Last modified: 25 Sep 2019 09:58
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.
|Session||First Sub Session||Credit Points||15 credits (7.5 ECTS credits)|
- 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.
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.
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 limits and continuity.
Information on contact teaching time is available from the course guide.
There are no assessments for this course.
|Assessment Weeks||Feedback Weeks|
|Knowledge Level||Thinking Skill||Outcome|
|Factual||Apply||Be able to state the main definitions and theorems of the course;|
|Factual||Remember||know about basic properties of the real numbers and what distinguishes them from the rational numbers;|
|Conceptual||Apply||Be able to establish the convergence of simple sequences and series|
|Factual||Remember||know precise definitions and basic properties of elementary functions;|
|Conceptual||Apply||Be able to use the theorems of the course in unseen situations;|
|Conceptual||Apply||Have developed the ability to prove elementary results, and be able to detect fallacious arguments;|
|Factual||Understand||be familiar with the concepts of limits and continuity.|