Summer School details
Sample problem
The aircraft’s flightpath goes through coordinates (1,2,0) and (23,-19,3). The top of the hill is at (18,-13,2).How close does the aircraft get to the top of the hill? Vectors make this an easy calculation. 
J Willard Gibbs, 1839-1903, was a pioneer in vector analysis. His family lived in Connecticut and Gibbs became Professor of Mathematical Physics at Yale in 1871 — rather surprisingly before he had published any work! He made major contributions to thermodynamics, the electromagnetic theory of light and statistical mechanics.Sample problem In a certain city, commuters go to work by car or bus. A study shows that from each year to the next year 20% of car users change to travelling by bus, while 15% of bus users change to travelling by car. What percentage of commuters travel by car, once things have settled down? Suspension bridges
The main cable of a suspension bridge naturally forms a curve called a catenary. When it is loaded with the horizontal road structure it deforms into a parabola. Unless very carefully designed, suspension bridges are susceptible to collapse from high winds or earthquakes.Sample problem
Let N be the number of individuals in a population. One model for studying N says that the rate of increase of N depends on both N itself (since the more individuals there are the more offspring will be produced) and on some residual amount M - N (since there will be competition for resources like food); so we have
for some constant M.Will the population die out or reach some maximum value? 
Gottfried Leibniz, 1646-1716, was one of the developers of calculus — the other was Isaac Newton. They used different approaches, and different notation. Leibniz also was a pioneer of mathematical logic. |
MATH160 Mathematics 1First Semester, Second Semester, Summer School, 18 points IntroductionAlgebra and Calculus form the basic tools used to produce most mathematical frameworks for modelling quantifiable phenomena. For example, to model the movement of an object through space we need first to create an algebraic structure in which to specify where our object is, and then we can study how that position changes with time (i.e. its movement) using calculus.Many other problems arising in areas such as Economics or Chemistry, can be examined in a mathematical way using the same basic ideas. For example, we may need to minimize a manufacturing cost, or the time for a chemical reaction to take place, or the effects of river pollution; in each case the techniques used for the minimization are based on a mixture of algebra and calculus theories. This course aims to develop skills with these tools both for use in other subjects and in preparation for further study of Mathematics. Paper detailsThis paper is the natural continuation of Year 13 Mathematics, and is divided between algebra and calculus.After a review of basic trigonometry, the algebra half focusses on three-dimensional vectors and their many uses (such as in geometry, computer graphics, surveying and even calculus). The vector representation of lines, planes and projections leads naturally to the discussion of linear systems of equations. The basic properties of matrices are studied together with some applications. Complex numbers and polynomials complete this section of the course. In the calculus half you will study the ideas and methods of differentiation and integration, using an approach that is intuitive and avoids excess formality. Applications will include optimization, related rates, the use of differentials, finding areas, the Taylor series, solving simple differential equations, and an introduction to partial derivatives. Potential studentsMath 160 is intended both for those with a main interest in studying Mathematics and/or Statistics, and those whose interest in Mathematics is mainly to support other areas of study. These might include the physical, health and biological sciences, computer and information science, engineering, surveying, architecture, economics and finance, and philosophy of science. An understanding of basic algebraic and differential and integral techniques is of benefit to all students exposed to the analysis of processes, whether involving one or several variables.The paper is suitable for students who have passed at least 12 credits in NCEA level 3 Calculus (or equivalent) or have passed at least 18 credits in NCEA level 3 Statistics. Weaker students should seek advice and might first consider taking Math 151. Main topicsAlgebra:
Calculus:
PrerequisitesNoneRequired textsMATH 160 Algebra Outline Notes (available from the Print Shop)Calculus by James Stewart (Truncated edition) (available from the University Book Shop) Useful referencesSeveral standard texts are suitable for reference. For example:Elementary Vector Algebra by A.M. MacBeath Algebra, Geometry and Trigonometry by M.V. Sweet Calculus with Analytic Geometry by Howard Anton (Wiley) Calculus by James Stewart (Full edition.) LecturersSemester 1:
LecturesCalculus: Mon and Wed and alternating FridaysAlgebra: Tues, Thurs and alternating Fridays
TutorialsAttendance at tutorials is voluntary. An open tutorial system operates where classes run for 8 hours per week, and students may attend as many as they need to and are able to.Internal AssessmentThere are ten marked assignments which make up your internal assessment mark (A).Five computer Skills Tests in each of Algebra and Calculus together make up 33.3% (T) of your final mark. You can check your marks by clicking on the Resources link at the top of this page. Terms RequirementYou have to fulfil the terms requirement in order to be allowed to sit the final exam. In this paper, to pass “terms” you need to gain at least 5/10 in each of the first four Skills Tests.Exam formatThe 3-hour final exam is answered in spaces provided on the question booklet. All questions should be attempted and the number of marks available for each question is indicated on the paper. There are usually from 15 to 20 questions for each of Algebra and Calculus. You may allocate your time between the two sections as you wish.Final markThe final mark F is calculated from:F = max { E, (3E + A)/4 } + T where E (exam mark) is out of 66.7, A (internal assessment) is out of 66.7, T (test mark) is out of 33.3.The “max” corresponds to plussage: if your internal assessment mark is greater than your exam mark then it is combined in the proportion shown. If it is less then it is ignored and the exam mark itself is used. So your internal assessment can boost your exam mark with a 1/4 weighting if that helps you. Note that the test component definitely counts and so the tests should be regarded as compulsory. PlagiarismStudents should make sure that all submitted work is their own. “Plagiarism is a form of dishonest practice. Plagiarism is defined as copying or paraphrasing another’s work and presenting it as one’s own” (University of Otago Calendar). In practice this means that plagiarism includes any attempt in any piece of submitted work (e.g. an assignment or test) to present as one’s own work the work of another (whether of another student or a published authority). Any student found to be responsible for plagiarism in any piece of work submitted for assessment shall be subject to the University’s dishonest practice regulations which may result in various penalties, including forfeiture of marks for the piece of work submitted, a zero grade for the paper, or in extreme cases exclusion from the University. The University of Otago reserves the right to use plagiarism detection tools.
While we strive to keep details as accurate and up-to-date as possible, information given here should be regarded as provisional. Individual lecturers will confirm teaching and assessment methods.
| |
