# Mathematics 1A20

## Unit aims

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

## Unit description

This unit is designed for students with a good grasp of A level mathematics who want a 20 credit-point unit on mathematical techniques.

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.

## Relation to other units

There are no follow-on units for this course.

## 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.

## Transferable skills

Mathematical techniques for application in the physical sciences.

## Syllabus

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

1. General introduction, Review of algebra and trigonometry. (3)
2. Sequences and series; limits of functions; continuous functions (3)
3. Functions and graphs: important examples, inverse functions. (4)
4. Exponential function; natural logarithm; hyperbolic functions (2)
5. Differential calculus, differentiability, basic methods, higher derivatives, Leibniz formula; differentiation of inverse functions (2)
6. 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)
7. Complex numbers; Argand diagram, polar form, complex exponential, complex roots (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)
Second order linear differential equations with constant coefficients, homogenous including simple harmonic motion, inhomogeneous including resonance. (4)
10. Full-range Fourier series in [-pi, pi]. (4)

Recommended, but 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.

Alternative texts which you may find 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.

Unit code: MATH11004
Level of study: C/4 (Open)
Credit points: 20
Teaching block (weeks): 1 (1-12)
Lecturer: Dr Yves Tourigny

## Pre-requisites

A-level Mathematics, or equivalent.

None

## Methods of teaching

The unit is based on lectures 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 to tutors for marking. Once the course is under way, students will attend weekly tutorials in which homework questions and additional problems will be covered.

Tutorials

Weekly tutorials will be held after the first or second week. You will be given the time of your tutorial. Experience shows that progress in mathematics depends crucially on regular work at examples. For this reason you are REQUIRED to attend the tutorials and to hand in the set work.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

• 100% from a 2 hour 30 minute exam in January*

*There are two parts to the exam; Part A consists of 8 shorter questions, Part B consists of 4 longer questions. ALL questions will be used for assessment. Part A contributes 40% and Part B contributes 60% of the overall mark for the paper.

NOTE: Calculators of an approved type (non-programmable, no text facility) are allowed.

Candidates may bring into the examination room one A4 double-sided sheet of notes.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.