### MAS3224 : The Foundations of Calculus

• Offered for Year: 2015/16
• Module Leader(s): Dr Oli King
• Owning School: Mathematics & Statistics
##### Semesters
 Semester 2 Credit Value: 10 ECTS Credits: 5.0
##### Pre Requisites
Code Title
MAS1041Mathematical Methods A
MAS1042Mathematical Methods B
MAS1242The Foundations of Analysis
###### Pre Requisite Comment

MAS2041, MAS2042, MAS2242 acceptable in place of MAS1041, MAS1042, MAS1242 respectively

N/A

#### Aims

To further develop an understanding of the role and importance of definition, proof and rigour in
Mathematics, in the context of analysis. To develop an understanding of the formal idea of a limit and the development of integration as a limiting process.

Module Summary

The differential calculus was discovered 350 years ago and ever since has been the single most important mathematical tool for the study of nature. From the beginning, even as the calculus was being applied in science and technology, there was concern about the apparent paradoxes and confusion about the properties of differentiation. It took leading mathematicians 200 years to formulate precise definitions of continuity and differentiability and to prove their fundamental properties. We extend these ideas to integrals and give a proper treatment of integration, explaining how the 'area under a curve' comes to be related to the 'opposite of differentiation'. We shall extend our ideas on series to power series, in the process extending notions of Maclaurin series representing functions. We shall discuss the range of values of x for which a power series converges.

#### Outline Of Syllabus

Sequences of real numbers. Bounds. Series and power series. Limits and continuity of real valued functions of one variable. Intermediate Value Theorem. Differentiability of real valued functions. Rolle's Theorem and the Mean Value Theorem. Riemann integration.

#### Learning Outcomes

##### Intended Knowledge Outcomes

Students will deepen and reinforce their understanding of the role and importance of definition, proof and rigour in mathematics, and the development of integration as a limiting process. Students in this module will be expected to demonstrate a deeper level of understanding than students in MAS2224

##### Intended Skill Outcomes

Students will be able to demonstrate an understanding of convergent and divergent sequences. They will be able to demonstrate an understanding of bounded and unbounded sets of real numbers, of upper bounds and lower bounds where applicable, including least upper bounds and greatest lower bounds. Students will be able to determine in many cases whether or not a series converges; they will be able to find the radius of convergence of certain power series. Students will be able to show that certain functions have limits and will be able to calculate limits in many cases. They will be able to show that certain functions are continuous or not. They will be able to show that certain functions are differentiable or not. They will be able to apply the Intermediate Value Theorem, Rolle’s Theorem and the Mean Value Theorem to examples. Students will be able to show that certain functions are Riemann integrable.

• Cognitive/Intellectual Skills
• Numeracy : Assessed
• Self Management
• Personal Enterprise
• Problem Solving : Assessed
• Interaction
• Communication
• Written Other : Assessed

#### Teaching Methods

##### Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision lectures
Scheduled Learning And Teaching ActivitiesLecture61:006:00Problem classes
Guided Independent StudyAssessment preparation and completion111:0011:00Revision for unseen Exam
Guided Independent StudyAssessment preparation and completion11:301:30Unseen Exam
Scheduled Learning And Teaching ActivitiesLecture221:0022:00Formal lectures
Guided Independent StudyAssessment preparation and completion55:0025:00Written assignments and CBAs
Scheduled Learning And Teaching ActivitiesDrop-in/surgery61:006:00Drop-ins in the lecture room
Guided Independent StudyIndependent study51:005:00Assignment review
Guided Independent StudyIndependent study121:3021:30Studying, practising and gaining understanding of course material
Total100:00
##### Jointly Taught With
Code Title
MAS2224The Foundations of Calculus
##### Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems. Drop-ins are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. Office hours provide an opportunity for more direct contact between individual students and the lecturer.

#### Assessment Methods

The format of resits will be determined by the Board of Examiners

##### Exams
Description Length Semester When Set Percentage Comment
Written Examination902A90unseen
##### Exam Pairings
Module Code Module Title Semester Comment
MAS2224The Foundations of Calculus2N/A
##### Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises2M10Written assignments and computer based assessments
##### Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. Coursework assignments (approximately 5 assignments of approximately equal weight) allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; this is thus formative as well as summative assessment. The coursework assignments may be written assignments, computer based assessments or a combination of the two, and in the case of combined assessments the deadlines for the two parts will not necessarily be the same.

N/A