# 2024/25 Undergraduate Module Catalogue

## MATH2017 Real Analysis

### 15 creditsClass Size: 330

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

**Taught:** Semester 2 (Jan to Jun) View Timetable

**Year running** 2024/25

### Pre-requisite qualifications

(MATH1010 or MATH1005 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 asingle 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

### Reading list

The reading list is available from the Library websiteLast updated: 29/04/2024 16:16:33

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- Undergraduate module catalogue
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