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# Mathematical Methods

Module title Mathematical Methods INT1202 2021/2 30
 Duration: Term Duration: Weeks 1 2 3 11 11
 Number students taking module (anticipated) 20

## Description - summary of the module content

### Module description

During your mathematics degree, you will be solving problems and proving theories in several branches of mathematics. Inevitably you need to be able to calculate. That is what gives the mathematics its great power. This module brings emphasis on the techniques rather than the applications of the techniques. You will study topics that include the geometry of conic sections, properties of functions such as continuity and differentiability, differential and integral calculus, limits and convergence of sequences and series including Power Series and Taylor Series. The module also develops the fundamentals of vector and matrix theory, multivariate calculus, and the classification of various types of differential equations as well as analytical methods for solving them.

This module develops methods that underpin the second year modules MTH2003 Differential Equations and MTH2004 Vector Calculus and Applications.

## Module aims - intentions of the module

This module aims to develop your skills and techniques in calculus, geometry and algebra. It is primarily focused on developing methods and skills for accurate manipulation of the mathematical objects that form the basis of much of an undergraduate course in mathematics. Whilst the main emphasis of the module will be on practical methods and problem solving, all results will be stated formally and each sub-topic will be reviewed from a mathematically rigorous standpoint. The techniques developed in this course will be essential to much of your undergraduate degree programme, particularly the second-year streams of Analysis, Differential Equations & Vector Calculus, and Mathematical Modelling. This module is equivalent to MTH1002 and students will join MTH1002 for lectures.

## Intended Learning Outcomes (ILOs)

### ILO: Module-specific skills

On successfully completing the module you will be able to...

• 1. Explain how techniques in differential and integral calculus are underpinned by formal rigour
• 2. Apply techniques in geometry and algebra to explore three dimensional analytic geometry
• 3. Perform accurate manipulations in algebra and calculus of several variables using a variety of standard techniques
• 4. Solve some specific classes of ordinary differential equations

### ILO: Discipline-specific skills

On successfully completing the module you will be able to...

• 5. Demonstrate a basic knowledge of functions, sequences, series, limits and differential and integral calculus necessary for progression to successful further studies in the mathematical sciences

### ILO: Personal and key skills

On successfully completing the module you will be able to...

• 6. Reason using abstract ideas, and formulate and solve problems and communicate reasoning and solutions effectively in writing
• 7. Use learning resources appropriately
• 8. Exhibit self-management and time-management skills

## Syllabus plan

### Syllabus plan

Geometry: lines; planes; conic sections.

Functions: single- and multivariate; limits; continuity; intermediate value theorem.

Sequences: algebra of limits; L'Hopital's rule.

Series: convergence/divergence tests; power series.

Differential calculus: simple and partial derivatives; Leibniz' rule; chain rule; Taylor approximation; implicit differentiation; minima and maxima.

Integral calculus: substitution; integration by parts; multiple integrals; applications.

Differential equations: linear and separable ordinary DEs; basic partial DEs.

Vectors, matrices: Gaussian elimination; transformations; eigenvalues/eigenvectors.

## Learning and teaching

### Learning activities and teaching methods (given in hours of study time)

Scheduled Learning and Teaching ActivitiesGuided independent studyPlacement / study abroad
1301700

### Details of learning activities and teaching methods

CategoryHours of study timeDescription
Scheduled Learning & Teaching activities66Lectures
Scheduled learning and teaching activities44Small group lessons
Scheduled learning and teaching activities10Large group lessons
Scheduled learning and teaching activities10Tutorials
Guided independent study170Lecture and assessment preparation, wider reading, completing exercises.

## Assessment

### Formative assessment

Form of assessmentSize of the assessment (eg length / duration)ILOs assessedFeedback method
Exercise sheets10 x 10 Hours1-8Annotated scripts with oral feedback from tutor
Mid-term Tests2 x 1 hour1-8Feedback on marked sheets

### Summative assessment (% of credit)

CourseworkWritten examsPractical exams
01000

### Details of summative assessment

Form of assessment% of creditSize of the assessment (eg length / duration)ILOs assessedFeedback method
Written examination - Closed book (Jan)502 hours1-8Via SRS
Written examination - Closed book (May)502 Hours1-8Via SRS

## Re-assessment

### Details of re-assessment (where required by referral or deferral)

Original form of assessmentForm of re-assessmentILOs re-assessedTimescale for re-assessment
As aboveWritten examination (100%)1-8During next examination period

### Re-assessment notes

Deferral– if you miss an assessment for reasons judged legitimate by the Mitigation Committee, the applicable assessment will normally be deferred. See ‘Details of reassessment’ for the form that assessment usually takes. When deferral occurs there is ordinarily no change to the overall weighting of that assessment.

Referral– if you have failed the module overall (i.e. a final overall module mark of less than 40%) you will be required to take a referral exam. Only your performance in this exam will count towards your final module grade. A grade of 40% will be awarded if the examination is passed.

## Resources

### Indicative learning resources - Basic reading

Core Text:

 Author Title Edition Publisher Year ISBN Thomas, G, Weir, M, Hass, J Thomas' Calculus 13th Pearson 2016 978-1292089799

Additional RecommendedReading for this module:

 Author Title Edition Publisher Year ISBN Tan, Soo T Calculus International edition Brooks/Cole Cengage Learning 2010 978-0495832294 Tan, T Soo Calculus Early Transcendentals International edition Brooks Cole/Cengage learning 2010 978-1439045992 Extended Reading: Stewart J. Calculus 5th Brooks/Cole 2003 000-0-534-27408-0 McGregor C., Nimmo J. & Stothers W. Fundamentals of University Mathematics 2nd Horwood, Chichester 2000 000-1-898-56310-1

### Indicative learning resources - Web based and electronic resources

Module has an active ELE page

### Key words search

Calculus; series; limits; convergence; divergence; integration; differential equations; conic sections; functions; continuity; vectors; matrices; eigenvalues; eigenvectors

Credit value 30 15 None None 4 No 14/08/2017 15/07/2021