EMT4801/101/0/2016
Tutorial letter 101/0/2016 Engineering Mathematics IV
EMT4801 Year module Department of Mathematical Mathematical Sciences
IMPORTANT INFORMATION: Please activate your myUnisa and my Life Life email addresses and ensure you have regular access to the my Unisa Unisa module site EMT4801-2016.
Note: This is an online module, and therefore your module is available on myUnisa. However,in order to support you in i n your learning process, you will also receive some study materials in printed format.
Contents 1. INTRODUCT INTRODUCTION ION AND WELCOME . . . . . . . . . . . . . . . . . . . . . . .
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1.1 Tutorial matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2. PURPOSE OF AND OUTCOMES OUTCOMES FOR THE MODULE
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2.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.2 Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3. LECTURER LECTURER AND CONTACT CONTACT DETAILS DETAILS
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3.1 Lecturers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.2 Department . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.3 University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.1 Prescribed bed and Recommended boo books . . . . . . . . . . . . . . . . . . . . . . .
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4.2 Recommended books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.3 Electronic reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6. MODULE MODULE SPECIFIC STUDY STUDY PLAN . . . . . . . . . . . . . . . . . . . . . . .
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4. MODULE MODULE RELATED RELATED RESOURCES RESOURCES
5. STUDENT STUDENT SUPPORT SUPPORT SERVICES SERVICES FOR THE MODULE MODULE
7. MODULE PRACTICAL WORK AND WORK WORK INTEGRATED INTEGRATED LEARNING 11 8. ASSESSMENT ASSESSMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8.1 Assessment Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8.2 General Assignment number bers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8.2.1 Unique assignment number bers . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8.2.2 Due dates of assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8.3 Submission of assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8.4 Assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9. EXAMINATIONS EXAMINATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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10. OTHER ASSESSMENT ASSESSMENT METHODS . . . . . . . . . . . . . . . . . . . . . . .
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11. FREQUENTL FREQUENTLY Y ASKED QUESTIONS QUESTIONS . . . . . . . . . . . . . . . . . . . . . .
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Open Rubric
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EMT4801/101/0/2016
1. INTRODUCTI INTRODUCTION ON AND WELCOME Welcome to the subject ENGINEERING MATHEMATICS IV (Theory) (EMT4801) at UNISA. This tutorial letter serves serves as a guideline guideline to this course. course. It provides provides you with general general administrative administrative information information as well well as specific information information about the subject. Read it carefully and keep it safe for future reference. We trust that you will enjoy this course. 1.1 Tutorial matter Some of this tutorial matter may not be availabl availablee when you register. register. Tutorial matter that is not available when you register will be posted to you as soon a possible, but is also available on myUnisa. 2. PURPOSE PURPOSE OF AND OUTCOMES OUTCOMES FOR FOR THE MODULE MODULE 2.1 Purpose This module is intended for students in final year of their electrical engineering qualifications.
2.2 Outcomes Specific outcome 1: Understand series and sequences
Assessment criteria 1. Understand Understand what is meant meant by the phrase convergenc convergencee of a sequence; sequence; 2. Know how to compute the limits of some basic convergen convergentt sequences (including (including ones that are defined by some given recursive formula); 3. Know what is meant by the terms arithmetic progression, progression, harmonic progression progression and geometric progression, and be able to identify such sequences.
4 Specific outcome 2:
Understand complex numbers in preparation for the work on complex analysis.
Assessment criteria
At the end of this unit the student should be familiar with the concept of a complex number, know how to perform basic operations with complex numbers, and know and be able to manipulate the definitions of Arg (z ), ez , and ln(z ).
Specific outcome 3:
Understand Understand Laplace transforms, transforms, and to extend extend these to possibly possibly complex complex variables. ariables. Also to introduce the initial value and final value theorems, to decribe the Laplace Transform of periodic functions, and to introduce the convolution theorem.
Assessment criteria 1. Familiarity amiliarity with the basic definitions definitions and properties properties of the Laplace Transform Transforms. s. 2. Understand Understand how this theory can be b e extended to complex variables. variables. 3. Understand and be able to apply the initial value and final value theorems (the student should in particular also be able to determine when these theorems are applicable and when not). 4. Be able to use tables of Laplace transforms transforms to compute both Laplace transforms transforms and inverse transforms. 5. Understand Understand the convolution convolution theorem theorem and be able to apply it in computing inverse inverse transforms.
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EMT4801/101/0/2016
Specific outcome 4:
Understand discrete time systems, the Z-transform, and its properties.
Assessment criteria 1. Familiarity amiliarity with the basic definitions definitions and properties properties of the Z-transform. Z-transform. 2. Understand Understand and be able to apply the initial value value theorem, final value value theorem, and convolution theorem. 3. LECTURER LECTURER AND CONTACT CONTACT DETAILS DETAILS
Always use your student number when you contact the university.
3.1 Lecturers You may contact your lecturers by post, e-mail, telephone or on myUnisa. Contact details: Dr. J.M. Manale Manale Corner of Christiaan de Wet Road & Pioneer Avenue Room 6-46 GJ Gerwel Building Department of Mathematical Sciences Universit University y of South Africa Science Campus, Florida 1709, Johannesburg, South Africa
Tel: el: +2 +27 7 11 670 670 9172 9172 / 9147 9147 Fax +27 11 670 9171 9171 E–mail:
[email protected] Online address: https://my.unisa.ac.za https://my.unisa.ac.za
6 and Dr. A.S. A.S. Kubek Kubeka Corner of Christiaan de Wet Road & Pioneer Avenue Room 6-647 GJ Gerwel Building Department of Mathematical Sciences Universit University y of South Africa Science Campus Florida 1709 Johannesburg South Africa
Tel: el: +27 11 670 670 917 9172 2 / 9147 9147 Fax +27 11 670 670 9171 9171 E–mail:
[email protected] Online address: https://my.unisa.ac.za https://my.unisa.ac.za
Functions of your lecturers and tutors
• Enquiries about technical content. • Set assignments. • Mark assignments. • Set examination papers. • Mark examination papers
Do N Do NOT OT submit submit your assignments to your lecturers or tutors.
3.2 Department You may contact the department by post or telephone.
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EMT4801/101/0/2016
Contact details:
Department of Mathematical Sciences Universit University y of South Africa Corner of Christiaan de Wet Road & Pioneer Avenue Room 6-46 GJ Gerwel Building Science Campus, Florida 1709, Johannesburg, South Africa
Tel: el: +2 +277 11 11 670 670 914 91477 Fax +27 11 670 9171 9171 E–mail:
[email protected] Online address: https://my.unisa.ac.za https://my.unisa.ac.za
3.3 University If you need to contact the University about matters not related to the content of this course, please consult the publication my studies @ Unisa that you received with your study material. material. This booklet contains contains information on how to contact contact the University University (e.g. to whom you can write for different queries, important telephone and fax numbers, addresses and details of the times certain facilities are open). Always have your student number at hand when you contact the University. 4. MODULE RELATED RELATED RESOURCES 4.1 Prescribed and Recommended books
The prescribed book for this module is DG Duffy, Duffy, Advanced Advanced Engineering Engineering Mathematics Mathematics with MATLAB MATLAB 3rd edition(or latest), Chapman-Hall/CRC Press, 2010 The library has a limited number of copies of this book.
8 4.2 Recommended books A further reference which students may find helpful, is the following book: KA Stroud (with additions by DJ Booth), Advanced Engineering Mathematics (4th ed), Palgrave Macmillan, 2003
4.3 Electronic reserves There are no electronic reserves for this module. 5. STUDENT STUDENT SUPPORT SUPPORT SERVICES SERVICES FOR THE MODULE For information on the various student support systems and services available at Unisa (e.g. student counseling, tutorial classes, language support), please consult the publication my studies @ Unisa that you received with your study material.
• Contact with fellow students • Study groups: It is advisable to have have contact contact with fellow students. students. One way way to do this is to form study study groups. groups. The address addresses es of studen students ts in your your area may be obtain obtained ed from the following department: Directorate: Directorate: Student Student Administrati Administration on and Registration Registration
•
P O Box 392 UNISA 0003
• myUnisa: If you have access to a computer that is linked to the internet, you can quickly access access resources and information information at the Universit University y. The myUnisa learning learning management system is Unisa’s online campus that will help you to communicate with your lecturers, with other students and with the administrative departments of Unisa – all through the computer and the internet.
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EMT4801/101/0/2016
To go to the myUnisa website, start at the main Unisa website, http://www.unisa.ac.za and then click on the “Login to myUnisa ” link on the right– hand side side of the screen. screen. This This should should take you you to the myUnisa webs website ite.. You can also go there directly by typing in http://my.unisa.ac.za. Please consult the publication my studies @ Unisa which you received with your study material for more information on myUnisa .
DO NOT CONTACT YOUR TUTOR REGARDING THESE QUESTIONS!!!! The Department of Despatch should supply you with the following tutorial matter for this module:
• Tutorial letter 101 – READ THIS FIRST • A study guide for EMT4801 Note: Note: Some Some of this tutorial tutorial matter may not be avail available able when when you you regist register. er. Tutorial utorial matter that is not available when you register will be posted to you as soon as possible. Once your registration is confirmed missing study material may also be downloaded in electronic format from https://my.unisa.ac.za. 6. MODULE SPECIFIC SPECIFIC STUDY STUDY PLAN
Module Module 1: Sequen Sequences ces and and Series Series
The material on sequences sequences and series is not covered covered in the prescribed prescribed bo ok. You should study this section section from the study guide. If you need addtional addtional background background you can refer to the textbook prescribed/recommended for MAT1581 and MAT2691, namely KA Stroud (with additions by Dexter J Booth), Engineering Mathematics, edition 5 or 6 or latest. (Edition 5 was published by Palgrave publishers in 2001 and edition 6 by Industrial press in 2007.)
10 This book has chapters chapters on both series and power power series which cover cover almost everythin everything, g, except the algorithms algorithms for estimating estimating the accuracy accuracy of a partial sum approximation, approximation, presented presented in section 1.2.5 of the study guide.
Module Module 2: Complex Complex Analy Analysis sis
This material may either be studied from the study guide or chapter 1 of the prescribed textbook. In the textbook you you do not need to do example example 1.9.4 and also also don’t need need to do any of the examples in section 1.10 (that is examples 1.10.1–1.10.3).
Module Module 3: Laplace Laplace Tran Transfor sforms: ms: Contin Continuou uouss Signals Signals and System Systemss
This material may either be studied from the study guide, or from chapter 6 of the prescribed scribed textbook. textbook. If you choose choose to study this from the prescribed prescribed textbook textbook you need to take note of the following points:
• You do not need to study sections 6.7 and 6.10 in the textbook. • Although the textbook does deal with transfer functions, it does not cover stability in enough enough detail. detail. It also does not deal with with the state space space approac approach. h. Hence Hence you will need to study section 3.5.2 and unit 3.6 from the study guide together with chapter 6 of the textbook.
Module 4: Z –transforms, –transforms, Discrete Signals and Systems
This material may either be studied from the study guide, or from chapter 7 of the prescribed scribed book. When When studyi studying ng this material material from the textbook you need to take note of the following following points: points: –transform in section section 7.3. • You do not need to know all the methods for inverting a Z –transform It is enough to study only the method based on partial fraction expansion.
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EMT4801/101/0/2016
textbook does not cover cover the state space space model. Hence Hence you will need need to study study • The textbook unit 4.5 from the study guide, together with chapter 7 of the textbook. 7. MODULE MODULE PRACTI PRACTICAL CAL WORK AND WORK WORK INTEGRA INTEGRATED TED LEARNING There are no practicals for this module. 8. ASSESSMENT ASSESSMENT Marks Marks will be allocated allocated for assignme assignment nts. s. The avera average ge of these these marks marks will form 20% of the final mark with 80% contributed by the final examination.
8.1 Assessment Plan
There are three three assignments assignments for this course, course, which which appear further on under section. section. The study material on which each one is based, is summarised in the assignment submission programme which appears further on this section.
8.2 General Assignment numbers
Assignments are numbered as 01, 02 and 03.
8.2.1 Unique assignment numbers
In addition to the general number each assignment has its own unique number which must be written on the assignment.
ASSI ASSIGN GNMEN MENTS TS
Uniqu Unique e number umberss
01
637879
02
834979
03
826478
12 8.2.2 Due dates of assignment THE CUT–OFF SUBMISSION DATES FOR THE ASSIGNMENTS Assignment 01
06 May 2016
Assignment 02
24 June 2016
Assignment 03
26 August 2016
8.3 Submission of assignments
Submit Submit at least one assign assignmen mentt before 06 May 2016.
There There is no way way around this this re-
quirem quiremen ent, t, which which is a conseq consequen uence ce of gov govern ernmen mentt regula regulation tions. s.
Assign Assignmen ments ts should should be
addressed addressed to:
The Registrar P O Box 392 UNISA 0003
You may submit your assignments either by post or electronicaly via myUnisa . Assi Assign gn-ments may not may not be submitted submitted by fax or e–mail. e–mail. For detailed information information and requiremen requirements ts as far as assignments are concerned, see the brochure my studies @ Unisa that that you received with your study material. To submit an assignment via myUnisa
• Go to myUnisa . • Log in with your student number and password.’ • Select the course. • Click on assignments in the left–hand menu. • Click on the assignment number you want to submit.
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EMT4801/101/0/2016
• Follow the instructions on the screen. IT IS VERY IMPORTANT TO CONSIDER THE FOLLOWING POINT :
FOR SUBSIDY SUBSIDY PURPOSES PURPOSES THE DEPAR DEPARTMENT TMENT OF EDUCATION EDUCATION • FOR REQUIRES STUDENTS TO GIVE INDICATION THAT THEY ARE ACTIVELY STUDYING THE MODULES THEY ARE REGISTERED FOR. FOR THIS REASON STUDENTS MUST BE SURE TO SUBMIT AT LEAST ONE ASSIGNMENT BEFORE OR ON 06 MAY 2016. AS FAR AS THIS THIS REQUI REQUIREM REMENT ENT IS CONC CONCERN ERNED, ED, NO EXTENEXTENSION CAN BE GIVEN.
Submission of assignments
You can either submit assignments by regular mail to the “Assignments Section”, or by dropping it off in one of the UNISA postboxes at the learning centres, or electronically via the internet. internet. When you submit submit take note of the following following points: points:
• ALLOW ENOUGH TIME FOR THE ASSIGNMENT TO REACH UNISA BEFORE THE CUT–OFF DATE.
• KEEP A CLEAR COPY OF THE ASSIGNMENT FOR YOUR OWN REFERENCE. THIS IS IMPORTANT, AS ASSIGNMENTS DO GET LOST. IF YOU YOU SUBM SUBMIT IT BY REGULA REGULAR R MAIL MAIL,, YOU YOU ARE ARE ADVI ADVISED SED TO • IF REGISTER THE MAIL.
• WHEN SUBMITTING VIA myUnisa IT IS ADVISABLE TO SUBMIT YOUR ASSIGNMENT IN PDF FORMAT.
To submit an assignment via myUnisa:
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• Go to myUnisa. • Log in with your student number and password. • Select the module from the orange bar. • Click on assignments in the left menu. • Click on the assignment number you want to submit. • Follow the instructions.
Feedback on assignments
Each assignment that is submitted will be returned together with a copy of the solutions of that assignment. If no assignment is submitted solutions will not be sent sent out automatically. automatically. Once exam admissions are finalised, solutions to the assignments will be made available on myUnisa.
Exam admission and the year mark
If you do gain entrance to the exam, your final mark for the module will be calculated from a year mark and the exam mark according to the following formula:
• The year mark contributes to 20%. • The examination mark contributes to 80%. The year mark is in turn calculated from the scores obtained for the assignments with each assignmen assignmentt contributing contributing to the year mark. Their contributio contribution n towards the year mark are as shown in the table below:
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ASSIGNMENT ENT
CONT ONTRIBUTION
NUMB NUMBER ER
TOW TOWARDS ARDS YEAR YEAR MARK MARK
01
25%
02
35%
03
40%
TOTAL
=100%
16 8.4 Assignments Assignment 01 Unique Unique numbe number: r: 637879 637879 Recommended closing date: 08 May 2016 Based among others, on Units 1, 2, 3, 4 of Module 1
Question 1 In each case find the limit as n → ∞ of the given sequence. 5
1.1 an =
(3n+11) 2 (n+5)3
1.2 a = n e n
1 n
(4)
−1
1.3 {xn } where x1 = 1 and x1 =
(4) 2x +3 4 n
for all n ≥ 1.
(You may assume { xn } converges.)
(3 ) [11]
Question 2 Give an example of an unbounded sequence which has a convergent subsequence. Justify your claims.
[3]
Question 3 3.1 Given Given ∞
n=1
n
(n + 1)(2n + 7)
,
apply the following convergence tests and say if the tests proves convergence, divergence, or does not confirm either. 3.1.1 D’Alemb ert’s ratio test.
(4 )
3.1.2 The integral test.
(5 )
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3.2 Next consider consider ∞
n=1
1 . n2 ln n
Apply the following convergence tests and say if the test proves convergence, divergence, or does not confirm either. 3.2.1 D’Alembert’s ratio test.
(4)
3.2.2 The comparison test.
(4) [17]
Question 4 Show that the sequence { xn } converges if and only if the series ∞
(x
n+1
− xn )
n=1
converges. converges. What will the sum be if the series conver converges? ges? ( Hint: Hint: Try to simplify its partial sums).
[4]
Question 5 Estimate the maximum error for the following series if the first six terms of the series are used to approximate the sum. ∞
4n − 3 n=1
5n
[5] Question 6 Determine the interval of convergence for the following power series. ∞
n=1
n n2
+1
(3 x − 1)n . [10] TOTAL: [50]
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Assignment 02 Unique Unique numbe number: r: 834979 834979 Recommended closing date: 26 June 2016 Based among others, on Units 1, 2, 3, 4, 5, 6 of Module 2 Question 1 1.1 Consider Consider the transformation transformation w = z 2 − 2z,
where z = x + iy and w = u + iv . Then determine u and v in terms of x and y . (6) 1.2 Determine Determine the image in the W -plane of the circle x2 + y 2 = 1.
(5) [11] Question 2 Show that the function u (x, y) = x 3 − 3xy 2 + y
is harmonic, and determine its harmonic conjugate v (x, y).
[8]
Question 3 Compute the Laurent expansions of the function f (z ) =
z2
z + 2 , + 5z + 6
valid for the region | z + 1 | < 1. That is, in powers of ( z + 1). Give at least three terms. [Hint: First use partial fractions to write the given function as a sum of simpler terms. Then expand these individually, where necessary]
[6]
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Question 4 Show how the real integral 2π
0
1 − cos θ dθ 5 − 2cos θ
may be converted to a contour integral for a suitable path C , then use the residue theorem to compute it. [12] Question 5 Show how the real integral ∞
−∞
x + 1 dx (x2 + 2)(x2 + 2x + 2
may be converted to a contour integral for a suitable path C , then compute it. Make sure you describe or sketch the path used. [14] TOTAL: [50]
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Assignment 03 Unique number: number: 826478 Recommended closing date: 28 August 2016 Based on all Units in Module 3 & 4 Question 1 Use Laplace transforms to solve the following system of differential equations;
y − y + 2 x = e −t , 3y − 2y + x = 0, where y(0) (0) = 1 and and x(0) = 0.
[Hint First Hint First solve for y . Then determine x using the above equations.] [15] Question 2 A system is characterised by the following equation:
x˙ = −70
0 x + 1 400.
1
x˙ 2
1
10 −40
0
x2
and the initial conditions are x1 (0) (0) = 0 and x2 (0) (0) = 0. Take Laplace Laplace transforms of the state equation and
solve.
[15]
Question 3 Solve the following state–space equation by taking a Z –transform –transform and using an inverse matrix, given that x1 (0) = x 2 (0) (0) = 0 and uk = 2.
x (k + 1)
=
yk
=
1
x2 (k + 1)
0 1 x (k) + 1 u x (k) 0 − − x (k ) [1 − 2] 1
1 8
3 4
k
2
1
x2 (k )
[20]
TOTAL: [50]
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9. EXAMINATIONS EXAMINATIONS EXAMINATION INFORMATION FOR ENGINEERING (MATHEMATICS IV (THEORY) EMT4801) Type of examination
Closed bo ok
Examination duration
3 hours
Examination language
English
Non–programmable calculators are allowed
Yes
Take note that the Engineering Applications in the study guide are primarly there for illustrativ illustrativee purposes. Hence in the exam the focus will be more on the actual mathematics mathematics underlying underlying the applications applications,, rather than the applications applications themselves. themselves. In addition some of the material in the study guide is there for the purpose of revision. Where material is included for the sake of revision, the focus will be more on that part of the work which which is truly new and not so much on the work being revised. revised. This means means that You u will will not be direct directly ly tested tested on matrix algebra algebra.. You merely merely need to be able able to • Yo use it to deal with state space equations.
• In the module on Laplace transforms the focus will be more on the material not dealt with in MAT301W, namely – the behaviour of F ( ( p) as p → ∞; – the initial–value theorem; – the final–value theorem; – solution of simultaneous of simultaneous linear linear de’s; – step and impulse functions, and the application of step functions to Laplace transforms of periodic functions; – all material relating to transfer functions – all material relating to convolution
22 – all material relating to the state space approach. In preparing preparing for the exam it is insufficien insufficientt to only do the assignment assignment questions. questions. Some of the exercises in the study guide and the textbook need to be attempted as well. Do as many of these as you need to be able to master the underlying techniques. Mathematics is a time consuming (but most enjoyable) subject once you have mastered it. You should should do so many many proble problems, ms, that once you’ve you’ve read the question question,, you you should should immediately immediately recognise the solution solution method. The Oct/Nov 2006 is included to assist you in your preparation.
Examination Paper INSTRUCTIONS:
• Pocket calculators may be used • Answer all the questions
QUESTION 1 Consider the series ∞
2n + 6 .
n=1
(n + 2)3
For each of the following convergence tests state with justification whether the test proves convergence, divergence, or does not confirm either: 1.1 Ratio test.
(3 )
1.2 Comparison test.
(4 ) [7]
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QUESTION 2 2.1 Estimate Estimate the maximum maximum error if the first four terms of the series ∞
5n − 4
n=1
4n
is used to estimate the sum.
(6)
2.2 Determine the interval of convergence of the pow power series ∞
(n + 2)2) (x + 2)
n=1
(12)
n
2n (3n + 2)
[18]
QUESTION 3 3.1 Consider Consider the transformation transformation w =
2 z + 2
from the z -plane onto the w-plane, where z = x + iy and w = u + iv. 3.1.1 Determine the equation of the image of the line y = x + 1 under this transformation.
(6)
3.1.2 Now compute the image of the points A (−1, 0) ; B (0, 1) ; C (1 (1, 2)
on the line in question 3.1.1.
(5)
3.2 Consider Consider u (x, y) = (x + 1)2 − y 2 . 3.2.1 Show that u is harmonic.
(4)
3.2.2 Determine Determine the harmonic conjugate conjugate v of u.
(5)
24 3.3 3.3.1 Show how how the real integral integral x2
∞
−∞
(x2 + 1) 1) (x2 + 4)
dx
may ma y be b e convert converted ed to a contour contour integr integral al for a suitab suitable le path. Mak Makee sure sure you describe or sketch the path used.
(7 )
3.3.2 Now determine determine ∞
−∞
x2
(x2 + 1) 1) (x2 + 4)
dx
using the contour integral described in (3.3.1).
(8 ) [36]
QUESTION 4 Suppose we are given a system with input u (t) and output x (t) described described by the equation x + 4x + 7x = 5u − 3u.
Assume also that the system is initially at rest (i.e. x (0) = x (0) = 0 = u (0)). 4.1 Write down the transfer function G ( p) of the system.
(3 )
4.2 Now write down a state-space model for the system (yielding the same transfer function).
(3 )
4.3 Use the initia initiall and final value theorems theorems to determ determine ine g (0+ ) and and lim lim g (t) where t→∞
g (t) = L −1 (G ( p)) .
(6) [12]
QUESTION 5 Use the method of convolution to find
25 1
−
L
p2
( p2 +1)2
EMT4801/101/0/2016
.
[7]
QUESTION 6 Suppose we have a system described by the difference equation with input { uk } 3yk+2 + 4 yk+1 + yk = u k+1 − uk which is initially in a quiescent state ( y0 = y 1 = 0 = u 0 ) . Write down the transfer function of the system and say if the system is stable or not. [5]
QUESTION 7 7.1 Solve Solve the follow following ing statestate-spa space ce equati equations ons by taking taking a Z -trans -transform form and using using an inverse matrix, given that 0 = x1 (0) = x 2 (0) (0) and and uk = {1, 0, 0, . . .} .
x (k + 1)
=
yk
=
1
x2 (k + 2)
1 0 x (k) + 1 u x (k ) 0 −2 3 x (k) 12 2 −3 1
k
2
1
(0.1)
x2 (k )
7.2 Determine Determine the values values y0 ; y1 ; y2 .
(3) [15] TOTAL: [100]
10. OTHER ASSESSMENT ASSESSMENT METHODS METHODS The are no other assessment methods in this module. 11. FREQUENTL FREQUENTLY Y ASKED QUESTIONS QUESTIONS
26 None. c UNISA 2016
