Undergraduate Programme and Module Handbook 2024-2025

# Module MATH1051: Analysis I

## Department: Mathematical Sciences

### MATH1051: Analysis I

Type | Open | Level | 1 | Credits | 20 | Availability | Available in 2024/2025 | Module Cap | Location | Durham |
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#### Prerequisites

- Normally grade A in A-Level Mathematics (or equivalent).

#### Corequisites

- Calculus I (MATH1061) and Linear Algebra I (MATH1071) OR Calculus I (Maths Hons) (MATH1081) and Linear Algebra I (Maths Hons) MATH1091). Note that for some students, this module may be taken as Level 1 course in the second year, but that such students will have taken Calculus I (MATH1061) and Linear Algebra I (MATH1071) OR Calculus I (Maths Hons) (MATH1081) and Linear Algebra I (Maths Hons) (MATH1091) in their first year.

#### Excluded Combination of Modules

- Maths for Engineers and Scientists (MATH1551), Single Mathematics A (MATH1561), Single Mathematics B (MATH1571).

#### Aims

- To provide an understanding of the real and complex number systems, and to develop calculus of functions of a single variable from basic principles using rigorous methods.

#### Content

- Numbers: real and complex number systems.
- sup and inf of subsets of R and of real valued functions.
- Convergence of sequences: Examples, Basic theorems.
- Bolzano-Weierstrass theorem.
- Convergence of series: Examples, tests for convergence, absolute convergence, conditional convergence.
- Limits and Continuity: Functions of a real and complex variable.
- Epsilon-delta definition of limit of a function.
- Continuity.
- Basic theorems on continuity.
- Intermediate Value theorem.
- Differentiability: Definition.
- Differentiability implies continuity.
- Basic theorems on differentiability.
- Proof of Rolle's theorem, Mean Value theorem.
- Step & regulated functions.
- Integration for regulated functions.
- Fundamental theorem of calculus.
- Basic theorems on integration.
- Pointwise and uniform convergence.
- Limit theorems.
- Real (and complex) power series: Radius of convergence, Basic theorems.
- Taylor series.

#### Learning Outcomes

Subject-specific Knowledge:

- By the end of the module students will: be able to solve a range of predictable or less predictable problems in Analysis.
- have an awareness of the basic concepts of theoretical mathematics in the field of Analysis.
- have a broad knowledge and basic understanding of these subjects demonstrated through one or more of the following topic areas: Numbers, supremum, infimum.
- Convergence of sequences and series.
- Limits, continuity, differentiation, integration.
- Real and complex power series.

Subject-specific Skills:

- students will have basic mathematical skills in the following areas: Spatial awareness, Abstract reasoning.

Key Skills:

- students will have basic problem solving skills.

#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

- Tutorials provide active engagement and feedback to the learning process.
- Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
- Weekly homework problems provide formative assessment to guide students in the correct development of their knowledge and skills. They are also an aid in developing students' awareness of standards required.
- The examination provides a final assessment of the achievement of the student.

#### Teaching Methods and Learning Hours

Activity | Number | Frequency | Duration | Total/Hours | |
---|---|---|---|---|---|

Lectures | 47 | 2 per week in term 1, 2 or 3 per week in term 2 alternating with Problems Classes and collection examination, 2 revision lectures in term 3 | 1 Hour | 47 | |

Tutorials | 14 | Weekly in weeks 2-10, fortnightly in weeks 13-19, and one in week 21. | 1 Hour | 14 | ■ |

Problems Classes | 4 | Fortnightly in weeks 14-20 | 1 Hour | 4 | |

Preparation and Reading | 135 | ||||

Total | 200 |

#### Summative Assessment

Component: Examination | Component Weighting: 90% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

Written examination | 3 hours | 100% | Yes |

Component: Continuous Assessment | Component Weighting: 10% | ||

Element | Length / duration | Element Weighting | Resit Opportunity |

Weekly written assignments during the first 2 terms. Normally, each will consist of solving problems and will typically be one to two pages long. Students will have about one week to complete each assignment. | 100% | Yes |

#### Formative Assessment:

45 minute collection paper in the beginning of Epiphany term.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University