MAT1581 101 2018 - gimmenotes · year of MAT2691: Engineering Mathematics, K. A. Stroud (with...

MAT1581/101/3/2018

Tutorial Letter 101/3/2018

Mathematics 1 (Engineering)

MAT1581

Semesters 1 and 2

Department of Mathematical Sciences This tutorial letter contains important information about your module. The information in this letter was correct on 15 July 2017. Any updates and follow-up tutorial letters will be posted on myUNISA under additional resources. Have you claimed your UNISA login and mylife e-mail? If not go to http://my.unisa.ac.za

BARCODE

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CONTENTS

Page

1 INTRODUCTION ............................................................................................................................ 4

1.1 Study material ................................................................................................................................. 4

2 PURPOSE AND OUTCOMES ........................................................................................................ 4

2.1 Purpose .......................................................................................................................................... 4

2.2 Outcomes ....................................................................................................................................... 4

3 LECTURER(S) AND CONTACT DETAILS .................................................................................... 5

3.1 Lecturer(s) ...................................................................................................................................... 5

3.2 Department ..................................................................................................................................... 5

3.3 University ........................................................................................................................................ 5

4 RESOURCES ................................................................................................................................. 6

4.1 Prescribed books ............................................................................................................................ 6

4.2 Recommended books ..................................................................................................................... 6

4.3 Electronic reserves (e-reserves) ..................................................................................................... 6

4.4 Library services and resources information .................................................................................... 6

5 STUDENT SUPPORT SERVICES ................................................................................................. 7

5.1 Tutor Support .................................................................................................................................. 7

5.2 Discussion classes ......................................................................................................................... 7

5.3 Science Foundation Project or the Extended Pathway ................................................................... 7

5.4 Other services @ Regional offices ................................................................................................. 7

6 STUDY PLAN ................................................................................................................................. 8

7 PRACTICAL WORK AND WORK-INTEGRATED LEARNING ..................................................... 9

8 ASSESSMENT ............................................................................................................................... 9

8.1 Assessment criteria ........................................................................................................................ 9

8.2 Assessment plan .......................................................................................................................... 11

8.3 Assignment numbers .................................................................................................................... 11

8.3.1 General assignment numbers ....................................................................................................... 11

8.3.2 Unique assignment numbers ........................................................................................................ 11

8.4 Assignment due dates .................................................................................................................. 11

8.5 Submission of assignments .......................................................................................................... 12

8.5.1 Written assignments ..................................................................................................................... 12

8.6 The assignments .......................................................................................................................... 13

8.6.1 Assignment 01 Semester 1 .......................................................................................................... 13

MAT1581/101/3/2018

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8.7 Other assessment methods .......................................................................................................... 18

8.8 The examination ........................................................................................................................... 19

9 FREQUENTLY ASKED QUESTIONS .......................................................................................... 19

10 SOURCES CONSULTED ............................................................................................................. 19

11 IN CLOSING ................................................................................................................................. 19

ADDENDUM ............................................................................................................................................. 20

12.1 Formula Sheets ............................................................................................................................ 20

4

Dear Student

1 INTRODUCTION

Welcome as a student to Mathematics I, MAT1581.

Check your registration papers now to confirm for which semester you are registered. Call the lecturer if in doubt or check on myUNISA. If you are registered for semester 1 you will write your final examination in May/June 2018 and qualify for this by doing assignments for semester 1. If you are registered for semester 2 you will write your final examination in October/November 2018 and qualify for this by doing assignments for semester 2.

1.1 Study material

The content of this module is covered in three study guides. The study guides of all the modules you are registered for can be downloaded from myUNISA. Please forward possible mistakes to the lecturer for errata to be posted on myUNISA.

You need to work on your mathematics regularly. See an example of a study plan further on in this letter. The amount of time you study is not important, but how efficient you use that time is very important.

2 PURPOSE AND OUTCOMES

2.1 Purpose

This module will be useful to students in developing basic skills which can be applied in the natural sciences and engineering sciences. Students credited with this module will have understanding of basic ideas of algebra and calculus in handling problems related to: Cramer’s rule to solve systems of linear equations, complex number system, binomial theorem, basic differentiation and integration. The focus is on building strong algebraic skills that will support the development of analytical skills that are crucial in problem solving in more advanced mathematics and related subjects.

This module will support students in their studies in the field of engineering and the physical sciences as part of a diploma.

2.2 Outcomes

1. Formulate and apply the binomial theorem to expand ) na b

2. Solve a system of linear equations using Cramer’s rule

3. Compute calculations with the complex numbers

4. Graph straight line, parabola, hyperbola, central hyperbola,ellipse

5. Apply basic differentiation techniques

6. Apply basic integration techniques

MAT1581/101/3/2018

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3 LECTURER(S) AND CONTACT DETAILS

Please have your student number at hand before contacting any department at UNISA.

If you have access to a computer that is linked to the internet, you can quickly access resources and information at the University. The myUnisa 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 .

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 of the screen. This should take you to the myUnisa website. You can also go there directly by typing in http://my.unisa.ac.za.

You can access myUnisa at your regional office learner centre (without any cost) if you do not have internet at home.

3.1 Lecturer(s)

Your lecturer at the time of compiling this tutorial letter (July 2017) is Miss LE Greyling, Science Campus, Florida, Roodepoort. If you experience any problems with the mathematical content you are welcome to contact the lecturer: 1) by e-mail ( [email protected] ) 2) by sending a message using the Questions and Answers function on myUNISA. 3) by telephone (011-471-2350) If you do not have sufficient funds on your phone you may

ask the lecturer to return your call. 4) by fax ( 086 274 1520) 5) or personally. For a personal visit you must make an appointment by telephone or e-mail.

You must be prepared to come to the Florida Campus in Roodepoort for a personal visit. You should spend time on a problem but do not brood over it, in most cases you only need a hint from the lecturer or tutor to solve the problem or to move forward with your studies. You can save valuable time if you contact your lecturer when needed. If you disagree with a solution in the study guide make contact so that we can work together to correct mistakes. You must mention your student number and the code MAT1581 in all enquiries about this module. Any question without your student number and code MAT1581 will be ignored.

3.2 Department

The department of Mathematical Sciences is situated on the UNISA Science Campus in Florida. The secretary of the department is available on 011-670-9147.

3.3 University

Read the brochure on Study@Unisa2018. Check this brochure on where to direct administrative enquiries. Your student number should be in the subject line of any e-mail to a service department at Unisa.

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Tip: Do not write any administrative requests like the change of contact details or examination venue in your assignment. The markers cannot help you. Likewise any messages for the lecturer must be directed to the lecturer by e-mail or telephone.

4 RESOURCES

4.1 Prescribed books

There are no prescribed books for this module. You need not buy any additional books, the study guides are sufficient.

4.2 Recommended books

You may consult the following book in order to broaden your knowledge of MAT1581 and next year of MAT2691: Engineering Mathematics, K. A. Stroud (with Dexter Booth) 5th ed. Palgrave ISBN: 0-333-91939-4. The sixth and seventh edition of this book is also suitable. Please consult your local library, you might find other interesting books on the topics covered in the lectures. (Dewey system 510-515.)

A limited number of copies are available in the Library and at learning centers.

MAT1581 2018

Recommended Books

Books supplied subject to availability

Engineering mathematics / K.A. Stroud, Dexter J. Booth. 7th ed. Basingstoke : Palgrave Macmillan, c2013. OR Engineering mathematics / K.A. Stroud, with additions by Dexter J. Booth.6th ed. New York : Industrial Press, c2007.

4.3 Electronic reserves (e-reserves)

There are no e-reserves for this module.

4.4 Library services and resources information

For brief information, go to www.unisa.ac.za/brochures/studies For detailed information, go to the Unisa website at http://www.unisa.ac.za/ and click on Library. For research support and services of personal librarians, go to http://www.unisa.ac.za/Default.asp?Cmd=ViewContent&ContentID=7102.

MAT1581/101/3/2018

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The library has compiled a number of library guides:

finding recommended reading in the print collection and e-reserves –http://libguides.unisa.ac.za/request/undergrad

requesting material – http://libguides.unisa.ac.za/request/request postgraduate information services – http://libguides.unisa.ac.za/request/postgrad finding, obtaining and using library resources and tools to assist in doing research –

http://libguides.unisa.ac.za/Research_Skills how to contact the library/finding us on social media/frequently asked questions –

http://libguides.unisa.ac.za/ask

5 STUDENT SUPPORT SERVICES

5.1 Tutor Support

After registration has closed tutorial services will send a sms informing you about your group, the name of your e-tutor and instructions on how to log onto MyUnisa in order to receive further information on the e-tutoring process.

Online tutorials are conducted by qualified E-Tutors who are appointed by Unisa and are offered free of charge. All you need to be able to participate in e-tutoring is a computer with internet connection. If you live close to a Unisa regional Centre or a Telecentre contracted with Unisa, please feel free to visit any of these to access the internet. E-tutoring takes place on MyUnisa where you are expected to connect with other students in your allocated group. It is the role of the e-tutor to guide you through your study material during this interaction process. For your to get the most out of online tutoring, you need to participate in the online discussions that the e-tutor will be facilitating.

There are modules which students have been found to repeatedly fail, these modules are allocated face-to-face tutors and tutorials for these modules take place at the Unisa regional centres. These tutorials are also offered free of charge, however, it is important for you to register at your nearest Unisa Regional Centre to secure attendance of these classes.

Note: E-tutors and classes at regional offices are organized by student support services. The lecturer is not responsible for organizing any tutorial activities.

5.2 Discussion classes

There are no discussion classes by the lecturer for this subject.

5.3 Science Foundation Project or the Extended Pathway

This project is financed by the Department of Education. Students identified as “AT-RISK” to fail will be registered on an extended program in which the same study material will be covered. You cannot register yourself on this program.

5.4 Other services @ Regional offices

For information on the various student support systems and services available at Unisa (e.g. student counselling, language support, academic writing), please consult the publication Study@Unisa2018 that you received with your study material.

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6 STUDY PLAN

To successfully prepare for submitting your assignments you have to work according to a time table. Use the following table or draw up your own table to schedule your studies for this subject. Be realistic. You need to add additional factors like work, family commitments and rest. If you do not schedule your “play time” you will feel guilty and stressed instead of relaxing and building up energy for your studies. Your study material is compiled in such a way that you should be able to complete a unit per day. The post-test should take you another day or a bit longer if you need to revise some of the work. Warning: Do not study selectively. To explain, do not take the assignment and try to find similar questions in the study guides and only work through those questions.

Chapter Learning

unit

CONTENT to study Week Comments

1

Binomial

Theorem

1 Binomial Theorem 1 Revise exponents

Post-test

2

Determinants

1 Properties

2 Value of n x n determinant 2

3 Cramer's Rule

Post-test

3

Partial

Fractions

1 Introduction 3

2 Proper Fractions

3 Improper Fractions

Post-test

4

Complex

Numbers

1 Imaginary and Complex

Numbers

4

5

Revise Grade 11 and

Grade 12 Trigonometry

2 Operations with Complex

numbers

3 Polar and exponential form

4 Operations in polar and

exponential form

Post-test

5

Co-ordinate

Geometry

1 Co-ordinate Systems 6 Together these topics

cover the conic sections.

Some of the content is

revision and some content

will be new. There are no

questions on this section

2 The Straight Line

3 The Parabola

4 The Rectangular Hyperbola

5 The Circle

6 The Ellipse

MAT1581/101/3/2018

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Chapter Learning

unit

CONTENT to study Week Comments

7 The Central Hyperbola in assignment 1. You will

be examined on this

section.

Post-test

6

Differentiation

1 Functional Notation 7

2 Limits

3 The Derivative

4 Standard Forms

5 Rules of Differentiation I

6 Rules of Differentiation II

7 Higher Order Derivatives 8

8 Applications I

9 Applications II: Maxima and

Minima

Post-test

7

Integration

1 Reverse of Differentiation I 9

2 Reverse of Differentiation II

3 Method of substitution

4 Standard Integrals

5 Partial Fractions 10

6 Trigonometric Functions

7 The Definite Integral

8 Areas

Post-test 10

7 PRACTICAL WORK AND WORK-INTEGRATED LEARNING

There are no practical work for this module

8 ASSESSMENT

8.1 Assessment criteria

Specific outcome 1: Formulate and apply the binomial theorem to expand ) na b

The expansion for n a positive integer is written down The expansion for n a negative integer is written down The expansion for n a rational number is written down The rth term of an expansion is written down

Specific outcome 2: Solve a system of linear equations using Cramer’s rule

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The value of a 2x2 and 3x3 determinant is found The properties of determinants are used to evaluate a determinant Cramer’s rule is written down in terms of determinants Cramer’s rule is used to solve a system of linear equations

Specific outcome 3: Calculations with the complex numbers

A number is expressed as a complex number The complex conjugate of a complex number is found A complex number is represented on an Argand diagram Any quadratic equation is solved Complex numbers are added, subtracted, multiplied and divided Equations that include complex numbers are solved Complex numbers are rewritten in rectangular, exponential and polar form A complex number is raised to a power De Moivre’s Theorem is used to raise a complex number to a power and to

find the nth roots of a complex number. Specific outcome 4: Graphs of conic sections.

Conversion between Cartesian and polar coordinates The graph of the straight line, the parabola, the rectangular hyperbola, the

circle, the ellipseand the central hyperbola Specific outcome 5: Apply basic differentiation techniques

A limit is explained and its value found The rate of change of a function is related to the gradient of the tangent at a

point Simple expressions are differentiated from first principles The following rules of differentiation are used to differentiate composite

functions: rule for sums and differences, product and quotient , chain rule Higher order derivatives are found Differentiation is applied to determine Limits of the form 0

0 with l’Hospital’s

rule Differentiation is applied to determine the gradient to a curve and find the

equations of the tangent and normal to a curve Differentiation is used to find the local maxima and minima of a function and

sketch the function. A real life problem is modelled and questions answered related to maximum

and minimum values using differentiation

MAT1581/101/3/2018

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Specific outcome 6: Apply basic integration techniques

Indefinite integrals are solved using a standard list of integrals The techniques of simplifying the integrand, using substitution, using partial

fractions and using trigonometric identities are applied. Definite integrals are solved Integration is applied to find the area under a function Practical problems are solved

8.2 Assessment plan

Your performance in your assignments plays a vital part in your final mark. Submission of an assignment gives you admission to the examination. Your assignment marks for assignment 1 and 2 will be used to calculate your year mark. Your year mark will form part of your final mark for the module.

Assignment % of Year mark Description and Instructions

01 50 Written assignment on study guide 1

02 50 Written assignment on Study guide 2

Your final mark will be calculated as follows: 20% Year mark + 80% Examination mark You need a final mark of 50% in order to pass the subject with a subminimum of 40% on your examination mark. A subminimum of 40% means that if you receive less than 40% in the exam you fail and in this case your year mark does not count. Each module in your study guides contains a post-test with solutions to help you prepare for the compulsory assignments and the final examination.

8.3 Assignment numbers

8.3.1 General assignment numbers

You must submit assignment 01 and 02 for the semester you are registered. Assignment questions per semester are given in 8.6 below.

8.3.2 Unique assignment numbers

All assignments have their own unique number per assignment and per semester given in the table below

8.4 Assignment due dates

Semester1 Unique nr. Semester 2 Unique nr. Assignment 01 Written 26 February 840218 14 August 698643 Assignment 02 Written 4 April 734238 11 September 821200

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8.5 Submission of assignments

Plan your programme so that study problems can be sorted out in time. The dates on myUNISA may differ from dates in your tutorial letter and messages from the university. You may confirm dates with the lecturer if a message seems suspect.

8.5.1 Written assignments

The rules for assignments are: - Please keep a copy of your answers. - Submit answers in numerical order. - Keep to the due dates. - Write clearly with a black pen. - Marks may be deducted or not given if answers are scratched out or difficult to read.

Write your answers down in the correct order and make sure that every answer is numbered clearly. Make sure that your answers are clear and unambiguous. Do not string a series of numbers together without any indication of what you are calculating. Be careful with the use of the equal sign (=). The correct units must be shown in you answer. Note that we are not only interested in whether you can get the correct answer, but also in whether you can formulate your thoughts correctly. Mere calculations are not good enough – you have to make sure that what you have written down consists of mathematically correct notation, which makes sense to the marker. Students must send in their own work. Of course, it is a good thing to discuss problems with fellow students. However, where copying has clearly taken place disciplinary action will be taken. An information sheet containing the formulas is enclosed at the end of this letter for your convenience. Keep this sheet at hand when completing your assignments. The same sheet will be supplied during the examination. Consult this sheet regularly, it may mean the difference between success and failure in this module. Make sure you know how to use the table of integrals in reverse to find derivatives. You need not memorize all formulas and can check memorized formulas. You may submit written assignments either by post or electronically via myUnisa. Choose one way to submit, do not use both. Double submissions waste not only my time but also cause extra work for the assignment department. If the marker cannot access an electronic submission we will send you an e-mail. Assignments may not be submitted by fax, e-mail or registered post. Assignments by post: Ensure that you complete the assignment cover. If the subject or assignment number is incorrect your assignment cannot be noted as received. Each assignment must have a separate cover with the unique number. Submit one assignment per envelope. All regional offices have Unisa post boxes. Only use the SA postal services if you cannot get to a regional office or one of the drop-off boxes listed in Study@Unisa2018. Assignments should be addressed to: The Registrar, PO Box 392, UNISA, 0003

MAT1581/101/3/2018

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Your marked assignment will be posted back to you. To submit via myUnisa: You can scan your handwritten assignment answers to be submitted electronically. Don’t scan the assignment cover as the system will create a cover for you when you upload the assignment. You assignment will be marked electronically. Please make sure that it is easy to read. Your assignment must be combined in one pdf document. Only one document can be uploaded per assignment. Login with your student number and password. Select the module. Click on assignments. Click on the assignment number you want to submit. Follow the instructions on the screen. Your marked assignment will be available on myUNISA for viewing.

8.6 The assignments

8.6.1 Assignment 01 Semester 1

ONLY FOR SEMESTER 1 STUDENTS Assignment 01

Due Date: 26 February Unique number: 840218

This assignment is a written assignment based on study Guide 1. Submission of this assignment by the due date will give you admission to the examination This assignment contributes 50% to your year mark. Source: Paper May 2011, October 2011 and October 2016

1. Find the first three terms in the binomial expansion of 152 1 .x (3)

2. Expand 42 3x x using the binomial theorem. Simplify each term. (3)

3. An alloy is composed of three metals X,Y and Z. The percentage of each metal

is given by the following set of equations:

100

2 0

4 0

X Y Z

X Y

X Z

Use Cramer’s rule to determine the percentage of metal X in the alloy. Use the percentage for X and the method of substitution to find the percentage of Y present in the alloy. (6)

4. Forces on a beam produce the equilibrium equations:

1 2

1 2

500 10 500 0

10 4 500 5 10 500 100 10 0

R R

R R

Use Cramèr's rule to calculate the forces R1 and R1 (in newton). (5)

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5. Resolve into partial fractions:

5.1 2

3

3 4 4

4

x x

x x

(5)

5.2

2

3

3 15 15

3

x x

x

(7)

6. Given 35 70V j and 7 5I j . Rewrite V and I in polar form and

then find .V

ZI

Leave your answer in polar form. (6)

7. Solve for x if 2 1 0x . Give the answers in the form a bj . (2)

8. Represent the complex number 2 2 i graphically and find the polar form of the

number. (3)

9. Two AC voltages are given by the expressions 31 and 7 3 .

2j y x j

If the voltages are equal what are the values of x and y? (6)

10. Identify and sketch the following curve:

2 2

116 25

x y (4)

TOTAL: 50


ONLY FOR SEMESTER 1 STUDENTS

Assignment 02 Due Date: 4 April

Unique number: 734238

This assignment is a written assignment based on Study Guide 2. This assignment contributes 50% to your year mark. Source: Paper May 2011, October 2011 and October 2016

1. Determine

1.1 2

2

2

4 4lim

4x

x

x

x

(3)

1.2 2

2

3 2lim

2 3x

x x

x x

(3)

MAT1581/101/3/2018

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1.3 0

limcosx

x

x (1)

2. Differentiate the following with regard to x :

2.1 4 secy x (3)

2.2 2 tan3 xy n x e by first using logarithm rules to simplify y . (4)

2.3 5 24 .cos(3 )y x x (3)

2.4 1

2

xy

x

Simplify your answer. (3)

3. The current through an inductance is given by 3 24 2 17.i t t

3.1 Find the times, 0t when the current is a relative maximum

or minimum. (4)

3.2 Find the current at these times. (2)

4. The displacement of a car is given by the formula 3 22 2 16 1s t t t where s is in metres and t is in seconds.

4.1 Find the formula for the velocity of the car at any time t. (2)

4.2 Find the acceleration of the car when 5t seconds. (4)

5. Determine the following integrals:

5.1 31x xe e dx

(2)

5.2 2

2 4

1

xdx

x x

(3)

5.3

2

2

2

2

x xdx

x

(5)

5.4 55 xe dxx

(2)

6. Determine the area enclosed between the curve 24 3 1y x x and the

x axis between 0x and 2x . (6)

TOTAL: 50

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Assignment 01 Due Date: 14 August

Unique number:698643

This assignment is a written assignment based on Study guide 1. Submission of this assignment by the due date will give you admission to the examination This assignment contributes 50% to your year mark. Source: Paper May 2011, October 2011 and May 2017

1. Find the first four terms in the binomial expansion of

1

3

1 .3

x

Simplify each term. (5)

2. Expand 21 x to three terms using the binomial theorem. (3)

3. The relationship between the displacement, s, velocity, v, and acceleration,

a, of a piston is given by the equations:

2 2 4

3 4 25

3 2 4

s v a

s v a

s v a

Find the displacement s using Cramèr's rule, given that

1 2 2

3 1 4 41.

3 2 1

(4)

4. Use Cramer’s rule to solve for x and y in the following system of equations and

then find z by substitution:

3 2 1

2 3 2

2 2 10

x y z

x y z

x y z

(9)

5. Resolve into partial fractions 4

2 21 1

x

x x (8)

6. Name the following curves and draw a sketch of each:

6.1 2 216 9 144x y (4)

MAT1581/101/3/2018

17

6.2 0419649 22 xy (4)

Show the vertices and foci clearly on your sketch.

7. Write the complex number 8 225 in exponential form. (2)

8. Solve for x and y if 22

3

jx yj

j

(6)

9. Solve the equation 2 4 20 0x x and plot your solutions on an Argand diagram.

Express these solutions in polar form. (5)

TOTAL: 50



Assignment 02 Due Date: 11 September Unique number:821200

This assignment is a written assignment based on Study Guide 2. Submission of this assignment by the due date will give you admission to the examination This assignment contributes 50% to your year mark. Source: Paper May 2011, October 2011 and May 2017

1. Determine

1.1 333

lim 3 xx

xn

(2)

1.2 2

25

5lim

5x

x

x

(3)

2. Differentiate the following with regard to x :

2.1 tan 2

tan 2

xy

x

(3)

2.2 2 2 2xy x a e (2)

2.3 342sec cosec y x x (3)

2.4 32 xy n x (4)

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3. The period T of a simple pendulum of length l is given by 2l

Tg

, where g

is a constant called the acceleration due to gravity. Find dT

dl (3)

4. Determine the relative maximum and/or minimum points for 21 4y x x (7)

5. Determine the following integrals:

5.1 3 1 2x x dx (2)

5.2 21 2

xdx

x

(2)

5.3 2

sin 2

1 sin

xdx

x

(2)

5.4 0

2 sin 2 cosx x dx

(4)

5.5 2

14

xn dx

n e

(3)

6. Determine the area bound by the straight line y x and the parabola 2y x . (5)

7. Suppose that liquid flows into a tank at a variable rate of q(t) litre per second.

It can be shown that the area under the graph of q against t represents the

volume of liquid in the tank. Consider a chemical storage tank with liquid entering

at a rate of 2

( ) 610 100

t tq t .

If at time t = 0 the tank is empty, calculate the volume of liquid in the tank at

time t = 20 seconds (5)

TOTAL: 50

8.7 Other assessment methods

There are no other assessment methods for this module.

MAT1581/101/3/2018

19

8.8 The examination

Particulars about the examination will be sent to you by the examinations division during the semester. To write the examination you must have one assignment registered on the student system before 30 March for semester 1 students and before 8 September for semester 2 students. Note that lecturers cannot give admission to the examination if you failed to obtain access to the examination nor can we change the examination date or your chosen venue. Please check your permission to write the examination on 16 April for semester 1 and 17 September for semester 2 on myUNISA. Also check your examination date and center. Copies of previous examination papers are not available on request from the lecturer. The most recent paper is available on MyUnisa under official study material without memorandum.

9 FREQUENTLY ASKED QUESTIONS

Question: Can I get an extension on the due date for my assignment 01?

Answer: No, this assignment gives you admission to the examination and for administrative purposes no extension can be given.

Question: May I use a calculator in the examination?

Answer: Yes, it must be non-programmable.

Question: Can I request more past papers from the lecturers?

Answer: No, past papers are on MyUnisa under official Study material.

Question: Where do I get memorandums for past papers?

Answer: No memorandums are supplied, but the lecturer may decide to post some solutions or answers only under additional resources on myUNISA.

Question: Why don’t I receive any follow-up Tutorial letters from UNISA?

Answer: All follow-up letters are only posted on myUnisa, so no post is sent to students.

Question: What does the letters FC after my module code mean?

Answer: FC stands for Financial Cancellation. You may continue your studies and even write the examination but no results will be released before your student account has been paid in full.

10 SOURCES CONSULTED

Study Guides and past papers for MAT1581

11 IN CLOSING

The semester system affords you more flexibility in planning your studies. The guideline is that you must be able to spend 72 minutes per day (including weekends) per module. Planning is the key to success.

May you achieve your dreams!

20

ADDENDUM

12.1 Formula Sheets

ALGEBRA

Laws of indices

n

nn

n

n

n

n

n mn

m

mnmnnm

nm

n

m

nmnm

b

a

b

a

baab

a

aa

aa

aa

aaa

aa

a

aaa

.8

.7

1.6

1and

1.5

.4

.3

.2

.1

0

Logarithms

Definitions: If xay then yx alog

If xey then ynx

Laws:

fefa

a

AA

AnA

BAB

A

BABA

ff

b

ba

n

a

nlog.5

log

loglog.4

loglog.3

logloglog.2

logloglog.1

Factors

2233

2233

babababa

babababa

Partial Fractions

dx

C

cbxax

BAx

dxcbxax

xf

bx

D

ax

C

ax

B

ax

A

bxax

xf

cx

C

bx

B

ax

A

cxbxax

xf

22

323

Quadratic Formula

a

acbbx

cbxax

2

4then

0If

2

2

DETERMINANTS

223132211323313321122332332211

3231

222113

3331

232112

3332

232211

333231

232221

131211

aaaaaaaaaaaaaaa

aaaa

aaaaa

aaaaa

aaaaaaaaaa

MAT1581/101/3/2018

21

SERIES

Binomial Theorem

11and

...!3

21

!2

111

and

..3

21

2

1

32

33221

x

xnnn

xnn

nxx

ab

..ba!

nnnba

!

nnbnaaba

n

nnnnn

Maclaurin’s Theorem

11

32

!1

0

!3

0

!2

0

!1

00 n

n

xn

fx

fx

fx

ffxf

Taylor’s Theorem

afn

haf

haf

hafhaf

axn

afax

afax

afax

afafxf

nn

nn

112

11

32

!1!2!1

!1!3!2!1

COMPLEX NUMBERS

212

1

2

1

212121

22

2

:Division.6

:tionMultiplica.5

andthen,If.4

:nSubtractio.3

:Addition.2

tanarg:Argument

:Modulus

1where

,sincos.1

r

r

z

z

rrzz

qnpmjqpjnm

dbjcajdcjba

dbjcajdcjba

a

barcz

bazr

j

rerjrbjaz j

1

1 1

7. De Moivre's Theorem

cos sin

8. has distinct roots:360

with 0, 1, 2, ,

9. cos sin

cos and sin

10. cos sin

11.

n n n

n

n n

j

j j

a jb a

j

r r n r n j n

z nk

z r kn

re r j

re r re r

e e b j b

n re n r j

22

GEOMETRY

1. Straight line:

11 xxmyy

cmxy

Perpendiculars, then 2

1

1

mm

2. Angle between two lines:

21

21

1tan

mm

mm

3. Circle:

222

222

rkyhx

ryx

4. Parabola: cbxaxy 2

axis at a

bx

2

5. Ellipse:

12

2

2

2

b

y

a

x

6. Hyperbola:

axis- round1

axis- round1

2

2

2

2

2

2

2

2

yb

y

a

x

xb

y

a

x

kxy

MENSURATION

1. Circle: ( in radians)

2

2

2

Area

Circumference 2

Arc length

1 1Sector area

2 21

Segment area sin2

r

r

r

r r

r

2. Ellipse:

ba

ab

nceCircumfere

Area

3. Cylinder:

2

2

22area Surface

Volume

rrh

hr

4. Pyramid:

height base area3

1Volume

5. Cone:

r

hr

surfaceCurved3

1Volume 2

6. Sphere:

3

2

3

4

4

rV

rA

7. Trapezoidal rule:

1321

2 nn yyy

yyhA

8. Simpsons rule:

RELFs

A 243

9. Prismoidal rule

321 46

AAAh

V

MAT1581/101/3/2018

23

HYPERBOLIC FUNCTIONS

Definitions:

xx

xx

xx

xx

ee

eex

eex

eex

tanh

2cosh

2sinh

Identities:

x

x

xxx

xxx

xx

xx

xx

xx

xx

2

2

22

2

2

22

22

22

sinh21

1cosh2

sinhcosh2cosh

coshsinh22sinh

12cosh2

1cosh

12cosh2

1sinh

cosech1coth

sechtanh1

1sinhcosh

TRIGONOMETRY Compound angle addition and subtraction formulae: sin(A + B) = sin A cos B + cos A sin B sin(A - B) = sin A cos B - cos A sin B cos(A + B) = cos A cos B - sin A sin B cos(A - B) = cos A cos B + sin A sin B

BA

BABA

BA

BABA

tantan1

tantantan

tantan1

tantantan

Double angles: sin 2A = 2 sin A cos A cos 2A = cos2A – sin2A = 2cos2A – 1 = 1 – 2sin2A sin2 A = ½(1 – cos 2A) cos2 A = ½(1 + cos 2A)

A

AA

2tan1

tan22tan

Products of sines and cosines into sums or differences: sin A cos B = ½(sin (A + B) + sin (A – B)) cos A sin B = ½(sin (A + B) – sin (A – B)) cos A cos B = ½(cos (A + B) + cos (A – B)) sin A sin B = –½(cos (A + B) – cos (A – B)) Sums or differences of sines and cosines into products:

2sin

2sin2coscos

2cos

2cos2coscos

2sin

2cos2sinsin

2cos

2sin2sinsin

yxyxyx

yxyxyx

yxyxyx

yxyxyx

TRIGONOMETRY

Identities

cos

sintan

tan- = )(-tan

cos = )(- cos

sin - = )sin(-

cosec = 1 +cot

sec = tan+ 1

1 cos sin

22

22

22

24

DIFFERENTIATION

h 0

1

2

1

1. lim

2. 0

3.

4. . . ' . '

. ' . '5.

6. ( ) ( ) . '( )

7. . .

n n

n n

f x h f xdy

dx hd

kdxd

ax anxdxd

f g f g g fdxd f g f f g

dx g g

df x n f x f x

dxdy dy du dv

dx du dv dx

8. Parametric equations

2

2

dydy dt

dxdxdt

d dyd y dt dx

dxdxdt

9. Maximum/minimum For turning points: f '(x) = 0

Let x = a be a solution for the above If f '' (a) > 0, then a is a minimum point If f ''(a) < 0, then a is a maximum point For points of inflection: f " (x) = 0 Let x = b be a solution for the above

Test for inflection: f (b – h) and f(b + h) Change sign or f '"(b) ≠ 0 if f '"(b)

exists.

1

2

1

2

12

1

2

1

2

1

2

1

'( )10. sin ( )

1 ( )

'( )11. cos ( )

1 ( )

'( )12. tan ( )

1 ( )

'( )13. cot ( )

1 ( )

'( )14. sec ( )

( ) 1

'( )15. cosec ( )

( ) 1

'( )16. sinh ( )

d f xf x

dx f x

d f xf x

dx f x

d f xf x

dx f x

d f xf x

dx f x

d f xf x

dx f x f x

d f xf x

dx f x f x

d f xf x

dx f

2

1

2

12

12

1

2

1

2

( ) 1

'( )17. cosh ( )

( ) 1

'( )18. tanh ( )

1 ( )

'( )19. coth ( )

1 ( )

'( )20. sech ( )

1 ( )

'( )21. cosech ( )

( ) 1

22. Increments: . . .

x

d f xf x

dx f x

d f xf x

dx f x

d f xf x

dx f x

d f xf x

dx f x f x

d f xf x

dx f x f x

z z zz x y w

x y w

23. Rate of change:

. . .dz z dx z dy z dw

dt x dt y dt w dt

INTEGRATION

b

a

b

a

b

a

dxyb-a

dxyb-a

F(aF(b)dxf(x)vduuv-udv

22 1)R.M.S.(.4

1= Mean value.3

).2:partsBy.1

MAT1581/101/3/2018

25

TABLE OF INTEGRALS

1

1

1 11

2 11

3

4

5

6 sin cos

7 cos sin

8

(n )n

nn

f(x) f(x)

f(x)f(x)

a x. ax dx c, n

n

f(x). f(x) .f'(x) dx c, n

n

f (x). dx n f(x) c

f(x)

. f (x).e dx e c

a. f (x).a dx c

n a

. f (x). f(x) dx f(x) c

. f (x). f(x) dx f(x) c

. f (x)

2

2

tan sec

9 cot sin

10 sec sec tan

11 cosec osec cot

12 sec tan

13 cosec cot

14

. f(x) dx n f(x) c

. f (x). f(x) dx n f(x) c

. f (x). f(x) dx n f(x) f(x) c

. f (x). f(x) dx n c f(x) f(x) c

. f (x). f(x) dx f(x) c

. f (x). f(x) dx f(x) c

sec tan sec

15 cosec cot cosec

. f (x). f(x). f(x) dx f(x) c

. f (x). f(x). f(x)dx f(x) c

MAT1581 101 2018 - gimmenotes · year of MAT2691: Engineering Mathematics, K. A. Stroud (with...

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