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Mathematics 1A20 (MATH 11004)

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Administrative Information

  1. Unit number and title: MATH 11004 Mathematics 1A20
  2. Level: C/4 (Open)
  3. Credit point value: 20 credit points
  4. Year: 10/11
  5. First Given in this form: 1997/8 (with a somewhat different syllabus)
  6. Unit Organiser: Richard Porter
  7. Lecturer: Dr Richard Porter
  8. Teaching block: 1
  9. Prerequisites: A-level Mathematics, or equivalent.

Unit aims

To consolidate, develop and extend the skills in single variable calculus introduced at A level.

General Description of the Unit

This unit is designed for students with a good grasp of A level mathematics who want a 20 credit-point unit on mathematical techniques. It consists of the first half of the unit Mathematics 1AM.

The unit begins with some basic ideas revising and extending school-level calculus, and then goes on to a thorough treatment of the calculus from the point of view of scientific applications. The subject is developed as far as differential equations and Fourier series. The mathematics is treated with enough logical precision to enable correct calculations and correct deductions to be made.

There is no other mathematics unit that can be taken as a sequel to this one. Students with an A level pass who wish to take more mathematics than 20cp should consider taking one of the 40 c.p. units Maths 1AM or 1AS instead of 1A20.

Relation to Other Units

This unit consists of the first half of the 40cp units Mathematics 1AM/1AS.

Teaching Methods

The unit is based on lectures, problems classes and tutorials on how to apply the techniques of the calculus in solving problems.

The lecturer will distribute problem sheets based on the work done in lectures, and will set specific problems which you will be required to hand in. During the first few weeks, weekly problems classes will be held and work handed in centrally. Later the problems  classes will be replaced by weekly tutorials and work will be handed in to tutors for marking.

Experience shows that progress in mathematics depends crucially on regular work at examples. For this reason you are REQUIRED to attend tutorials and to hand in the set work. See the section Formal Requirements of the Unit below.

Tutorials

Weekly tutorials will be held after the first few weeks. You will be given the time of your tutorial.

Learning Objectives

After taking this unit, students should have a thorough grasp of one-variable calculus and complex numbers, including simple differential equations and Fourier Series.

Assessment Methods

The final assessment mark for Mathematics 1A20 is calculated as follows:

  • 10% from a midsessional examination in January,
  • 90% from an examination in May/June.

More information is given below.

Use of Calculators and Notes

Candidates may bring into the examination room a calculator of the approved type (briefly: no graphics, programming, text storage, complex numbers, matrices, or symbolic algebra).

Candidates may bring into the examination room one A4 sheet of notes; both sides of the sheet may be used.

Details of the Summer Examination

The summer examination lasts 3 hours, and is in two sections.

  • Section A has 10 short questions, all of which should be answered; it carries 40% of the marks for this paper.
  • Section B has 6 longer questions, of which you should do FOUR. If you do more than four, your best four answers from this section will be assessed. Section B carries 60% of the marks for this paper.

January examinations

The January midsessional examinations are right at the start of the second term. This term begins on Friday 14th January 2011, and the Maths 1A20 examination may be on Friday 14th January or Saturday 15th January. IT IS YOUR RESPONSIBILITY to ensure that you are in Bristol to sit the examination; otherwise your mark will be zero (unless you have a certified illness or other special circumstances). You will be notified of the date, time and place of the January examination before the end of the first term.

The midsessional written examination lasts 1 1/2 hours, and is in two sections:

  • Section A has 5 short questions, all of which should be answered; it carries 40% of the marks for this paper. Section B has 3 longer questions, of which you should do TWO. If you do more than two, your best two answers from this section will be assessed. Section B carries 60% of the marks for this paper.

September examinations

If you fail this unit in June, you may (depending on which Faculty you are in and how you have done in your other units) be allowed to resit it in September. The September examination paper has the same structure as in June. If you are offered a resit, you must take the resit examination.

Award of Credit Points

Formal requirements of the unit

You must gain 120 credit points each year in order to be allowed to continue on your degree programme. There are also conditions on how much you must pass: details will be available from your Faculty Handbook or Departmental Handbook. We explain below what you must do (1) to pass this unit, and (2) to gain the 20 credit points for this unit.

Note: we will make allowances for illness and other such good reasons, PROVIDED that you follow the Department of Mathematics procedures and inform the Undergraduate Coordinator in Mathematics and fill out the Extenuating Circumstances forms supplied by the Department, which will also require the submission written documentation of your case (e.g. a doctor's certificate, specifying the date(s) you were unable to undertake academic work).

(1) Passing the unit

You will pass if your final assessment mark is 40 or more.

(2) Gaining credit points for the unit

You will be awarded the 20 credit points for the unit if

  • either you pass the unit with a mark 40 or over,
  • or you score a mark between 30 and 39 inclusive, and
  • you have attended at least 75% of the tutorials; and
  • you have made a serious attempt in at least 75% of the weekly homework assignments and
  • you have attended the January exam.

Transferable Skills

Mathematical techniques for application in the physical sciences.

Texts

Recommended Texts:

The following book is recommended, but it is not essential.

Jordan, D.W. & Smith, P. Mathematical Techniques: An introduction for the engineering, physical, and mathematical sciences (4th edition), Oxford University Press, Oxford, 2008.

Supplementary Booklist

These are alternative texts. They should be available in the library, and you may find them useful in different ways, as discussed below.

  1. Stewart, J., Calculus - Early Transcendentals, Brooks/Cole
    A very clearly written and comprehensive introduction to calculus, going beyond the Maths 1AM course. Includes vectors but not matrices. Recommended - if you can afford it. There are many similar textbooks in the library.
  2. Gilbert, J. and Jordan, C., Guide to Mathematical Methods, Palgrave (Macmillan) 2002.
    Introduces topics in a fairly elementary way, but does not cover all the material.
  3. Berry, J., Northcliffe, A., & Humble, S., Introductory mathematics through science applications, Cambridge University Press, Cambridge.
    Introduces topics in a fairly elementary way. May be useful if you feel you need to strengthen your basic skills.
  4. Boas, M.L., Mathematical methods in the physical sciences, Wiley,
    Useful for the second-year physics course: you may find it too demanding at the beginning of the 1AM course.
  5. Jeffrey, A., Mathematics for engineers and scientists, Chapman & Hall, London
    Covers most of the syllabus, and a good deal more besides, in a terse style..
  6. Jeffrey, A., Essentials of engineering mathematics, Chapman & Hall, London
    Similar in style to the previous book, though with slightly less extensive coverage.

Syllabus

The numbers of lectures (shown in brackets) are a rough guide only.

  1. General introduction, Review of algebra and trigonometry. (2)
  2. Functions and graphs: important examples, inverse functions. (2)
  3. Sequences and series; limits of functions; continuous functions (3)
  4. Exponential function; natural logarithm; hyperbolic functions (2)
  5. Complex numbers; Argand diagram, polar form, complex exponential, complex roots (4)
  6. Differential calculus, differentiability, basic methods, higher derivatives, Leibniz formula; differentiation of inverse functions (3)
  7. Taylor approximations; Taylor series; convergence of the series; ratio test for power series; applications of Taylor series: maxima and minima; l'Hospital's rule for limits (4)
  8. Integration: integrals as antiderivatives and as area; standard techniques; infinite integrands; infinite ranges of integration. (4)
  9. Differential equations: 1st-order separable and first order linear differential equations. (2)
  10. 2nd order linear differential equations with constant coefficients, homogenous including simple harmonic motion, inhomogeneous including resonance. (4)
  11. Full-range Fourier series in [-pi, pi] and general intervals. (4)

Advice for Students

This is a 20 credit-point unit given in the first half of the year, which means that you should spend a third of your working time on it during the first half-year. The university expects students to work roughly 40 hours a week, which means 13 or 14 hours per week spent on mathematics in the first 12 weeks. Some students may need to spend more time than this in order to master the subject, some may need less.

Remember that you will not be allowed to remain in the University unless you work well enough to obtain 20 credit points from this course; see the section Formal Requirements of the Unit for more details.

Studying mathematics is different from many other subjects. Mathematicians use words in a way different from everyday life: mathematical terms have very precisely defined meanings, which may sometimes take a bit of work to understand fully. You must read your texts and lecture notes very carefully, thinking about the meaning of every word and every symbol until you have it all clear in your mind. If you just skim instead of reading carefully, the subject will soon become a vague blur in your mind, and you will not make much progress.

Another feature of mathematics is the way that each topic builds on knowledge of the previous material. If you don't have a really firm grasp of earlier material, you will not be able to grasp later parts of the course.

It is essential to practice doing mathematics, in order to build speed and confidence in mathematical techniques; that is an important part of this unit. Think of it as like training for a sport: if you don't do training and practice, you can't expect to perform well. We provide you with exercises to do; you should work them carefully. If you get stuck with a question, it is helpful to go back to your lecture notes and textbook: make sure that you understand the mathematical ideas behind the question, and then look for worked examples that might help.

We encourage you to discuss the exercises with other students - working in small groups is fun, and is a good way of learning. But the work that you hand in should be your own write-up, even if some of the ideas were generated in group discussion.

Tutorials and, to a lesser extent, problems classes give you a chance to discuss detailed difficulties you have in working the exercises or understanding the material from the lectures. You can also discuss more general issues of how to approach studying mathematics: the kind of issue discussed in the previous few paragraphs. Tutorials are an essential part of our teaching - make the most of them, by bringing up points for discussion which will help you in mastering the course.

Calculators

Don't feel that you have to buy a graphics calculator - they are not essential, and graphic calculators may not be used in the examination. An ordinary scientific calculator (cost £6 upwards) is sufficient, but you should check that it satisfies our requirements on calculators used in examinations.