taylor-MacLaurin series, and integration. The teaching and learning methods that enable students to achieve this learning outcome are: Lectures, tutorials, problem sheets and independent study. The assessment methods that measure the achievement of this learning outcome are: Problem sheets and unseen examination.

Key skills: …understand formal mathematical definitions and theorems, and apply them to prove statements about real-valued functions. The teaching and learning methods that enable students to achieve this learning outcome are: Lectures, tutorials, problem sheets and independent study. The assessment methods that measure the achievement of this learning outcome are: Problem sheets and unseen examination.

### 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
- L100 - Year 1, L116 - Year 1, LM1D (LLD2) - Year 1, V7ML - Year 1, L1L8 - Year 1, LA99 - Year 1, R9L1 - Year 1, R3L4 - Year 1, R4L1 - Year 1, R2L4 - Year 1, R1L4 - Year 1
- Pre or Co-requisites
- A-level Mathematics or the equivalent
- 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 (summer) (80%)
- Coursework Details
- Problem Set 1 (4%), Problem Set 2 (4%), Problem Set 3 (4%), Problem Set 4 (4%), Problem Set 5 (4%), 2 hour exam (summer) (80%)
- Exam Timing
- Summer

### Exam Rubric

Time Allowed: 2 Hours

Read all instructions carefully- and read through the entire paper at least once before you entering your answers.

There is ONE Section in this paper. Answer THREE questions (25 marks each).

Approved pocket calculators are allowed.

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

You should not submit answers to more than the required number of questions. If you do, we will mark the questions in the order that they appear, up to the required number of questions in each section.

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.