Mathematics - 2007

Extension Program Mathematics for High Achieving Students

Introduction

This subject is designed for students who enjoy mathematics and are mathematically talented. If you are such a student you may have already been involved in an acceleration or enrichment program during your earlier secondary school program, though this is not necessary. Topics build on those in Specialist Mathematics and thus they provide you with the opportunity to extend your knowledge, as well as to experience and appreciate some of the depth and complex beauty of higher level mathematics.

Career Opportunities

Graduates in Science who have majored in Mathematics and Statistics find employment in areas such as: biology (population modelling); business (quality management and improvement); demography and behavioural science (survey sampling, analysis and prediction of human populations); in finance, commerce and economics (the study and forecasting of production, consumer behaviour and stock prices); engineering (design and testing of structures, flood control and prediction, analysis of computer networks); education (secondary and tertiary teaching); and the medical sciences (epidemiology and analysis of medical research data).

Objectives

On completion of Extension Program Mathematics for High Achieving Students, you should be able to comprehend:

• some of the nature of the different types of numbers used;
• the intuitive notion of limits as used in continuity, differentiation and integration;
• the notion of integral as area;
• the fundamental ideas in the calculus of functions of several variables;
• the extension of the notion of vectors in two or three dimensions to any finite number of dimensions;
• the theoretical treatment of systems of simultaneous linear equations.

Have developed:

• an ability to manipulate complex numbers and to use them to solve problems;
• an ability to use differential calculus to solve extremal problems;
• an ability to compute a wide range of integrals;
• an ability to use integration to compute area, length and volume;
• an ability to solve arbitrary systems of simultaneous linear equations.

Appreciate:

• the role of proof and logical reasoning in mathematics;
• the use of complex numbers; the role of limits in both the differential and integral calculus;
• the practical uses of calculus; the use of the ideas of linear algebra in dealing with the solution of simultaneous linear equations.

VCE Prerequisites

Completion of Mathematical Methods 3/4 in Year 11 and enrolment in Specialist Mathematics 3/4 in Year 12; or concurrent enrolment in Mathematical Methods 3/4 and Specialist Mathematics 3/4 after completion of Mathematical Methods 1/2 and General Mathematics* (Specialist Mathematics orientation) 1/2.

*Applicants with a B+ average in General Mathematics will be considered. Other prerequisite subjects require A/A+ average.

A B+ average across non prerequisite subjects is required.

Teaching Locations

Extension Program Mathematics is taught at School Centres only. You must list two choices, in order of preference, on your application form. Classes will run at listed locations subject to minimum enrolments.

*Classes at School Centre scheduled during school hours.

Contact Hours

• Total class hours per week: 2-3 depending on School Centre
• Total days per week: 1-3 depending on School Centre

In addition there are regular exercises to be completed each week and a number of assignments spread throughout the year.

Credit Arrangements

See Credit Policy for further information.

Subject Description

620-120 Extension Program Mathematics for High Achieving Students

This subject must be successfully completed in order to obtain the ENTER increment.

Content

• Foundations: Sets, integers, mathematical induction; real numbers; complex numbers, polar form, de Moivre’s theorem, complex exponential.
• Calculus: Functions of one real variable (including limits and continuity), derivatives; curve sketching; maxima and minima, curvature; antiderivatives and the definite integral; trigonometric functions and their inverses, logarithm, exponential function, hyperbolic functions and their inverses; systematic integration; approximate integration; applications of integration, areas, arc length, surface areas and volumes of solids of revolution. Taylor polynomials and binomial series.
• Multivariable calculus: Functions of several variables, level curves, heights; partial derivatives, commutation of mixed partial derivatives; total derivative, gradient vector, directional derivatives and applications; chain rule; coordinate transformations, Jacobi matrix and determinant; Hessian matrix, maxima and minima of functions of several variables; introduction to double and triple integrals and applications.
• Vectors and linear equations: Vectors in three-dimensional space, dot and cross products, triple products, determinants; linear dependence; equations of lines and planes, geometrical applications; bases and co-ordinates, dimension; row-reduction, rank, inverse, solution of linear equations, geometrical interpretation.

Course Materials

1. Exercise Sheets

The booklet of exercise sheets contains 10 separate parts which are numbered 'Sheet 1' through to 'Sheet 10'. These correspond to the different sections of the course, and each comprehensively covers the material you are expected to know.

Exercise sheets are supplied to students free of charge.

2. Student Material

There are two books, both produced by the Department of Mathematics and Statistics, which complement the exercise sheets. It is strongly recommended that students purchase these (the total cost will be $30). Your teacher will give you details of how to order them.

The material includes:

1. Answers to all exercises; Solutions to selected exercises; Examination papers, Department of Mathematics and Statistics. This book includes answers to all of the problems on the exercise sheets as well as fully worked solutions to a selection of problems. It also includes past examination papers from the first year course 620-121 and Extension Program examination papers - two papers each year. Solutions to all examination papers will be made available to you through your teacher later in the year.
2. Course Notes, Department of Mathematics and Statistics. This is a set of brief topic notes and summaries which cover a large part of this course.

3. Textbooks

There are no prescribed texts for 620-120 Mathematics for High Achieving Students. That is, it is not compulsory for you to purchase a textbook. However, the books listed below are strongly recommended for reference. There should be at least one copy of a Calculus text and one copy of a Linear Algebra text available in your school library so that you have easy access to them, and you are encouraged to use them.

If you are considering buying a text, you may wish to consider that the textbooks that are prescribed for the first year maths courses at the University of Melbourne are the two listed here:

• Thomas' Calculus, M.D. Weir, J. Hass and F.R. Giordano, 11th edition, Pearson 2005
• Elementary Linear Algebra, H. Anton & C. Rorres, Applications Version, 9th edition, Wiley 2005
  

The books listed below cover most but not all of the topics in 620-120 UMEP Mathematics for High Achieving Students. All are good references and sources of worked examples and additional problems.

• Calculus and Analytic Geometry, 9th edition, by G. B. Thomas and R. L. Finney, Addison-Wesley, 1996 (the 7th and 8th editions are just as good as the 9th)
• Calculus with Analytical Geometry, by H. Anton, 4th edition, J. Wiley & Sons, 1992
• Elementary Linear Algebra with Applications, 7th edition, by H. Anton and C. Rorres, J. Wiley & Sons, 1994
• Calculus, 5th edition, by Stanley I. Grossman, Saunders College (Harcourt Brace), 1992 (a student Solutions manual is available for this text)
• Elementary Linear Algebra, 5th edition, by Stanley I. Grossman, Saunders College (Harcourt Brace), 1994 (a student Solutions manual is available for this text)
• Calculus: Concepts and Contexts, by James Stewart, Brooks/Cole, 1998 (a student Solutions manual is available for this text)
• Elementary Linear Algebra, 4th edition, by Venit and W. Bishop, Brooks/Cole, 1996 (a student Solutions manual is available for this text)

Assessment

Up to 50 pages of written assignments, a 2 hour written examination and a 3-hour written examination, both being final examinations in November.

More Information

Dr Deborah King
Tel: 8344 8052

Department of Mathematics and Statistics
The University of Melbourne VIC 3010

Web: www.ms.unimelb.edu.au