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MODULE HANDBOOK Mathematics II TEACHING PLAN (2012-13) Semester 1 Name Tom Lunney Room MG121D Ext 75388 Module Mathematics II (COM420; CRN 46532) Semester 1 Course BEng(Hons) Computing & Electronics/BEng(Hons) Computer Games Lecturer(s): Dr Tom Lunney; Arrangements for Lectures/Tutorials/Practicals: Classes occur on a weekly basis. Venue and times: Lectures: Tuesday 10.15- 12.05 (Room MF131); Tutorials: Tuesday 13.15- 15.05 (Room MB104); Practicals: Tuesday 9.15- 10.05 (Room MF116_PC); Presentation Schedule: Week 1/2 – Differentiation Basic ideas and definitions Differentiation of a function of one variable The product quotient and function of a function rules Rates of change, maximum and minimum Week 3/4/5 – Integration Integration as the inverse of differentiation Definite and indefinite integration Integration by inspection, substitution, partial fractions, by parts Week 6 – Numerical Integration Introduction to Numerical Integration Trapezium Rule Simpson’s Rule Week 7/8 – Differential Equations First order differential equations Solution by direct integration separating the variables use of an integration factor Standard linear equations 1 Week 9 – Complex Numbers Addition, subtraction, multiplication of complex numbers Powers of complex numbers Graphical representation of complex numbers Polar form of a complex number, phase angle and modulus calculations Week 10 – Vectors Introduction to three dimensional vectors Addition, subtraction of vectors Dot (scalar) products of vectors Cross (vector) product of vectors Week 11 – Laplace Transforms Explanation and Definition Simple transforms Inverse transforms Application to the solution of differential equations Week 12 – Revision and Examination Preparation TEACHING AND LEARNING METHODS Lectures will be used to expose the students to new material. Tutorials will be used to help students become familiar with material delivered in the lectures and will take a variety of forms including tutor-led problem-solving sessions, solving assigned problems in groups and singly in order to allow the student to engage with the tutor and other class mates. Additional backup online support material is provided to help with student understanding and provide context for the different topics covered during the module. This material will be introduced during the laboratory sessions. Students will be directed to read relevant text indicated below in order to gain more practice on topics if required. Students will be expected to complete Tutorials and Class Tests as appropriate. The module is web blended and all material relating to the module will be available online. ASSESSMENT METHODS 2 Coursework Activity(Typical) Coursework 1 (75% of coursework marks) Biweekly Multiple-choice Tests are administered during the module to facilitate timely feedback and contribute to the coursework marks. Each test will last approximately one hour. Coursework 2 (25% of coursework marks) This will typically be a two hour open-book written test occurring towards the latter stages of the module, where students will be required to develop full mathematical solutions. This coursework aims to give students a good indication of their overall progress in the module. Examination: A written examination lasting three hours is completed by the student at the end of the semester (during the January examination period) and this contributes 75% of the overall marks for the module (see Appendix A). READING LIST Essential STROUD, K.A., BOOTH, D.J, 2007, Engineering Mathematics, (6th ed.), Palgrave Macmillan Required BIRD, J.O., 2005, Basic Engineering Mathematics, Oxford: Newnes BIRD, J.O., MAY, A.J.C., 1999, Mathematical Formulae, (3rd ed), Pearson Longham. BOOTH, D.J., 1998, Foundation Mathematics, (3rd ed), Addison-Wesley Longman Ltd. CROFT, A., DAVISON, R., 2003, Foundation Mathematics, (3rd ed), Pearson Education Ltd. CROFT, A., DAVISON, R., 2004, Mathematics for Engineers: A Modern Interactive Approach, Pearson Education Ltd. CROFT, A., DAVISON, R., HARGREAVES, M., 2001, Engineering Mathematics: a Foundation for Electronic, Electrical, Communications and Systems Engineers, (3rd ed), Pearson Education Ltd. GREENBERG, M. D., 1998, Advanced Engineering Maths, (2nd ed), Prentice Hall Inc JAMES, G., 2008, Modern Engineering Mathematics, (4th ed), Pearson Education 3 Ltd. JEFFERY, A., 2005, Essentials of Engineering Mathematics: Worked Examples and Problems, (2nd ed), Chapman and Hall JORDAN, D.W., SMITH, P., 2002, Mathematical Techniques: An Introduction for the Engineering, Physical and Mathematical Sciences, (3rd ed), Oxford University Press. THOMPSON, S.P., GARDINER, M., 1998, Calculus made easy, Palgrave TUMA, J., 1998, Engineering Mathematics Handbook, (4th ed), McGraw Hill SUMMARY DESCRIPTION This module introduces students to the essential mathematics required for embarking on further study in engineering, computing or a related discipline. It develops the students mathematical skills required to solve problems that arise in the context of their undergraduate study. The module content is introduced in a pragmatic way and then related to real world problems, which enhances understanding and makes the concepts more meaningful and relevant for the student. The module also aims to generate in the student a spirit of mathematical investigation and discovery leading to the development of mathematical confidence. Also note that the full set of Lecture Notes, Tutorials, Support Material, Module Specification and other associated module material is available via the link (http://www.infm.ulst.ac.uk/~tom/) 4 Appendix A – Sample Examination Paper UNIVERSITY OF ULSTER UNIVERSITY EXAMINATIONS Semester One Module Code: COM420 CRN: Title: Mathematics II Time Allowed: 3 hrs Use of Dictionaries: Dictionaries are not permitted Examination Aids: Candidates may use any programmable graphics calculator Instructions to Candidates: Candidates should read this section carefully before commencement of the examination. There are two sections to this paper. Section A contains ten compulsory questions each worth 4 marks. Section B contains five questions of which the candidates should answer three. Students are advised to write their registration number and desk number only on any attachment, e.g. graph paper, or any other documentation being submitted with their examination script book(s). [insert name(s) of Module Co-ordinator(s)] 5 Section A – Compulsory Section – Answer all questions (Each question is worth 4 marks) 1. Differentiate y 10 cos ecx 2. Using the quotient rule, differentiate y 3. d 2y 3 2 Find the second derivative of y 2x x 10 x 2 dx 4. Find 5. Using the substitution method, find the integral 6. Using integration by parts, find the integral x cos( x )dx 7. Using direct integration, find the general solution of the differential dy 6x 4 3 equation x dx 8. By separating the variables, solve the differential equation 9. Find 3 j 42 j 5 10. sin x 2 x 1 x x 1 dx dx 2x 1 dy 2 x dx y Given the vectors p = 2i + 3j + 5k and q = 4i +j+6k find the scalar product p.q of the two vectors. 6 Section B – Answer any 3 questions (Each question is worth 20 marks). Question 1 a) Differentiate y 4 x 3 sin x using the product rule. (4 marks) x 5 b) Differentiate y sin using the chain rule followed by the quotient rule. 2x 4 (10 marks) c) The volume v of a sphere of radius r is given by v 4 3 r . If the radius of a soap 3 bubble is increasing by 0.1cm per second, find the rate of increase of its volume when the radius is 2cm. (6 marks) Question 2 2 1 a) Find the definite integral cos xdx 2 0 b) After resolving 2x 2x dx into partial fractions, find 2 x x 2 x x 2 2 (6 marks) (14 marks) Question 3 OA a i j 4k Given OB b 8i 2k OC c 5i 2 j 11k Find: a) c-b b) b-a c) the magnitude of the vector OB , i.e. OB (2 marks) (2 marks) d) the magnitude of the vector OC , i.e. OC (4 marks) e) f) the scalar product the scalar product (4 marks) (4 marks) a.c b.c (4 marks) 7 Question 4 a) Determine the Laplace transform of 2e t t , i.e. find L 2e t t b) Determine the Laplace transform of e 3t (t 2 4) , i.e. find L e 3t (t 2 4) (4 marks) c) d) 8 Find the inverse Laplace transform for F (s ) 2 , s 64 8 L1 2 i.e. find s 64 9s 8 L1 2 s 2s Determine (2 marks) (4 marks) (10 marks) Question 5 2 a) 1 cos 2 xdx Evaluate using the Trapezium rule with 8 intervals to 3 decimal 0 places. (10 marks) x 2 b) Evaluate 3 x 2 dx using Simpson's rule with 4 intervals to 3 decimal 1 places. (10 marks) 8 MATHEMATICAL FORMULAE Standard Derivatives: Standard Integrals: y dy dx y ydx f(x) xn axn sin ax a cos ax sin ax cos ax a sinax cos ax tan ax a sec 2 ax tan ax sec ax a sec ax tan ax sec 2 ax cot ax a cos ec 2ax cos ec 2ax cos ecax a cos ecax cot ax sec ax tan ax eax aeax sec ax tan ax ln x 1 x cos ecax cot ax a n 1 x n 1 1 cos ax a 1 sin ax a 1 ln(sec ax ) a 1 tan ax a 1 cot ax a 1 sec ax a 1 sec ax a 1 cos ecax a 1 ax e a sin1 x a cos 1 x a 1 a x 1 2 2 a2 x 2 eax 1 x ln x cos x sin x 1 1 y2 1 a 2 ln sin x tan 1 y x2 1 a 2 x2 1 x 2 a2 arcsin x a arccos x a 1 x arctan a a 9 Laplace Transforms Function f ( t ) 1 a t tn eat te at sinat cos at sinh at cosh at u( t c ) f ( t c ) Transform F( s) 1 s a s 1 s2 n! sn 1 1 sa 1 (s a)2 a 2 s a2 s 2 s a2 a 2 s a2 s 2 s a2 e csF(s) Inverse Laplace Transforms Function F( s) a s 1 sa n! sn 1 1 sn a s a2 Transform f ( t ) a e at tn t n1 (n 1)! sinat 2 10 s 2 s a2 a 2 s a2 s 2 s a2 cos at sinh at coshat Product Rule for Differentiation If y uv where u and v are functions of x, then dy du dv v u dx dx dx Quotient Rule for Differentiation u If y where u and v are functions of x, then v dy dx v du dv u dx dx 2 v Chain Rule for Differentiation If u is a function of x and y is a function of u, then dy dy du . dx du dx ‘Integration by Parts’ Rule for Integration If u and v are functions of x, then dv du u dx dx uv v dx dx Trapezium Rule for Numerical Integration b ydx 2 y 0 2y1 2y 2 ...... 2y n1 y n . h a Simpson’s Rule for Numerical Integration b h ydx 3 y 0 4( y1 y 3 y 5 .......) 2( y 2 y 4 y 6 .......) y n . a 11