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MA2009: ANALYSIS I (2017-2018)

Last modified: 23 Aug 2017 15:27


Course Overview

Analysis provides the rigourous, 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), continuity, and differentiation, 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 rigourous mathematical proofs, valid reasoning, and the avoidance of fallacious arguments.

Course Details

Study Type Undergraduate Level 2
Session First Sub Session Credit Points 15 credits (7.5 ECTS credits)
Campus Old Aberdeen Sustained Study No
Co-ordinators
  • Dr William Turner

Qualification Prerequisites

  • Programme Level 2

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

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?

No

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.
- Differentiation of functions of one variable: basic definitions and properties, chain rule, basic results on differentiable functions, Rolle's Theorem, Mean Value Theorem.

 

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 and Conditional 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.
  • Differentiation: Definitions and properties, Standard rules for differentiation, Extrema, Mean value theorem, Monotonicity and Convexity.

 

Further Information & Notes

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, continuity of functions, differentiation and integration. Some applications will also be given.
 
Learning Objectives
By the end of the course the student should:
_ be able to state the main definitions and theorems of the course;
_ know about basic properties of the real numbers and what distinguishes them from the rational numbers;
_ be able to establish the convergence of simple sequences and series;
_ know precise definitions and basic properties of elementary functions;
_ be able to use the theorems of the course in unseen situations;
_ have developed the ability to prove elementary results, and be able to detect fallacious arguments;
_ be familiar with the concepts of continuity, differentiability and Riemann integrability.

Degree Programmes for which this Course is Prescribed

  • BSc Applied Mathematics
  • BSc Computing Science-Mathematics
  • BSc Mathematics
  • BSc Mathematics with French
  • BSc Mathematics with Gaelic
  • BSc Physics with Geology
  • MA Business Management - Mathematics
  • MA Mathematics
  • MA Mathematics with Gaelic
  • Master of Physics with Complex Systems Modelling
  • Mathematics Joint
  • Mathematics Major
  • Mathematics Minor

Contact Teaching Time

43 hours

This is the total time spent in lectures, tutorials and other class teaching.

Teaching Breakdown


Assessment

1 two-hour written examination (80%); in-course assessment (20%).

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact Course Coordinator for feedback on the final examination.

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