Faculty of Science

School of Mathematics and Statistics

MATH1013: Mathematical Modelling

Semester 2, 2014 | 3 Credit Points | Coordinator: Dr David Easdown(david.easdown@sydney.edu.au)

1 Introduction

MATH1013 is a 3 credit point unit of study that covers the topic of Mathematical Modelling.

Mathematical models are used in almost all branches of the natural and social sciences. Amathematical model uses various types of equations to capture relevant aspects of somesystem in order to better understand its properties, or, to make predictions about its behaviour.One of the most challenging and exciting aspects of modelling is to create a model that issimple enough to use, but sophisticated enough to describe interesting properties of the system.

In this course, we will look at the two most common ways in which systems are modelled. Thefi rst uses functions to describe how some property changes with time (for example, using anexponential function to describe the increase in carbon-dioxide concentrations). The seconduses sequences of values to describe the system at equally spaced intervals in time (forexample, using a geometric progression to describe the annual population of Australia).

1.1 Assumed Knowledge and Prohibitions

Assumed knowledge: HSC Mathematics or MATH1111

Prohibition: May not be counted with MATH1003, MATH1903, MATH1907.

2 Course Aims, Learning Objectives andGraduate Attributes

2.1 Course Aims

In the MATH1013 course students will learn and use the following mathematical techniques

In the MATH1013 course students will learn and use the following mathematical techniquesand skills.

Recognise simple models for growth and decay, including differential equations forlinear and exponential growth and recurrence relations for arithmetic and geometricgrowth;

1.

Write down general and particular solutions to simple models of growth and decay;2.Determine the order of a differential equation or recurrence relation;3.Find the equilibria (steady state and fixed point solutions) of simple differentialequations and recurrence relations, and analyse their stability using both graphicalmethods and slope conditions;

4.

Recognise separable fi rst-order equations and apply separation of variables to determinethe general solution;

5.

Use partial fractions and separation of variables to solve certain nonlinear differentialequations, including the logistic equation;

6.

Use a variety of graphical and numerical techniques to locate and count solutions toequations, including the use of sign-change tests and monotonic functions;

7.

Solve equations numerically by fi xed-point iteration, including checking if an iterationmethod is stable;

8.

Explore sequences numerically, and classify their long-term behaviour, including cycles;9.Determine the general solution to linear second-order equations or simultaneous pairs offirst order equations, including recognising when the solutions are trigonometric.

10.

Students enrolling in MATH1013 should be able to:

Solve quadratic equations;1.Rearrange and solve simple equations involving powers, logarithms and exponentialfunctions;

2.

Solve pairs of simultaneous equations in two variables;3.Differentiate and integrate simple functions, including powers, exponentials, logarithmsand trigonometric functions;

4.

Sketch simple functions using intercepts, asymptotes, turning points and points ofinflection;

5.

Verify solutions to equations by substitution and verify inde nite integrals bydifferentiation;

6.

Distinguish between absolute and relative rates of change.7.

2.2 Learning Outcomes

After successfully completing this unit, you should be able to:

1. classify, interpret and construct simple mathematical models;2. compare and discuss the results of applying different models to the same data or situation;3. understand the limitations of models and mathematical methods, including when and why they fail;4. recognise the same information or model when presented in different forms, and convert or transformbetween equivalent forms;5. extract useful information from a model or equation without solving it exactly, including the use of graphicalarguments;6. apply simple techniques in unfamiliar situations, including generalising from simple to complex systems and

6. apply simple techniques in unfamiliar situations, including generalising from simple to complex systems andverifying the generalisations;7. combine two or more techniques or steps to complete a complex task, including using simple models asbuilding blocks;8. use numerical exploration to aid in the understanding of the behaviour of models, including using tools, suchas calculators, efficiently to estimate and approximate.

2.3 Graduate Attributes

Graduate Attributes are generic attributes that encompass not only technical knowledge butadditional qualities that will equip students to be strong contributing members of professionaland social communities in their future careers. The overarching graduate attributes identifiedby the University relate to a graduates attitude or stance towards knowledge, towards theworld, and towards themselves. These are understood as a combination of five overlappingskills or abilities, the foundations of which are developed as part of specific disciplinary study.For further details please refer to the Science faculty website at:http://www.itl.usyd.edu.au/graduateAttributes/facultyGA.cfm?faculty=Science

Graduate Attributes LearningOutcomes

A Research and Inquiry

A1. Apply scientific knowledge and critical thinking to identify, define andanalyse problems, create solutions, evaluate opinions, innovate andimprove current practices.

1, 2, 3, 4, 5, 6, 7, 8

A2. Gather, evaluate and deploy information relevant to a scientific problem. 1, 3, 4, 5, 6, 7, 8

A3. Design and conduct investigations, or the equivalent, and analyse andinterpret the resulting data.

1, 2, 3, 4, 5, 6, 7, 8

A4. Critically examine the truth and validity in scientific argument anddiscourse, and evaluate the relative importance of ideas.

1, 2, 3, 4, 5, 6, 7, 8

A5. Disseminate new knowledge and engage in debate around scientificissues.

1, 2, 3, 4, 5, 6, 7, 8

A6. Value the importance of continual growth in knowledge and skills, andrecognise the rapid, and sometimes major, changes in scientificknowledge and technology.

1, 2, 3, 4, 5, 6, 7, 8

B Information Literacy

B1. Use a range of searching tools (such as catalogues and databases)effectively and efficiently to find information.

1, 2, 3, 4, 5, 6, 7, 8

B2. Access a range of information sources in the science disciplines, forexample books, reports, research articles, patents and companystandards.

1, 2, 3, 4, 5, 6, 7, 8

B3. Critically evaluate the reliability and relevance of information in ascientific context.

1, 2, 3, 4, 5, 6, 7, 8

B4. Consider the economic, legal, social, ethical and cultural issues in thegathering and use of information.

1, 2, 3, 4, 5, 6, 7, 8

B5. Use information technology to gather, process, and disseminate scientificinformation.

1, 2, 3, 4, 5, 6, 7, 8

C Communication

C1. Explain and present ideas to different groups of people in plain English. 1, 2, 3, 4, 5, 6, 7, 8

C2. Write and speak effectively in a range of contexts and for a variety ofdifferent audiences and purposes.

1, 2, 3, 4, 5, 6, 7, 8

C3. Use symbolic and non-verbal communication, such as pictures, icons andsymbols as well as body language and facial expressions, effectively.

1, 2, 3, 4, 5, 6, 7, 8

C4. Present and interpret data or other scientific information using graphs,tables, figures and symbols.

1, 2, 3, 4, 5, 6, 7, 8

C5. Work as a member of a team, and take individual responsibility within thegroup for developing and achieving group goals.

1, 2, 3, 4, 5, 6, 7, 8

C6. Take a leadership role in successfully influencing the activities of a grouptowards a common goal.

1, 2, 3, 4, 5, 6, 7, 8

C7. Actively seek, identify, and collaborate with others in a professional andsocial context.

1, 2, 3, 4, 5, 6, 7, 8

D Ethical, Social and Professional Understanding

D1. Demonstrate an understanding of the significance and scope of ethicalprinciples, both as a professional scientist and in the broader socialcontext, and a commitment to apply these principles when makingdecisions.

1, 2, 3, 4, 5, 6, 7, 8

D2. Appreciate the importance of sustainability and the impact of sciencewithin the broader economic, environmental and socio-cultural context.

1, 2, 3, 4, 5, 6, 7, 8

D3. Demonstrate empathy with, and sensitivity towards, another's situation,feelings and motivation.

1, 2, 3, 4, 5, 6, 7, 8

E Personal and Intellectual Autonomy

E1. Evaluate personal performance and development, recognise gaps inknowledge and acquire new knowledge independently.

1, 2, 3, 4, 5, 6, 7, 8

E2. Demonstrate flexibility in adapting to new situations and dealing withuncertainty.

1, 2, 3, 4, 5, 6, 7, 8

E3. Reflect on personal experiences, and consider their effect on personalactions and professional practice.

1, 2, 3, 4, 5, 6, 7, 8

E4. Set achievable and realistic goals and monitor and evaluate progresstowards these goals.

1, 2, 3, 4, 5, 6, 7, 8

E5. Demonstrate openness and curiosity when applying scientificunderstanding in a wider context.

1, 2, 3, 4, 5, 6, 7, 8

2.4 Threshold Learning Outcomes

The Threshold Learning Outcomes (LTOs) are the set of knowledge, skills and competenciesthat a person has acquired and is able to demonstrate after the completion of a bachelor degreeprogram. The TLOs are not equally weighted across the degree program and the numberingdoes not imply a hierarchical order of importance.

Threshold Learning Outcomes LearningOutcomes

1 Understanding science

1.1 Articulating the methods of science and explaining why current scientificknowledge is both contestable and testable by further inquiry

1, 2, 3, 4, 5, 6, 7, 8

1.2 Explaining the role and relevance of science in society 1, 2, 3, 4, 5, 6, 7, 8

2 Scientific knowledge

2.1 Demonstrating well-developed knowledge in at least one disciplinary area 1, 2, 3, 4, 5, 6, 7, 8

2.2 Demonstrating knowledge in at least one other disciplinary area 1, 2, 3, 4, 5, 6, 7, 8

3 Inquiry and problem solving

3.1 Gathering, synthesising and critically evaluating information from a rangeof sources

1, 2, 3, 4, 5, 6, 7, 8

3.2 Designing and planning an investigation 1, 2, 3, 4, 5, 6, 7, 8

3.3 Selecting and applying practical and/or theoretical techniques or tools inorder to conduct an investigation

1, 2, 3, 4, 5, 6, 7, 8

3.4 Collecting, accurately recording, interpreting and drawing conclusionsfrom scientific data

1, 2, 3, 4, 5, 6, 7, 8

4 Communication

4.1 Communicating scientific results, information or arguments, to a range ofaudiences, for a range of purposes, and using a variety of modes

1, 2, 3, 4, 5, 6, 7, 8

5 Personal and professional responsibility

5.1 Being independent and self-directed learners 1, 2, 3, 4, 5, 6, 7, 8

5.2 Working effectively, responsibly and safely in an individual or teamcontext

1, 2, 3, 4, 5, 6, 7, 8

5.3 Demonstrating knowledge of the regulatory frameworks relevant to theirdisciplinary area and personally practising ethical conduct

1, 2, 3, 4, 5, 6, 7, 8

For further details on course learning outcomes related to specific topics see LMS site andCourse Handbook.

3 Study Commitment

The current standard work load for a 3 credit point unit of study is 3 hours per week of

The current standard work load for a 3 credit point unit of study is 3 hours per week offace-to-face teaching contact hours (2 lectures and 1 tutorial) and an additional 3 hours perweek of student independent study. Below is a breakdown of our expectations for this unit. Itshould be noted that Independent Study is based on what we believe to be the amount of timea typical student should spend to pass an item of assessment. Times are a guide only.

In class activities Hours

Lectures (26 @ 1 hr each) 26

Tutorials(12 @ 1 hr each) 12

Total 38

Independent Study Hours

Preparation for lectures (26 @ 0.5 hr each) 13

Review and self assessment (12 weeks@ 1 hr each) 12

Preparation for tutorials (12 @ 1 hr each) 12

Total 37

Study Tips

You are now in control of your own study strategy, and as an adult learner it is up to you todevise a study plan that best suits you. Many resources are available to assist your learning,including a set of independent study exercises for you to complete.

Any questions?

Before you contact us with any enquiry, please check the FAQ page at http://www.maths.usyd.edu.au/u/UG/JM/FAQ.html

Where to go for help

For administrative matters, go to the Mathematics Student Office, Carslaw room 520.For help with mathematics, see your lecturer, or your tutor. Lecturers guarantee to be availableduring their indicated office hour.If you are having difficulties with mathematics due to insufficient background, you should goto the Mathematics Learning Centre (Carslaw room 441).

4 Learning and Teaching Activities

WEEKLY SCHEDULE

LECTURES

There are two different lecture streams. You should attend one stream (that is, two lectures perweek) as shown on your personal timetable. Lectures run for 13 weeks and the last lecture willbe on Friday 28 October.

Times Location Lecturer Consultation8am Thurs & Fri to be advised to be advised to be advised

11am Thurs & Fri to be advised to be advised to be advised

TUTORIALS

One tutorial per week, starting in week 2. You should attend the tutorial given on yourpersonal timetable. Attendance at tutorials will be recorded. Your attendance will not berecorded unless you attend the tutorial in which you are enrolled.

Tutorial sheets

The tutorial sheets for a given week will be available on the MATH1013 webpage by theFriday of the previous week. You must take the current weeks sheet to your tutorial. Thesheet must be printed from the web.Solutions to tutorial exercises for week n will usually be posted on the web by the afternoon ofthe Friday of week n.

WEEK-BY-WEEK OUTLINE

Week Topic

1 1. intro to differential equations2. general and particular solutions (differential equations)

2 3. equilibrium (steady-state) solutions for differential equations4. stability of equilibria for differential equations (graphical method)

3 5. separation of variables6. simple linear models

4 7. partial fractions8. the logistic function

5 9. Applications of logistic models10. ... continued

6 11. intro to difference equations (recurrence relations)12. general and particular solutions (difference equations)

7 13. equilibrium (fixed-point) solutions for difference equations14. stability of fixed points

8 15. numerical solution of equations16. fixed-point iteration (Gregory-Dary method)

9 17. behaviour of logistic map18. applications of logistic map MID SEMESTER BREAK

10 19. second-order equations20. the characteristic quadratic (real roots case only first)

11 21. pairs of first-order differential equations22. pairs of first-order difference equations

12 23. trigonometric solutions24. ...continued

13 25. review of the unit of study26. review of past exam

5 Teaching Staff and Contact Details

Unit Coordinator Email

Dr DavidEasdown david.easdown@sydney.edu.au

Teaching Staff Email Room Phone Note

to be advised to be advised to be advised

6 Learning Resources

Textbook

No prescribed textbook.

Course notes

Poladian L. Mathematical Modelling. School of Mathematics and Statistics, University ofSydney, Sydney, NSW, Australia.

Available from Kopystop, 55 Mountain St., Broadway.

References

Adler FR, 2007. Modelling the Dynamics of Life. 2nd Edition. Brooks/Cole Publishers, USA.

Banner A, 2007. The Calculus Lifesaver. Princeton University Press.

Bittinger M, Brand N and J. Quintanilla, 2006. Calculus for the Life Sciences. Pearson.

Cohen D and J. Henle, 2005. Calculus: the Language of Change. Jones and BartlettPublishers, USA.

Hughes-Hallett et al., 2002. Calculus. 3rd edition. John Wiley and Sons Inc.

Web Site

It is important that you check the Junior Mathematics web site regularly.It may be found through Blackboard, by following links from the University of Sydney frontpage, or by going directly to http://www.maths.usyd.edu.au/u/UG/JM/

Important announcements relating to Junior Mathematics are posted on the site, and there is alink to the MATH1013 page. Material available from the MATH1013 page may includeinformation sheets, the Junior Mathematics Handbook, notes, exercise sheets and solutions,and previous examination papers.

7 Assessment Tasks

You are responsible for understanding the University policy regarding assessment andexamination.

Formative and Summative Assessment

Assessment in this unit will be both formative (for feedback) and summative (for marks). Quizzes and assignments incorporate both formative and summative assessment. Formativeassessment provides feedback on your performance, and summative assessment comprisesmarks for performance in assignments, quizzes and examinations, which will count towards afinal unit mark.

7.1 Summative Assessments

Assessment Task Percentage Mark Due Date Learning Outcomes

Quiz 1 15 Week 6 (week starting Sunday, 31 August 2014)

1, 2, 3, 4, 5, 6, 7, 8

Quiz 2 15 Week 12 (week starting Sunday, 19 October 2014)

1, 2, 3, 4, 5, 6, 7, 8

Homeworks 5 Weekly 1, 2, 3, 4, 5, 6, 7, 8

Assignment 5 Week 11 (week starting Sunday, 12 October 2014)

1, 2, 3, 4, 5, 6, 7, 8

Final exam 55 Exam Period 1, 2, 3, 4, 5, 6, 7, 8

7.2 Assessment Grading

Your final raw mark for this unit will be calculated as follows:

Exam at end of semester: 55%

Exam at end of semester: 55%Quiz 1 mark (using the bettermark principle): 15%Quiz 2 mark (using the bettermark principle): 15%Homeworks: 10%Assignment 5%

The bettermark principle means that for each quiz, the quiz counts if and only if it isbetter than or equal to your exam mark. If your quiz mark is less than your exam markthe exam mark will be used for that portion of your assessment instead. So forexample if your quiz 1 mark is better than your exam mark while your quiz 2 mark isworse than your exam mark then the exam will count for 70%, quiz 1 will count for15%, the homework will count for 10% and the assignment 5% of your overall mark.The homework and assignment marks count regardless of whether they are betterthan your exam mark or not. Final grades are returned within one of the following bands:

High Distinction (HD), 85100: representing complete or close to completemastery of the material;Distinction (D), 7584: representing excellence, but substantially less thancomplete mastery;Credit (CR), 6574: representing a creditable performance that goes beyondroutine knowledge and understanding, but less than excellence;Pass (P), 5064: representing at least routine knowledge and understandingover a spectrum of topics and important ideas and concepts in the course.

A student with a passing or higher grade should be well prepared to undertake furtherstudies in math- ematics on which this unit of study depends.

8 Learning and Teaching Policies

For full details of applicable university policies and procedures, see the Policies Online siteat https://sydney.edu.au/policy

Academic Policies relevant to student assessment, progression and coursework:

Academic Dishonesty in Coursework. All students must submit a cover sheet for allassessment work that declares that the work is original and not plagiarised from thework of others. The University regards plagiarism as a form of academic misconduct,and has very strict rules that all students must adhere to. For information see thedocument defining academic honesty and plagiarism at:

https://sydney.edu.au/policies/showdoc.aspx?recnum=PDOC2012/254&RendNum=0

Coursework assessment policy. For information, see the documents outlining theUniversity assessment policy and procedures at:

https://sydney.edu.au/policies/showdoc.aspx?recnum=PDOC2012/266&RendNum=0 andhttps://sydney.edu.au/policies/showdoc.aspx?recnum=PDOC2012/267&RendNum=0.

The Faculty process is to use standards based assessment for units where grades arereturned and criteria based assessment for Pass / Fail only units. Norm referencedassessment will only be used in exceptional circumstances and its use will need to bejustified to the Undergraduate Studies Committee. Special consideration for illness ormisadventure may be considered when an assessment component is severelyaffected. Details of the information that is required to be submitted along with theappropriate procedures and forms is available at:

https://sydney.edu.au/science/cstudent/ug/forms.shtml#special_consideration

Start by going to the Faculty of Science Webpage, and downloading the SpecialConsideration pack at the link above.

Special Arrangements for Examination and Assessment. In exceptionalcircumstances alternate arrangements for exams or assessment can be made. Howeverconcessions for outside work arrangements, holidays and travel, sporting andentertainment events will not normally be given. The policy, guidelines and applicationform including examples of circumstances under which you might be awarded a specialarrangement for an examination or assessment task can be found at:

https://sydney.edu.au/science/cstudent/ug/forms.shtml#special_arrangements

Student Appeals against Academic Decisions. Students have the right to appeal anyacademic decision made by a school or the faculty. The appeal must follow theappropriate procedure so that a fair hearing is obtained. The formal application form canbe obtained at:

https://sydney.edu.au/science/cstudent/ug/forms.shtml#appeals

Relevant forms are available on the Faculty policies websiteat https://sydney.edu.au/science/cstudent/ug/forms.shtml

Special consideration and special arrangements

Students who suffer serious illness or misadventure that may affect their academicperformance may request that they be given special consideration in relation to thedetermination of their results.Students who are experiencing difficulty in meeting assessment tasks due to competingessential community commitments may request that special arrangements be made in respectof any or all factors contributing to their assessment.The Faculty of Science policies on these issues apply to all Mathematics and Statistics units ofstudy. Information relating to these policies, including the Application Packs and instructionson how to apply, can be obtained from the Faculty of Science website.Before applying for special consideration, please read the Faculty Policy, and the rest of thissection, to determine whether or not you are eligible. Note that occasional brief or trivial

illness will not generally warrant special consideration.

How to apply

Applications for special Consideration must be made within 5 working days of the date forwhich consideration is being sought.Applications for special arrangements must be submitted at least seven days BEFORE the duedate of the assessment or examination for which alternative arrangements are being sought.

The procedure is as follows.

Obtain the application forms from the Faculty of Science website or from the StudentInformation Office of the Faculty of Science.Take the original paperwork, plus one copy for each piece of assessment for whichconsideration is being sought, to the Student Information Office of the Faculty ofScience. Note that applications are to be lodged with the Science Faculty, regardless ofthe faculty in which you are enrolled. Your copies will be stamped at the Faculty StudentInformation Office.Take the stamped documentation to the Mathematics Student Services Office, Carslawroom 520 opposite the lifts on Carslaw Level 5). Your personal information must becompleted on all the forms, including the Academic Judgement form, before the formwill be accepted.

Note that an application for special consideration or special arrangements is a request only,and not a guarantee that special consideration will be granted or special arrangements made.Applications are considered in the light of your participation in the unit during the semester,and your academic record in mathematics.

Special consideration relating to assignments

Applications for special consideration relating to assignments will not be accepted.Exemptions from submission of assignments are not generally granted. If serious illness ormisadventure during the period prior to the due date prevents you from submitting anassignment on the due date then you should do the following:

Contact the Mathematics Student Office (by phone or email, or in person) to request anextension. Unless there are exceptional circumstances you must do this before the duedate.

1.

If you are granted an extension, take your assignment to the Mathematics Student Officeby the extended due date. (Do not put the assignment in the collection boxes.)

2.

Submit some supporting documentation (for example, a medical certificate) when youhand in your assignment.

3.

Late assignments will only be accepted if you have an approved extension, or in the followingcircumstance:Should you be ill on the due date only, and unable to submit your assignment, then you maysubmit it the following day, accompanied by supporting documentation (for example, amedical certificate). In this case, your assignment should be taken to Mathematics Student

Office. (Do not put the assignment in the collection boxes.)

Special consideration relating to quizzes

If you miss a quiz due to illness or misadventure, then you must go to the Mathematics StudentServices Office as soon as possible afterwards. Arrangements may be made for you to sit thequiz at another time. If that is not possible then you may be eligible to apply for specialonsideration.If your application for special consideration relating to missing a quiz is successful then apro-rata mark for that quiz will be awarded, based on your final examination mark in the unitof study.

Special consideration relating to end-of-semester examinations

If you believe that your performance on an exam was impaired due to illness or misadventureduring the week preceding the exam, then you should apply for 28 special consideration. Ifyour application is successful then your mark may be adjusted, or you may be offered theopportunity to sit a supplementary exam.Please note that illness or misadventure during the week preceding the exam is not anacceptable reason for missing an exam. If you miss an exam due to illness or misadventure onthe day of the exam then you should apply for special consideration. If your application issuccessful you will be granted the opportunity to sit a supplementary examination.Students who have participated only minimally in the unit throughout the semester will not begranted supplementary exams.

Special consideration relating to attendance

The Faculty policy applies. Note that special consideration will not be granted for brief illnessor minor misadventure that causes you to miss a tutorial. Unless a quiz was held during thetutorial, applications for special consideration in such cases will not be accepted. Jury duty, military service, national sporting and religious or cultural commitments

Students who will miss an assessment due to commitments such as these may apply for specialarrangements to be made. The Faculty of Science Special Arrangements Policy applies for allfirst year mathematics units. Note that an application for special arrangements must be madeat least seven days before the date of the assessment concerned.

Replacement assessments for end of semester examinations

Students who apply for and are granted either special arrangements or special consideration forend of semester examinations in units offered by the Faculty of Science will be expected to sitany replacement assessments in the two weeks immediately following the end of the formalexamination period. Later dates for replacement assessments may be considered where theapplication is supported by appropriate documentation and provided that adequate resources

application is supported by appropriate documentation and provided that adequate resourcesare available to accommodate any later date.