## MATH2017 Real Analysis

### 15 creditsClass Size: 310

Module manager: Professor Martin Speight
Email: J.M.Speight@leeds.ac.uk

Taught: Semester 2 View Timetable

Year running 2019/20

### Pre-requisite qualifications

(MATH1010 and MATH1026) or (MATH1050 and MATH1055) or equivalent

### This module is mutually exclusive with

 MATH2016 Analysis

This module is not approved as a discovery module

### Module summary

Calculus is arguably the most significant and useful mathematical idea ever invented,with applications throughout the natural sciences and beyond. This module develops the theory ofdifferential and integral calculus of real functions in a precise and mathematically rigorous way. Particularemphasis will be put on sequential notions.

### Objectives

a) To develop a mathematically rigorous theory of differential and integral calculus for real functions of a
single variable, including all the standard foundational results.
b) To develop the theory of power series, and explore the difference between smoothness and analyticity.
c) To introduce the concept of uniform convergence, and use it to analyze suitable double limit problems.

Learning outcomes
On completion of this module, students should be able to:
a) Define precisely the central objects of calculus (limits, derivatives, the Riemann integral).
b) Determine whether a given function is continuous, differentiable, integrable.
c) Compute derivatives and integrals from first principles.
d) Understand the difference between differentiable, smooth and analytic functions.
e) Compute Taylor expansions with controlled errors.
f) Construct rigorous proofs of (a selection of) the theorems presented.

### Syllabus

a) Review of sequential definitions of continuity and limits, epsilon-delta criteria, their equivalence.
b) Differentiability, elementary properties (differentiability implies continuity, linearity, the product and
chain rules).
c) Functions differentiable on an interval: the Mean Value Theorem, L'Hospital's Rule, Taylor's Theorem with
remainder,
d) The Riemann Integral: dissections, upper and lower sums. Continuous functions are integrable. Monotonic
functions are integrable. The algebra of integrable functions. The Fundamental Theorem of the Calculus.
e) Uniform convergence versus pointwise convergence, double limits.
f) Power series: differentiability, smoothness. Smooth functions which are not analytic.

### Teaching methods

 Delivery type Number Length hours Student hours Workshop 10 1.00 10.00 Lecture 33 1.00 33.00 Private study hours 107.00 Total Contact hours 43.00 Total hours (100hr per 10 credits) 150.00

### Opportunities for Formative Feedback

Written, assessed work throughout the semester with feedback to students.

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Problem Sheet . 15.00 Total percentage (Assessment Coursework) 15.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 30 mins 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated