The University of Southampton
Courses

# GENG0002 Mathematics B

## Module Overview

This module offers an introduction to the differential and integral calculus that underpins engineering mathematics.

### Aims and Objectives

#### Module Aims

• To present the mathematical methods of differential and integral calculus and to provide and introduction to differential equations and to vectors • To enable you to build a working toolbox of mathematical techniques for differentiating and integrating functions, for solving differential equations and for working with vector quantities. • To emphasise the meaning and purpose of these techniques and their use in solving Engineering and Physics problems

#### Learning Outcomes

##### Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• The mathematical methods of differential and integral calculus and of some simple solution methods for various types of differential equations
• Vector operations
##### Transferable and Generic Skills

Having successfully completed this module you will be able to:

• Apply problem solving techniques to familiar and unfamiliar problems
##### Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• Select and apply appropriate mathematical methods to solve abstract and real-world problems
• Show confidence in manipulating mathematical expressions, setting up and solving equations and constructing simple proofs

### Syllabus

Differentiation • understand the gradient of a curve at a point as the limit of the gradients of a sequence of chords • use the derivative of xn, lnx, ex, sinx, cosx, tanx and constant multiples, sums/differences of these • find gradient of a curve at a point • find equation of tangent/normal to a curve at a point • use the product and quotient rules • use the chain rule to differentiate functions of the form f(g(x)) • understand that a derivative gives a rate of change • find the second derivative of a function • understand and be able to use the relationship dydxdxdy1= • find the first derivative of a function which is defined parametrically • find the first derivative of a function which is defined implicitly • use logarithmic differentiation • locate stationary points and distinguish between maxima and minima (by any method) Integration • understand integration as the reverse of differentiation; integrate xn (including n =-1) ex sinx cosx sec2x together with sums/differences and constant multiples of these • recognise integrands of the form kf ‘(x)/f(x) and kf(x)f ‘(x) and integrate • integrate expressions requiring linear substitution e.g. sin(ax+b) and double angle formulae e.g. cos2x • use integration to find a region bounded by a curve and two ordinates or by two curves • recognise where an integrand may be regarded as a product and use integration by parts to integrate • integrate rational functions using partial fractions • integrate simple functions in parametric form • use the mid-ordinate rule, the trapezium rule and Simpson’s rule to obtain approximate values for definite integrals • apply integration to find volumes of revolution about the x-axis, centroids of uniform laminae, mean and rms values of functions Differential Equations • formulate a simple statement involving a rate of change as a differential equation • find by integration a general form of a solution for a differential equation in which the variable are separable • use initial condition(s) to find a particular integral Vectors • know difference between a scalar and a vector • use vector notation to locate points in 3 dimensions • add, subtract and multiply by a scalar • use unit vectors, position vectors and displacement vectors • find modulus of a vector • calculate scalar product and use to find angle between two vectors and to show that vectors are parallel or perpendicular • calculate vector product, interpret as area of a parallelogram and as a vector perpendicular to two others

### Learning and Teaching

#### Teaching and learning methods

Learning activities include • Individual work on examples, supported by tutorial/workshop sessions/extra support sessions. • Elements of the coursework module GENG0015 may support your learning in this module. Teaching methods include • Lectures, supported by example sheets. • Tutorials/Workshops/support sessions. • Printed notes will be available through Blackboard and/or through your module lecturer

TypeHours
Tutorial36
Revision6
Preparation for scheduled sessions35
Follow-up work35
Lecture36
Total study time150

Stroud, Palgrave (2001). Engineering Mathematics.

Thompson, Macmillan (1999). Calculus made easy.

Understanding Pure Mathematics Sadler & Thorning.

Croft & Davison (1998). Mathematics for Engineers.

### Assessment

#### Assessment Strategy

External repeat students will have marks carried forward from the previous year for tests (5%), and therefore exam will contribute 95% of total assessment.

#### Summative

MethodPercentage contribution
Exam  (120 minutes) 95%
Test 1.25%
Test 1.25%
Test 1.25%
Test 1.25%

#### Referral

MethodPercentage contribution
Exam 95%
Test marks carried forward 5%

#### Repeat Information

Repeat type: Internal & External

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.