Mathematics 1

LEARNER

GUIDE

2016 FACULTY OF APPLIED SCIENCES DEPARTMENT OF MATHEMATICS QUALIFICATIONS: ND: Electrical Engineering - NDELC2/NDEIN1/NDECM1 ND: Electrical Power Engineering - NDEPE2 ND: Mechanical Engineering - NDMCH2/NDMCT1 ND: Industrial Engineering - NDIND2 ND: Computer Systems - NDCSY2 ND: Civil Engineering - NDCVL2 ND: Surveying - NDSUR2 ND: Analytical Chemistry - NDACH2 ND: Chemical Engineering - NDCME2 ND: Pulp & Paper – NDPPT1/NDPPT2

SUBJECT: MATHEMATICS I SUBJECT CODE: MATH101 SAQA CREDITS: 12 Date Revised: JULY 2016 Revised by: N ALLY 1

MATHEMATICS I Welcome to Mathematics I. Successful Engineering and Science students at DUT should have a competent grasp of two languages, that is, English and Mathematics. In this subject students will be exposed to the universal “language” of mathematics and will be provided with a “tool kit” of mathematical techniques which can be applied to problems they will encounter in their Engineering and Science subjects. It is important to realize that Mathematics is a life-long learning experience. From birth we come across the concept of size, shape and form. Initially numbers are learned by observation and during formal schooling these concepts are formalized into structures - arithmetic, algebra, geometry, calculus etc. At tertiary level we continue to build on prior knowledge, introducing and using additional techniques that are useful in solving real life engineering and science problems. In this course the emphasis will be on problem solving and lectures will be based on the assumption that all students have a copy of (or access to) the prescribed textbook and that all assigned reading and tutorial work has been completed before the next lecture takes place.

Very Important: Please note that all information and requirements relating to this subject are detailed in this study guide and must be read carefully.

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CONTENTS: Departmental location Contact details Pre-requisites Recognition of Prior Learning (exemptions) General outcomes of this subject Course meeting times: Learning and teaching strategy Assessment Scanning of scripts Absenteeism Scheme of work Study tips Test and examination protocol Declaration forms DEPARTMENT LOCATION The Department of Mathematics is located on the Steve Biko Campus: Block S3 - level 1. Mathematics Notice Boards are located immediately outside this location and all information pertaining to tests etc will be found on these. CONTACT DETAILS Head of Department Department Secretary Department Phone Department Fax Lecturers

Prof D B Lortan Mrs D Day 031-3732075 031-3732723 Different Mathematics lecturers are assigned to different diploma groups. If you need to consult your lecturer outside of formal contact time, you will find at the departmental reception counter a list of lecturer’s office phone numbers and an internal telephone on which you can phone your lecturer. Otherwise it is advisable to make appointments after any lecture session.

Please try to ensure you ask your lecturer any concerns you may have and do not wait for the day before the test. E-mail address

Each lecturer has their own email address and you can request this from your lecturer or obtain them from the website addresses: www.dut.ac.za

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PRE-REQUISITES In order to register for Mathematics I you need to have passed senior certificate mathematics at the level required by your programme department. Your admission to the programme will indicate that you have shown competences in the following outcomes:Assessment Criteria

Specific Outcome Factorize algebraic expressions

Difference of two squares Trinomial factorization Grouping Sum and difference of two cubes

Operate with fractions

Numeric and algebraic fractions, added and /or subtracted, multiplied and/or divided . Fractional equations solved Powers with the same base can be multiplied/divided/ roots found & raised to powers . Equations solved using factorization, formula and elimination of variables

Manipulate powers

Solve simultaneous equations Manipulate formulae

Formulae can be operated on by: substitution and changing the subject Algebraic functions (polynomial, exponential) and trigonometric functions can be defined and used to solve problems. Inverse functions can be found where applicable Algebraic polynomials can be differentiated using the standard form axn.

Define and use algebraic and trigonometric functions

Differentiate simple algebraic functions. Perform co-ordinate geometry operations

Co-ordinate geometry problems including: Distance and gradient between points, finding equation of straight lines, parallel and perpendicular relationships, locus situations can be solved.

** If you have passed Mathematics at an FET institution please follow the procedure outlined on the following page:

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RECOGNITION OF PRIOR LEARNING (EXEMPTIONS) If you have passed Mathematics I or equivalent at another institution of higher learning you must bring your original certificate together with the DUT form requesting exemption/credit (obtained from the Engineering Faculty Office) to the Mathematics Department. You may also be required to bring a syllabus outline. The Head of Department ( or any other designated member of staff) will then decide whether you can be accepted directly into Mathematics II or exempted from doing Mathematics I. GENERAL OUTCOMES OF THIS SUBJECT After completing this subject the student should be competent to use mathematical tools to:identify, analyze, describe and solve discipline-related problems. Some overlapping examples are listed in the table below:EXAMPLE DISCIPLINE Electrical and Electronic Engineering

Utilize Cramer’s determinant rule to solve a given system of equations obtained from a circuit. Solve trigonometric equations and give the amplitude and phase angle for compound sine and cosine functions. Substitute into a mechanical formula and/or change the subject of a formula. Utilize Cramer’s determinant rule to solve a given system of equations obtained from a framework. Differentiate and integrate standard mathematical equations using the rules, techniques and standard forms used in elementary calculus.

Mechanical and Industrial Engineering Civil Engineering and Surveying

Chemical Engineering, Chemistry and Pulp and Paper

COURSE MEETING TIMES: LEARNING AND TEACHING STRATEGY Five contact periods per week are allocated. These periods are used for lectures, tutorial work and classroom assessments. All lectures and tutorials are compulsory. During lecture periods students will be expected to take their own notes. References to textbooks will be made where applicable. Problems/tutorial questions will be given relating to the work covered in the lectures. It is essential that tutorial work is completed by the end of each week as new work is built on what has gone before. Lecturers are available to provide assistance and feedback to students during contact periods and by appointment outside of formal contact time.

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Students are expected to work, on average, four hours per week outside of formal lecture time on Mathematics. Self discipline plays an important part in your success. Therefore it is important to recognize at the start of the course that you are going to have to work hard and consistently in order to pass. Sometimes it will be difficult to deprive yourself of leisure and do Mathematics instead, but remember This course is only one semester, it is not a life sentence so keep up to date and to maximize your chances of success. Additional student support may be provided during some lunch hours and on Saturdays, when available. Your lecturer will provide more details. ASSESSMENT There is no examination at the end of this course. The final mark is calculated from assignments/tests written during the semester as follows:-

TYPE OF ASSESSMENT MAJOR TEST 1

PROVISIONAL DATE

WORK TO BE TESTED

9 SEPTEMBER

% OF FINAL MARK

Programmes: F2, F3, F5, F6, P4, F7, F8, F12 (F10) , {P1&P2}

40

MAJOR TEST 2

21 OCTOBER

Programmes: P3, F10 (F11), P7, P8, F11(F12)

40

CLASS MARK :

Multi-choice 1: 19 AUGUST Multi-choice 2: 7 OCTOBER Lecturer’s tutorial tests: Dates to be advised TO BE CONFIRMED

This will be given to you by your lecturer

20

3 Minor marks NO MAKE UP TESTS FOR MINOR TESTS SPECIAL TEST

WHOLE SYLLABUS

THE ONUS IS ON THE STUDENT TO ENSURE THAT HE/SHE IS AWARE OF THE WORK TO BE TESTED, THE CONFIRMED TEST DATE, TIME AND VENUE. ALL SUCH INFORMATION WILL BE DISPLAYED ON THE MATHS NOTICE BOARDS SITUATED AT S3 LEVEL 1.

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FINAL MARK = (Major Test 1 x 0.4) + (Major Test 2 x 0.4) + (Class Mark x 0.2) The pass mark is 50%. NOTE: 1. If a student obtains a final result from 45% to 49% inclusive for Maths I he/she will be eligible to write a 3 hour SPECIAL test covering the whole syllabus. The date for the special test will be displayed on the Math notice board (S3 Level 1). If a student passes this he/she will be allocated a final result of 50%. However, if the student fails this special test, the original results will stand. If a student who is eligible for the test does not write it for any reason he/she will not be eligible for any further test and will have to re-register for the subject. TEST RESULTS The results of each Major Test will be displayed on the ITS Student Portal. Kindly check the portals for your results two weeks after the test has been written. Please report any errors to your lecturer timeously. SCANNING OF SCRIPTS - APPEALS PROCESS Learners have 1 week after the results of each test have been released to scan their Major Test scripts. Each lecturer will inform their learners during which periods they may scan their scripts, Thereafter the scripts will be collected back and stored in the archives and will not be available for scanning. The purpose of the scanning is to provide the learner with feedback and to give the learner the opportunity to check that all questions they have answered have been marked and that the addition is accurate. In the case of a minor test, the results will be released within 2 weeks of the test being written. In the case of assignments and projects the results will be released on or before 3 weeks prior to the commencement of examinations. ABSENTEEISM If a student misses a Major Test 1 or Major Test 2, satisfactory evidence, eg: DUT Medical Certificate attached at the end of this guide, must be submitted to the Secretary, WITHIN SEVEN CALENDAR DAYS of the test being written. If this is not forthcoming a student will be given 0 % for the test. If a student has produced satisfactory evidence for his/her absence he/she will be required to write an Aegrotat Test. Note: Aegrotat tests apply only to major test 1 and major test 2. There will be no Aegrotat tests for the multi choice tests. Both Major 1 and Major 2 Aegrotat Tests will be written on the same day and at the same time as the Special Test mentioned above. Furthermore, results from the Aegrotat Test will be used to obtain the final mark and no further tests will be conducted thereafter irrespective of the final result. If a student misses the Aegrotat tests Rule G13(3)(b)(iii) will apply. 7

NB: In terms of DUT general rule G13 3(a)(i) a doctor’s Certificate will only be accepted if you are examined on or immediately before the date of the Major test you have missed. (No retrospective Doctor’s certificates will be accepted). POLICIES AND RULES DUT and departmental Policies and Rules will be adhered to. Students are referred to the relevant Programme Handbooks and the DUT website. In the event of assignments being submitted for part of the DP mark the student is reminded of the DUT policy on plagiarism (Refer to Rule G13 (1)(o)).

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SCHEME OF WORK All page and frame references in the table are from the same prescribed book viz:Stroud, K.A., 2007. Engineering Mathematics. 5th(or 6th) ed. Great Britain: Palgrave. Other textbooks, Maths software and a website which can be referenced in the library are:1. Jeffrey, A., 1985. Mathematics for Engineers and Scientists. 3rd ed. Great Britain: Billing & Sons Limited Worcester. 2. Abbott, P., 1985. Teach yourself Calculus. Great Britain: Hodder and Stoughton Ltd. 3. James, Glyn, Modern Engineering Mathematics, 3rd Edition, Essex, England 4. Scientific Workplace 5. Engineering Mathematics Personal Tutor - CD (KA Stroud) 6. www.khanacademy.org

Lectures may also give out their own hand outs and texts for reference. Specific Outcomes Manipulation of algebraic functions

Perform Log manipulations

Assessment Criteria Algebraic functions can be manipulated

Utilize the log rules [log a product, log of a quotient, log of a power, change of base] Natural log and exponential functions can be manipulated

Theory (6th Edition)

Tutorial (6th Edition)

Week

F2: Fr 1 to 78 (Fr 1 to 80)

Pg 116 Fr 80 (Fr 82) Nos 1 a & e, 2 to 8.

1&2

F3: Fr 1 to 61 (Fr 1 to 62)

Pg 146 Fr 63 (Fr 64): Nos 1 c, 2 & 4

F5: Fr 1 to 25

Pg 194 (Pg 202) Fr 27: Nos 6 & 17

F6: Fr 1 to 39

Pg 217 (Pg 225) Fr 41: Nos 1a, 2a, 3a

Evaluate determinants of 2x2 and 3x3 ‘square’ matrices

Simultaneous equations can be manipulated and solved using determinants (specifically Cramer’s rule).

P4: Fr 1 - 6 Class notes on Cramer’s rule. Using Cramer’s rule: Fr 10, 12, 14 & 15 Fr 17 to 27 & 32. Fr 59 to 66

Pg 533 (Pg 552) Fr 69: Nos 2a, 5, 9, 10, 12, 16 & 19

Split an algebraic fraction into partial fractions

Quotients can be broken partial fractions.

F7: Fr 1 - 47

Pg 239 (Pg 247) 4 Fr 49: Nos 1, 2, 9, 10, 12, 13, 16, 17, 20 & 21

into

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3

Perform Trig operations

Trig identities can be simplified. Trig equations can be solved

F8: Fr 4 - 8

5 Pg 266 (Pg 274) Fr 46: Test exercise No 9

Fr 31 - 45 Pg 267 (Pg 275) Fr 47: Nos 3, 4, & 5 Manipulate algebraic functions and inverse functions

Functions and inverse functions can be manipulated.

F12: Fr 1 to 72 & 75 (F10: 1 to 73) Class notes on SinP +/- sinQ = 0 and cosP +/- cosQ =0

Pg 414 (Pg 356) Fr 81(Fr 82) : Nos 1 to 5 & 7 to 12 Pg 415 (Pg 357) Fr 82 (Fr 83): Nos 1,2, 4, 6, 7, 9, 10a & 13 Solve: 1) sinax + sin3x = 0 2) sin4A = sin 3A 3) cos5P = cos 2P

Identify and use complex numbers

Complex numbers can be:- written in rectangular, polar and exponential format. - added, subtracted, multiplied, divided and roots found.

P1: Fr 1 to 53 (Fr 1 to 55) Fr 54 & 55 (Fr 56 & 57) (read only) Fr 56 to 59 (Fr 58 to 61) P2: Fr 1 to 48

Pg 445 (Pg 464) 7&8 Fr 63 (Fr 65): Nos 1, 3, 6, 7, 8, 10, 11, 14, 18, 19 & 20 Pg 472 (Pg 492) Fr 59: Nos 1 to 7 Pg 473 (Pg 493) Fr 60: Nos 1, 3, 5, 6, 8, 11 & 13

Define and use hyperbolic functions

Hyperbolic functions can be defined in terms of exponential functions Hyperbolic functions can be manipulated and simplified

P3: Fr 1 to 38 (Students who did complex nos read Fr 39 to 47)

Pg 498 (Pg 518) Fr 53: Nos 1 to 7 Fr 54: Nos 1, 2, 5, 6, 10 & 16

Differentiate simple algebraic, trig and hyperbolic functions

Concepts (derivative, slope, velocity, rate of change )

F10: Fr 1 to 48 (F11: Fr 1 to 45)

Pg 331 (Pg 399) 10 & Fr 51 (Fr 70): Nos 1 to 11 6 Pg 332 (Pg 400) Fr 52 (Fr 71): Nos 1 to 5

Algebraic, trig and hyperbolic functions can be differentiated using the following:- standard rules - the chain (function of a function) rule - the product rule - the quotient rule - the log technique - implicit functions - parametric functions The differentiation techniques above can be used to find:- the equations of tangents and normals to a curve.

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6

9

P7: Fr 1 to 36 Pg 617 (Pg 637) Fr 38 all problems Fr 39: Nos 1, 2, 3, 5, 6, 7, 9, 13, 19 & 20 P8: Fr 1 to 23

Pg 640 (Pg 660) Fr 46: Nos 1 to 4 Fr 47: Nos 1, 2, 3, 4 & 6

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Integrate simple algebraic, trig and hyperbolic functions

-Introduce as a Riemann sum; Algebraic, trig and hyperbolic functions can be integrated using the following:- standard forms - constant rule - sum and/or difference rule - algebraic substitution - partial fractions Integration can be used to find areas.

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F11 (F12): Fr 1 to 44

Pg 364 (Pg 432) Fr 45: all problems Pg 365 (Pg 433) Fr 46: all problems

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STUDY TIPS 1. Theory: Read the theory pages carefully, making sure you understand the work. Work through the worked examples on your own after going through them in the book and check your answer carefully. Do the revision exercises at the end of the chapter. 2. Tutorials: Do as many of the Test Exercise and Further Problems questions for that section that you can manage. The examples given in the Scheme of Work have been chosen as representative of the chapter, but are the bare minimum of what you should be doing. 3. Time: Use time efficiently. When tackling problems on your own don’t spend more than 15 minutes on any one problem. If by this stage you are not getting anywhere, leave the problem and move on to the next one. The next lecture/tut you have you can ask your lecturer for help. 4. Recap: At the end of each week take a quick look back at the weeks work. If you don’t feel confident about any part of it speak to your lecturer as soon as possible. 5. Co-operative learning: Try to assist your fellow students with problems they are experiencing - you will find by explaining to others you understand the work much better yourself. You will also find that other views sometimes clarifies aspects of the work you may be struggling with. As a registered student for the course Mathematics I you will sign a declaration/contract to remind you that you agreed to the following:

1.

I am in receipt of a Learner Guide for Mathematics I.

2.

I have understood the rules pertaining to tests and assessments.

3.

I agree to abide by the Rules of the University.

4.

I will work from the very beginning and not wait for the day before the test and will consult my lecturer whenever I encounter problems.

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TO BE COMPLETED BY A MEDICAL PRACTITIONER (registered with the South African Medical and Dental Council), Sister or Psychologist from the University’s Student Health Clinic (registered with the South African Medical and Dental Council), Homoeopath or Chiropractor (registered with the Chiropractors, Homoeopaths and Allied Health Service Professions Council of South Africa). It is hereby certified that

Mr



Mrs 

Miss 

Ms



(applicable block)

Full names of student: ............................................................................................................................. of (address) ............................................................................................................................................... who states that he/she is a registered student of Durban University of Technology, was attended by me on

(please give precise date(s))................................................................................................

on which day(s) I found him/her to be suffering from (please print as precise a diagnosis as possible) .................................................................................................................................................................... I,.................................................................................................................................................................. (please print your full name) hereby certify that this illness rendered the student in question unfit to be examined by the University on the following day(s) ..........................................................................to ............................................................(Inclusive). (please state these dates precisely) SIGNATURE: .........................................................

DATE: .................................................

DESIGNATION: ......................................................

TEL NO: ...............................................

ADDRESS: ...................................................................................................................................................... .................................................................................................................................................................... PLEASE ENDORSE WITH YOUR OFFICIAL STAMP

Receipt of medical certificate (For proof of application of special/make-up test) Student Name :_______________________Student Number__________________________ Math 1 Math2 Math 3 (tick appropriate box)

Date received :___________________ Received by: _______________________________ 13

DECLARATION FORM

Fill in this sheet and return to your lecturer immediately I, (Print SURNAME, INITIALS in Block letters) (Print STUDENT NUMBER) (Print QUALIFICATION Registered for) (Print LECTURER’S Name)

DECLARE THAT I AM: 1.

A registered student for the course Mathematics I

2.

In receipt of a Learner Guide for Mathematics I

3.

I have understood the rules pertaining to tests and assessments

4.

Agree to abide by the Rules of the University.

5.

I will work from the very beginning and not wait for the day before the test and will consult my lecturer whenever I encounter problems.

SIGNATURE

DATE

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