MA2009: ANALYSIS I (2021-2022)

Last modified: 31 May 2022 13:05

Course Overview

Analysis provides the rigorous, foundational underpinnings of calculus. It is centred around the notion of limits: convergence within the real numbers. Related ideas, such as infinite sums (a.k.a. series) and continuity are also visited in this course.

Care is needed to properly use the delicate formal concept of limits. At the same time, limits are often intuitive, and we aim to reconcile this intuition with correct mathematical reasoning. The emphasis throughout this course is on rigorous mathematical proofs, valid reasoning, and the avoidance of fallacious arguments.

Course Details

Study Type	Undergraduate	Level	2
Term	First Term	Credit Points	15 credits (7.5 ECTS credits)
Campus	Aberdeen	Sustained Study	No
Co-ordinators	Dr Alexey Sevastyanov Professor Benjamin Martin

Qualification Prerequisites

Programme Level 2

What courses & programmes must have been taken before this course?

MA1005 Calculus 1 (Passed)
Any Undergraduate Programme (Studied)
MA1508 Calculus II (Passed)

What other courses must be taken with this course?

None.

What courses cannot be taken with this course?

None.

Are there a limited number of places available?

Course Description

- Fundamental properties of real numbers: field operations, order, completeness.

- Sequences and limits: convergence, basic examples, methods of deducing convergence, properties of convergent sequences, the Bolzano-Weierstrass Theorem.

- Infinite sums (series): convergence, convergence tests.

- Functions of one real variable: limits and continuity, methods of deducing limits, Extreme Value Theorem, Intermediate Value Theorem, uniform continuity.

Syllabus

Properties of the real numbers: Field operations, Order, Completeness, Density of the real numbers.
Sequences: Convergence (epsilon-delta), Properties of limits, Monotone Convergence Criterion, Subsequences, Bolzano-Weierstrass theorem.
Series: Partial sums, Convergence, Properties of series, Criteria and tests for convergence, decimal representation of real numbers, Absolute convergence.
Sets of real numbers: Closed and open sets.
Continuous functions: Limits and continuity, Basic results on continuous functions, Uniform continuity, Extreme and intermediate value theorems, Points of discontinuity.

Course Aims

To put on a sound footing many of the results, procedures, and concepts used in Calculus. It will include a discussion of fundamental properties of real numbers, sequences and limits, series, and continuity of functions. Some applications will also be given.

Contact Teaching Time

Information on contact teaching time is available from the course guide.

Teaching Breakdown

More Information about Week Numbers

Details, including assessments, may be subject to change until 30 August 2024 for 1st term courses and 20 December 2024 for 2nd term courses.

Summative Assessments

10x Weekly multi-choice or short answer online quizzes - 1% each

3x Standard assignments - 30% each

Alternative Resit Arrangements

Resubmission of failed elements (pass marks carried forward)

Formative Assessment

There are no assessments for this course.

Course Learning Outcomes

Knowledge Level	Thinking Skill	Outcome
Conceptual	Apply	Be able to use the theorems of the course in unseen situations;
Factual	Remember	know about basic properties of the real numbers and what distinguishes them from the rational numbers;
Conceptual	Apply	Have developed the ability to prove elementary results, and be able to detect fallacious arguments;
Factual	Apply	Be able to state the main definitions and theorems of the course;
Conceptual	Apply	Be able to establish the convergence of simple sequences and series
Factual	Remember	know precise definitions and basic properties of elementary functions;
Factual	Understand	be familiar with the concepts of limits and continuity.