*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

##### 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

##### Transferable and Generic Skills

Having successfully completed this module you will be able to:

- Manage your own learning
- Apply problem solving techniques to familiar and unfamiliar problems

### 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

Type | Hours |
---|---|

Preparation for scheduled sessions | 35 |

Lecture | 36 |

Tutorial | 36 |

Follow-up work | 35 |

Completion of assessment task | 2 |

Revision | 6 |

Total study time | 150 |

#### Resources & Reading list

Understanding Pure Mathematics Sadler & Thorning.

Croft & Davison (1998). Mathematics for Engineers.

Stroud, Palgrave (2001). Engineering Mathematics.

Thompson, Macmillan (1999). Calculus made easy.

### 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

Method | Percentage contribution |
---|---|

Examination (2 hours) | 100% |

#### Referral

Method | Percentage contribution |
---|---|

Examination | 100% |

#### Repeat Information

**Repeat type: Internal & External**