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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 Level Undergraduate 2 First Term 15 credits (7.5 ECTS credits) Aberdeen No Dr William Turner

Qualification Prerequisites

• Programme Level 2

None.

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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

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

10 x Weekly Quiz

Assessment Type Weighting Summative 10
Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

Homework

Assessment Type Weighting Summative 20
Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

Exam

Assessment Type Weighting Summative 50
Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

Homework

Assessment Type Weighting Summative 20
Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

Formative Assessment

There are no assessments for this course.

Resit Assessments

Exam

Assessment Type Weighting Summative 100 Best of (resit exam mark) or (resit exam mark with carried forward CA marks)
Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

Course Learning Outcomes

Knowledge LevelThinking SkillOutcome
ConceptualApplyBe able to use the theorems of the course in unseen situations;
ConceptualApplyHave developed the ability to prove elementary results, and be able to detect fallacious arguments;
FactualApplyBe able to state the main definitions and theorems of the course;
FactualRememberknow about basic properties of the real numbers and what distinguishes them from the rational numbers;
ConceptualApplyBe able to establish the convergence of simple sequences and series
FactualUnderstandbe familiar with the concepts of limits and continuity.
FactualRememberknow precise definitions and basic properties of elementary functions;

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