Module MATH1051: Analysis I

Department: Mathematical Sciences

MATH1051: Analysis I

Prerequisites

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

Corequisites

• Calculus and Probability I (MATH1XX1) and Linear Algebra I (MATH1XX1) -- 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 and Probability I (MATH1XX1) and Linear Algebra I (MATH1XX1) in their first year.

Excluded Combination of Modules

• Maths for Engineers and Scientists (MATH1551), Single Mathematics A (MATH1561), Single Mathematics B (MATH1571) and Foundation Mathematics (MATH1641).

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.
• Intermediate Value theorem.
• Differentiability: Definition.
• Differentiability implies continuity.
• Basic theorems.
• Proof of Rolle's theorem, Mean Value theorem.
• Integration: Discussion of Riemann sums.
• Fundamental theorem of calculus.
• Basic theorems.
• Issues of convergence.
• Real and complex power series: Radius of convergence, Basic theorems.
• Taylor series.

Learning Outcomes

• 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.

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

• students will have basic problem solving skills.

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

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Tutorials provide active engagement and feedback to the learning process.
• 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

Summative Assessment

Formative Assessment:

- One written assignment each teaching week. Normally it will consist of solving problems from a Problem Sheet and typically will be about 2 pages long. Students will have about one week to complete each assignment. - 45 minute collection paper in the first week 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