# Mathematics

## ECM1702 - Calculus and Geometry (2015)

MODULE TITLE CREDIT VALUE Calculus and Geometry 15 ECM1702 Ms Aileen MacGregor (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 weeks 0 0
 Number of Students Taking Module (anticipated) 280
DESCRIPTION - summary of the module content

By taking this module, you will develop knowledge and skills in two and three dimensional analytic geometry and differential calculus. As a result, you will be able to quickly and accurately perform calculus on simple functions using a variety of standard techniques.

The module will introduce you to MAPLE, a symbolic manipulation and programming language used to investigate the properties of a function. You will also learn how to accurately sketch the graphs of functions. The module gives an informal treatment of theorems from analysis.

It will also teach you how to reason using abstract ideas, and how to formulate and solve problems and communicate reasoning and solutions effectively in writing. As with the other modules, it will develop your self management and time-management skills and broaden your use of learning resources, including the use of IT.

AIMS - intentions of the module

This module aims to develop your knowledge and skills in calculus and geometry. It is about developing methods and skills for calculus-related manipulation of the mathematical objects that form the basis of much of an undergraduate course in mathematics. Furthermore, it aims to introduce some formal definitions and statements of theorems from analysis, but with only limited formal proofs.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
1 demonstrate knowledge of basic two and three dimensional analytic geometry and differential calculus;
2 show understanding of basic concepts concerning functions, sequences, series and limits;
3 perform accurate calculus manipulations using a variety of standard techniques;
4 sketch the graphs of a variety of functions of one variable.
Discipline Specific Skills and Knowledge:
5 demonstrate a basic knowledge of functions, sequences, series, limits and differential calculus necessary for progression to successful further studies in the mathematical sciences;
Personal and Key Transferable/ Employment Skills and  Knowledge:
6 reason using abstract ideas, formulate and solve problems and communicate reasoning and solutions effectively in writing;
7 use learning resources approrpriately;
8 exhibit self management and time management skills.

SYLLABUS PLAN - summary of the structure and academic content of the module

- introduction to sets and inequalities, intervals;
- introductory guidance for self-study of the MAPLE package;
- two and three dimensional analytic geometry as loci;
- coordinate forms of equations for lines, circles and planes
- normals and tangents;
- formulae for parabola, ellipse, hyperbola, planes, lines, spheres, cones;
- functions: dependent and independent variables;
- injections, surjections and bijections, image and preimage;
- graphs of real functions;
- even, odd functions, sums, differences, products, quotients, composition;
- inverse functions;
- trigonometric and hyperbolic functions;
- sequences and series: finite and infinite sequences;
- geometric and arithmetic progressions;
- definition of a limit of a sequence;
- partial sums and series;
- comparison and ratio tests;
- absolute convergence;
- continuity of functions: definition of a limit, limits from right and left, manipulation and calculation of limits;
- finite limits as the variable tends to infinity, infinite limits;
- continuity;
- statement of intermediate value theorem;
- differentiation: definitions, properties and applications of differentiation;
- first principles calculations;
- rules for differentiation;
- implicit differentiation;
- linearisation of functions;
- higher order derivatives;
- Leibniz's rule;
- logarithmic differentiation;
- introduction to integration;
- applications of differentiation: maxima and minima of functions;
- L'Hopital's rule;
- rolle and mean value theorems;
- Taylor and Maclaurin series with remainder;
- applications in geometry and curve sketching: tangents, normals, asymptotes, turning points.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 49 101 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 33 Lectures Scheduled learning and teaching activities 5 Problem class Scheduled learning and teaching activities 11 Tutorials Guided independent study 101 Lecture and assessment preparation, wider reading

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Fortnightly exercise 5 x 3 hours All Annotated scripts with oral feedback from tutor
Mid-term test 40 minutes All Peer-to-peer marked

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 0 100
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam - Closed book 100 2 hours All Annotated scripts

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

Basic reading: Any A-level on mathematics and further mathematics

ELE: http://vle.exeter.ac.uk