# EC119: Mathematical Analysis

### Principal Aims

This module provides students with a strong background in pure mathematics, particularly the theory of sets and functions, the real number system, 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 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, online quizzes and independent study. The assessment methods that measure the achievement of this learning outcome are: Problem sheets, online quizzes and unseen examination.

Subject knowledge and understanding: … demonstrate an understanding of basic topics in the analysis of real-valued functions, including convergence of sequences and series, 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, online quizzes and independent study. The assessment methods that measure the achievement of this learning outcome are: Problem sheets, online quizzes 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, online quizzes and independent study. The assessment methods that measure the achievement of this learning outcome are: Problem sheets, online quizzes and unseen examination.

### Syllabus

The module will typically cover the following topics:Set theory (notation, basic concepts), Real numbers (basic properties, interval notation), Functions (injectivity, surjectivity, composition), Sequences and Series (convergence, divergence, boundedness), Limits of functions (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, V7ML - Year 1, L1L8 - Year 1, LA99 - Year 1, R9L1 - Year 1, R3L4 - Year 1, R4L1 - Year 1, R2L4 - Year 1, R1L4 - Year 1, R2L5 - Year 1, R4LA - Year 1, R1L5 - Year 1, L1CA - Year 1
- Pre or Co-requisites
- A-level Mathematics or the equivalent

### Assessment

- Assessment Method
- Coursework (30%) + In-person Examination (70%)
- Coursework Details
- In-person Examination (70%) , Problem Set 1 (5%) , Problem Set 2 (5%) , Problem Set 3 (5%) , Problem Set 4 (5%) , Test 1 (2%) , Test 2 (2%) , Test 3 (2%) , Test 4 (2%) , Test 5 (2%)
- Exam Timing
- Summer

### Exam Rubric

Time Allowed: 2 Hours

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

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

Answer each whole question in a separate booklet.

Approved scientific (non-graphical) 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.