Math 1903 3j4ru

THE UNIVERSITY OF SYDNEY Semester 2, 2011 Information Sheet for MATH1903 Integral Calculus and Modelling (Advanced) Web Sites It is important that you regularly check both the Junior Mathematics web site http://www.maths.usyd.edu.au/u/UG/JM/ and the MATH1903 web site http://www.maths.usyd.edu.au/u/UG/JM/MATH1903 Lectures Times Location 8 am Thu & Fri Carslaw 157 Lecturer Weeks 1–6, 13: Dr J Parkinson, Carslaw room 614 Weeks 7–13: Dr D Daners, Carslaw room 715 Consultation Thursdays , 1-2pm Wednesdays, 1-2pm Lectures run for 13 weeks. The last lecture will therefore be on Friday 28 Octob er. Tutorials Tutorials (one per week) start in week 2. You should attend the tutorial given o n your personal timetable. Attendance at tutorials will be recorded. Your attendance will not be recorded unless you attend the tutorial in which you are enrolled. Tutorial sheets The tutorial sheets will be available on the MATH1903 webpage. You must take the current week’s sheet to your tutorial. The sheet must be printed from the web. Solutions to tutorial exercises for week n will usually be posted on the web by the afternoon of the Friday of week n. Course notes Parkinson, Luckock, Dullin. Lecture notes for MATH1903: Integral Calculus and Modelling (Advanced), School of Mathematics and Statistics, University of Sydney, 2010. Available from KOPYSTOP. Reference book James Stewart. Calculus. Brooks/Cole Publishing Company. ISBN 053459493. Availab le from the CO-OP BOOKSHOP. 1

Assessment Your nal raw mark for this unit will be calculated as follows: 70%: 20%: 10%: Exam at end of semester 2. Quiz mark. Assignment mark. Your nal raw mark is then scaled to produce your nal mark. Marks are scaled so tha t the distribution of grades is consistent with the quality of the class, and the di cul ty of the unit, as required by the University. Examination There is one examination of 1.5 hours’ duration during the examination period at t he end of semester 2. Further information about the exam will be made available at a later date. Quizzes There are two quizzes, each worth 10% of your nal raw mark. Quizzes are held duri ng tutorials, in week 6 (beginning 29 August) and week 11 (beginning 10 October). You should put those dates in your diary now! You must sit for the quiz during t he tutorial in which you are enrolled. Your quiz mark will not be recorded if you sit for the q uiz in a tutorial in which you are not enrolled (unless you have made an arrangement with the Mathema tics Student O ce). If you miss a quiz, then you must go to the Mathematics Student O ce as soon as possible afterwards. Assignments Two assignments will be set and marked. Each assignment is worth 5% of your nal r aw mark. Assignments will be due on Tuesday 16 August and Tuesday 4 October. Please see p age 26 of the Junior Mathematics Handbook for details relating to the submission of ass ignments. Any questions? Before you us with any enquiry, please check the FAQ page: http://www.maths.usyd.edu.au/u/UG/JM/FAQ.html Where to go for help For istrative matters, go to the Mathematics Student O ce, Carslaw room 520. For help with mathematics, see your lecturer, or your tutor. Your lecturer guara ntees to be available during his indicated o ce hour, but may well be available at other times as well. 2

Aims and Learning Outcomes This unit of study builds on your rst semester calculus course, and is an importa nt part of your transition into intermediate units of study. The unit has two components: Integr al Calculus (weeks 1-6) and Di erential Equations and Modelling (Weeks 7-12), with week 13 reserved f or revision lectures. The integral calculus component of the course puts the notion of the Riemann int egral onto a solid footing and builds on the intuitive understanding you have of integration from high school. This part of the course also contains an introduction to sequences and series, a nd a careful study of Taylor polynomials and Taylor series. This is a beautiful area of mathematics , with many remarkable formulae and theorems, and it is essential for your later studies in both pure and applied mathematics. The component on di erential equations and modelling introduces the notion of a di e rential equations and applies them to a variety of modelling problems. Di erential equatio ns are the fundamental tools in describing change. The solution of a di erential equation is a family of functions, depending on arbitrary constants that are xed by initial or boundary c onditions. In some cases solving a di erential equation can be reduced to integration, and a var iety of such methods will be derived. By the end of semester, students should: • Appreciate the signi cance of Integral Calculus, and in particular appreciate the powerful interplay between Integral and Di erential Calculus (via the Fundamental Theorem o f Calculus). • Be comfortable working with functions de ned by integrals, and have an intuitive u nderstanding of how these functions behave. • Have a basic understanding of sequences and series. • Understand the notion of convergence of sequences and series, and be able to con struct proofs of convergence. • Appreciate the beauty of calculus by working through tutorial exercises on some mathematical gems: π is irrational, er is irrational for all r ∈ Q\{0}, Stirling’s asymptotic formu la for n!, Wallis’ product formula for π, Leibnitz’ series, Euler’s series, and many more. • Understand the concept of a di erential equation geometrically (direction eld) and analytically (general solution vs. particular solution). • Appreciate the power of modelling with di erential equations, and be familiar with a variety of examples of modelling in the sciences. • Be able to solve simple rst order di erential equation by either separation of vari ables or using an integrating factor. • Understand how to change variables in a di erential equation. • Solve 2nd order linear equations with constant coe cients (homogeneous and inhomog eneous) with ease. • Have a basic understanding of systems of rst order di erential equations.

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Week-by-week outline Week 1 Topic Riemann sums and integrals 2 Fundamental Theorem of Calculus 3 Applications of the integral 4 Improper integrals 5 Sequences and series 6 Taylor polynomials and series. 7 Introduction to Di erential Equations and modelling 8 1st order separable DEs 9 1st order linear DEs 10 2nd order DEs 11 2nd order linear inhomogeneous DEs 12 Systems of di erential equations 13 Contents Evaluation of Riemann sums. De nition of the Riemann integral. Fundamental Theorem of Calculus. Functions de ned by integrals. Areas, volumes,

lengths and surface areas. Integrals of unbounded functions. Integrals over unbounded intervals. Comparison tests; p-integrals Convergence of sequences and series. Comparison tests for sequences and series. p-series. The remainder term. Taylor’s Theorem. Taylor series. Applications of Taylor series. Exponential growth. Order, general solution. Simple solutions. Direction Fields. Visualisation of solution curves. Separable Equations. Uniqueness. Equilibrium solutions. Logistic model. Partial Fractions. Classi cation. Integrating Factor. Superposition principle. Change of variables. Families of curves and exact DEs IVP and BVP. Linear DEs: Superposition principle. Linear homogeneous constant coe cients. Method of undetermined coe cients. Simple harmonic motion without and with damping, resonance. Predator-prey system. Reduction to 2nd order. Eigenvalues and stability. Matrix exponential Review 4