Unit information: Applied Analysis A in 2024/25

Unit name	Applied Analysis A
Unit code	MATH10023
Credit points	20
Level of study	C/4
Teaching block(s)	Teaching Block 1 (weeks 1 - 12)
Unit director	Dr. Tourigny
Open unit status	Not open
Units you must take before you take this one (pre-requisite units)	none
Units you must take alongside this one (co-requisite units)	none
Units you may not take alongside this one	none
School/department	School of Mathematics
Faculty	Faculty of Science

Unit Information

Why is this unit important?

Analysis is the study of the continuum. It is central to mathematics and its applications, encompassing calculus and the theory of functions. Applied Analysis A provides a rigorous foundation for the theory of functions of a single real variable. Its companion unit, Applied Analysis B, provides an analogous foundation for functions of several real variables.

The emphasis is on calculation and problem-solving, underpinned by theory and computation. On the theoretical side, methods of calculus are developed systematically; definitions and theorems are formulated precisely, and some theorems are proved, so that the scope of the theory is clearly established. On the computational side, paper-and-pencil calculations are complemented by computer laboratories and demonstrations.

How does this unit fit into your programme of study

Analysis is central to mathematics and its applications including data science, physical sciences, economics and other disciplines. Its principles underlie differential and integral calculus, optimisation, convexity, probability and statistics, and differential equations. The unit introduces methods of proof and logical rigor.

Your learning on this unit

An overview of content

Topics below are organised thematically rather than in order of presentation.

• Sets, logical quantifiers, methods of proof.
• Number systems. Real numbers and the least upper bound property.
• Sequences: Boundedness, convergence, limits. The Bolzano-Weierstrass theorem and the Cauchy criterion.
• Series: Convergence, geometric and harmonic series, convergence tests. Absolute and conditional convergence.
• Continuous functions: Definitions and properties. Intermediate and extreme value theorems.
• Differentiable functions: Definition and properties. Rules for differentiation. Inverse functions. Minima and maxima. Mean value theorems. Higher derivatives and Taylor’s theorem.
• Elementary transcendental functions: Exponential and logarithm. l’Hopital’s rule. Trigonometric functions.
• Convex functions: Properties, optimization, inequalities.
• Integration: Fundamental theorem of calculus. Methods of integration. Improper integrals.
• Brief introduction to ordinary differential equations.

Learning Outcomes

At the end of the unit, the students should be able to:

• apply and synthesise a range of core methods
• formulate definitions of core concepts
• apply and/or prove core theorems
• recognise and/or construct examples and counterexamples

How you will learn

The unit will be taught through a combination of

• lectures
• independent activities such as problem sheets
• tutorials
• lab sessions

How you will be assessed

Assessment for learning/Formative assessment:

• problem sheets set by the lecturer and marked by the students' tutors.

Assessment of learning/Summative assessment:

• Timed examination (80%)
• Coursework (20%)

Resources

If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.

If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATH10023).

How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours of study to complete. Your total learning time is made up of contact time, directed learning tasks, independent learning and assessment activity.

See the University Workload statement relating to this unit for more information.

Assessment
The Board of Examiners will consider all cases where students have failed or not completed the assessments required for credit. The Board considers each student's outcomes across all the units which contribute to each year's programme of study. For appropriate assessments, if you have self-certificated your absence, you will normally be required to complete it the next time it runs (for assessments at the end of TB1 and TB2 this is usually in the next re-assessment period).
The Board of Examiners will take into account any exceptional circumstances and operates within the Regulations and Code of Practice for Taught Programmes.