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### 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 Level Undergraduate 2 First Sub Session 15 credits (7.5 ECTS credits) Old Aberdeen No Dr William Turner

### Qualification Prerequisites

• Programme Level 2

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

• MA1508 Calculus II (Passed)
• MA1005 Calculus 1 (Passed)

None.

None.

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.

### 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.

### 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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