mat1581 101tutletter2014 semester engyear of mat2691: engineering mathematics, k. a. stroud (with...

MAT1581/101/3/2014

Tutorial letter 101/3/2014

Mathematics I (Engineering)

MAT1581

Semesters 1 & 2

Department of Mathematical Sciences

IMPORTANT INFORMATION:

This tutorial letter contains important information about your module.

2

CONTENTS Page

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

1.1 Study Material ................................................................................................................................. 4

2 PURPOSE OF AND OUTCOMES FOR THE MODULE ............................................................... 5

2.1 Purpose .......................................................................................................................................... 5

2.2 Outcomes ....................................................................................................................................... 5

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

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

3.2 Department ..................................................................................................................................... 6

3.3 University ........................................................................................................................................ 6

4 MODULE-RELATED RESOURCES .............................................................................................. 6

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

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

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

5 STUDENT SUPPORT SERVICES FOR THE MODULE ................................................................ 7

5.1 Tutor Classes .................................................................................................................................. 7

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

5.3 Science foundation Project ............................................................................................................. 7

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

6 MODULE-SPECIFIC STUDY PLAN ............................................................................................... 7

7 MODULE PRACTICAL WORK AND WORK-INTEGRATED LEARNING ................................... 10

8 ASSESSMENT ............................................................................................................................. 10

8.1 Assessment plan .......................................................................................................................... 10

8.2 General assignment numbers ....................................................................................................... 10

8.2.1 Unique assignment numbers ........................................................................................................ 10

8.2.2 Due dates for assignments ........................................................................................................... 10

8.3 Submission of assignments .......................................................................................................... 11

8.3.1 Multiple choice assignments ......................................................................................................... 11

8.3.2 Written assignments ..................................................................................................................... 11

8.4 Assignments ................................................................................................................................. 13

8.4.1 Assignment 01 Semester 1 .......................................................................................................... 13




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8.5 Marking and Feedback on Assignments ....................................................................................... 28

9 OTHER ASSESSMENT METHODS ............................................................................................. 28

10 EXAMINATION ............................................................................................................................. 28

10.1 Preparation Paper I (May/June 2010) ........................................................................................... 29

10.2 Preparation Paper II (October 2010) ............................................................................................ 30

10.3 Preparation Paper III (May 2011) ................................................................................................. 33

10.4 Preparation Paper IV (October 2011) ........................................................................................... 35

10.5 Preparation Paper V (May/June 2012) ......................................................................................... 37

11 FREQUENTLY ASKED QUESTIONS .......................................................................................... 40

12 SOURCES CONSULTED ............................................................................................................. 40

13 CONCLUSION .............................................................................................................................. 40

14 ADDENDUM ................................................................................................................................. 40

14.1 Errata Study guides ...................................................................................................................... 40

14.2 Formula Sheets ............................................................................................................................ 46

4

1 INTRODUCTION

Dear Student

Welcome as a student to Mathematics I, MAT1581. You will see the old codes MAT181Q and WIM131U on some of your material. This is correct as the study material did not change only the code to reflect that it is now a semester course. You must always use the code 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 2014 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 2014 and qualify for this by doing assignments for semester 2.

1.1 Study Material

NOTE: THE COVER OF YOUR STUDY GUIDES SHOULD HAVE THE WORDING:

MATHEMATICS I LINKED TO

MATHEMATICS MINING I

MAT181QE/WIM131UE

You will receive three study guides. Corrections are given in addendum 14.1 at the end of this letter. Ignore the codes MAT181Q and WIM131U in these study guides. Your code is MAT1581. Please forward possible mistakes to the lecturer to be included in future letters to students. You need to work on your mathematics regularly. The amount of time you study is not important, but with absolute concentration and maximum effort your time will be used efficiently. See an example of a timetable further on in this letter. If you have access to the Internet, you can view the study guides and tutorial letters for the modules for which you are registered on the University’s online campus, http://my.unisa.ac.za

Some of this tutorial matter may be out of print when you register. Tutorial matter that is not available when you register will be posted to you as soon as possible, but is available on myUnisa. You need not wait to receive the study material by post to start studying.

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2 PURPOSE OF AND OUTCOMES FOR THE MODULE

2.1 Purpose

To serve the mathematical needs of learners engaged in a first course on engineering. Learners from other fields like computing, accounting and business management who want to broaden their mathematical knowledge will also benefit from this course.

2.2 Outcomes

On completion of this module the learner should have a basic knowledge of algebra, trigonometry, complex numbers, analytic geometry, differentiation and integration. Refer to the study plan for more detail.

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 if you do not have internet at home.

3.1 Lecturer(s)

Your lecturer at the time of compiling this tutorial letter (July 2013) is Miss LE Greyling, Florida Campus. 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 the service is available you can leave a voicemail

message. The message must contain your name, the subject and a telephone number where you can be reached. You may also ask the lecturer to call you if she is in the office and you do not have sufficient funds on your phone.

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 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.

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3.2 Department

The department of Mathematical Sciences will be moving to the Florida Campus in September 2013. No contact numbers in Florida are currently available.

3.3 University

Read the brochure on Mystudies@Unisa2014. 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. Tip: Do not write any requests like the change of contact details or extra stationary in your assignment. The markers cannot help you. Likewise any messages for the lecturer must be directed to the lecturer by telephone, fax or e-mail.

4 MODULE-RELATED 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 2014

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.

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5 STUDENT SUPPORT SERVICES FOR THE MODULE

Important information appears in your MyStudies@Unisa brochure.

5.1 Tutor Classes

Contact your nearest regional office to register for the tutorial program. The regional offices organize tutorial classes if enough students are interested. Your tutor will be a qualified mathematician approved by the Department of Mathematical Sciences. As part of student support you may be assigned to an e-tutor. The e-tutor will contact you on your mylife e-mail through myUNISA. The lecturer is not responsible for any tutorial classes organised by student support services.

5.2 Discussion classes

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

5.3 Science foundation Project

This project is financed by the Department of Education. Extra classes in mathematics will be offered to students identified as “AT-RISK” to fail. The regional offices are responsible to contact you if you are one of these students as classes will be at the regional office. This is separate from the ordinary tutorial services rendered by the regional offices. This service is also using e-tutors.

5.4 Other services @ Regional offices

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

You can access myUnisa at your regional office learner centre.

6 MODULE-SPECIFIC 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. 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.

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Take note of the comments as some work is revision of what you have done in Grade 12 or earlier. You should be able to work through those units quickly. From the final closing date of registration to the due date of your last assignment you have about 9 weeks. Concentrate on new work. The lecturer marked parts that she thinks can be regarded as revision, but you must judge for yourself.

MODULE UNIT CONTENT to study Week Comments

1

Basic Concepts

1 Arithmetic 1

Revision

2 Algebra

3 Linear Equations

4 Simultaneous equations

5 Quadratic Equations

Post-test

Complete Assignment 1

2

Exponents and

Logarithms

1 Exponents Revision

2 Scientific Notation

3 Logarithms

4 Graphs

5 Equations

Post-test

3

Binomial

Theorem

1 Binomial Theorem 2

Post-test

4

Determinants

1 Properties

2 Value of n x n determinant

3 Cramer's Rule

Post-test

5

Partial

Fractions

1 Introduction 3

2 Proper Fractions

3 Improper Fractions

Post-test

6

Trigonometry

1 Trigonometric Ratio’s Revision Grade 12

2 Radian Measure

3 Graphs Revision

4 Trigonometric Equations

5 Identities and Equations

Posttest

7

Co-ordinate

Geometry

1 Co-ordinate Systems 4

2 The Straight Line Revision

3 The Parabola Revision

4 The Rectangular Hyperbola Revision

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MODULE UNIT CONTENT to study Week Comments

5 The Circle

6 The Ellipse

7 The Central Hyperbola

Post-test

8

Complex

Numbers

1 Imaginary and Complex

Numbers

5

2 Operations with Complex

numbers

3 Polar and exponential form

4 Operations in polar and

exponential form

Post-test

Complete Assignment 2

9

Differentiation

1 Functional Notation 6 Very important

2 Limits

3 The Derivative

4 Standard Forms

5 Rules of Differentiation I

6 Rules of Differentiation II

7 Higher Order Derivatives 7

8 Applications I

9 Applications II: Maxima and

Minima

10 Graphs: Intercepts, Symmetry,

Asymptotes and Concavity

Take note only

Post-test

10

Integration

1 Reverse of Differentiation I 8

9

Very important

2 Reverse of Differentiation II

3 Method of substitution

4 Standard Integrals

5 Partial Fractions

6 Trigonometric Functions

7 The Definite Integral

8 Areas

Post-test

Complete assignment 3 and 4

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7 MODULE PRACTICAL WORK AND WORK-INTEGRATED LEARNING

There are no practicals for this module.

8 ASSESSMENT

8.1 Assessment plan

Students must complete and submit all assignments. Assignment 01 gives you admission to the examination. Your assignment marks will be used to calculate your year mark. Your year mark will form part of your final mark for the subject. Your performance in your assignments thus plays a vital part in your final mark.

Assignment % of Year mark Description and Instructions

01 10 20 Multiple choice questions on basic concepts

02 50 Written assignment on Module 3,4, 5, and 8.

03 40 25 Multiple choice questions on all modules.

04 0 Written assignment on Module 9 and 10 Optional submission. Solutions on myUNISA for self-evaluation.

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.

8.2 General assignment numbers

You must submit assignment 01, 02 and 03 for the semester you are registered. Assignment questions per semester are given in 8.4 below.

8.2.1 Unique assignment numbers

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

8.2.2 Due dates for assignments

Semester1 Unique nr. Semester 2 Unique nr. Assignment 01 MCQ 12 February 295723 30 July 890695 Assignment 02 Written 19 March 358067 8 September 878379 Assignment 03 MCQ 23 April 198498 8 October 817985 Assignment 04 Written 25 April 340300 10 October 842734

Warning: Plan your programme so that study problems can be sorted out in time. No extension will

MAT1581/101

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be given by the lecturer without valid reasons that can be verified. 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.

8.3 Submission of assignments

For Mathematics I you have to send in three assignments to the university before the given closing dates. You will not be permitted to write the examination if you do not send in assignment 01. You may submit assignment 04. Submit the assignments linked to your registration period (semester 1 or 2). Note: use the correct unique number, assignment number and module code.

8.3.1 Multiple choice assignments

Prepare your answers before deciding on how to submit your assignment. Select the correct answer from the 3 possible answers given. You can submit your assignment in one of three ways:

1) Preferred method online through myUNISA. Simply follow the instructions on screen. 2) Using an internet-enabled mobile phone. When submitting this way use the UNISA mobile

application. Read how to download it to your phone and submit your MCQ assignment answers: http://www.unisa.ac.za/mobileapp.

3) Using the mark reading sheet provided. For detailed instructions refer to my Studies @ Unisa 2014. Remember: - Only use the orange mark-reading sheet that you received with your study material.

No other sheets will be accepted. - Only use an HB pencil. - Do not attach a barcode sticker. - Mark-reading sheets should not be put in an assignment cover and stapled. - A mark-reading sheet that is filled in incorrectly, damaged or folded cannot be

marked.

8.3.2 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 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

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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. Assignments may not be submitted by fax, e-mail, registered post or courier. Assignments by post: Make sure that you complete the assignment cover. If the subject or assignment number is incorrect your assignment can not 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 my Studies @ Unisa 2014. Assignments should be addressed to: The Registrar, PO Box 392, UNISA, 0003 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. Your assignment must be combined in one document. Only one document can be uploaded per assignment. Log in 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.

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8.4 Assignments

8.4.1 Assignment 01 Semester 1

ONLY FOR SEMESTER 1 STUDENTS Assignment 01 (Compulsory)

Due Date: 12 February Unique number: 295723

This assignment contributes 10% to your year mark.

Select the correct answer from the possible answers given.

Evaluate the expressions, correct to 2 decimal places where necessary:

(Decimal comma used as in study guides)

QUESTION 1: 27 5 3

[1] 12 [2] 96 [3] 96 (1)

QUESTION 2: 7,1 5,4 2,7 4 2 1,7

[1] 3,2 [2] 10,3 [3] 3,9 (1)

QUESTION 3: 3 2

4 5

[1] 5

9 [2]

23

20 [3]

5

20 (1)

QUESTION 4: 1 2

38 5

[1] 125

16 [2]

5

4 [3]

53

16 (1)

QUESTION 5: 3

25,68

[1] 614,4 [2] 1,56 [3] 9,6 (1)

QUESTION 6: 9 1

[1] 4 [2] 3,16 [3] 100 (1)

QUESTION 7: 23x

[1] 2 9x [2] 9x [3] 9x (1)

QUESTION 8: 23x

[1] 2 9x [2] 2 6 9x x [3] 2 6 9x x (1)

14

QUESTION 9: Solve for x : 21 4 4 0x x

[1] 2 2 2 [2] 1

2 [3]

1 2

2 2 (1)

QUESTION 10: True or false: The equations 5 0x and 5 0x x are equivalent.

[1] True [2] False (1)

QUESTION 11: True or false: The equations 5 0x and 7 5 0x are equivalent.


QUESTION 12: 2 2x x

[1] 2 4x [2] 2 2x [3] 2 2x (1)

QUESTION 13: 3 24 12

4

x x x

x

[1] 2

34

xx [2] 3 11x x [3]

224 12

4

xx x (1)

QUESTION 14: 6 9

3

x y x y

x y

[1] 2 3x [2] 2 3x y [3] 2 9x y (1)

QUESTION 15: 3 2

4 2

3 3

x x

x x x

[1] 2

3 2

3

x

x x

[2]

2

2

4 2

3

x x

x x

[3]

3

2

4 2

3

x x

x x

(1)

QUESTION 16: One factor of 6 2( 6)x y y is

[1] 6x [2] 12y [3] 2x (1)

QUESTION 17: Factoring 4 81x gives

[1] 2 9 3 3x x x [2] 2 23 3x x [3] 2 29 9x x (1)

QUESTION 18: Solve for x: 6 5 3 5

4 5 2 5

x x

x x

[1] 5

6 [2] 1 [3] 0 (1)

QUESTION 19: If 4x then 2

4

x

x

is

[1] undefined [2] 0 [3] (1)

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2

x

x

is

[1] undefined [2] 0 [3] (1)

MAXIMUM: 20


ONLY FOR SEMESTER 1 STUDENTS

Assignment 02(Compulsory) Due Date: 19 March

Unique number: 358067

This assignment is a written assignment based on Module 3,4,5 and 8


QUESTION 1

Use the binomial theorem to expand 3 1 2x to four terms,

Simplify your answer as far as possible. [8] QUESTION 2 In a mechanical system the relationship between the displacement s, velocity v and acceleration

a is given by the simultaneous equations:

5 8 1

3 2 4 3

10 16 2 3

s v a

s v a

s v a

Use Cramer’s rule to find s, v and a. [12]

QUESTION 3

Resolve into partial fractions: 5 2

2

2

1

x x

x. [12]

QUESTION 4

4.1 Write the Cartesian coordinates 1, 2 in polar form (3)

4.2 Simplify 2 3

2

j j

j j

(6)

4.3 Determine two distinct roots of 1 3 j . Write your answer in exponential form. (7)

[16]

Overall presentation and logic [2]

MAXIMUM: 50

16



Assignment 03 (Compulsory) Due Date: 23 April


This assignment is a written assignment based on all Study Guides



QUESTION 1: log10000 6log 10

[1] 6

10000

10 [2] 1 [3] 0 (1)

QUESTION 2: Solve for x if 4 4log 6 2 logx x

[1] 4x [2] 2x [3] 8x (1)

QUESTION 3: Solve for x if 2 1 32 8x x

[1] 10x [2] 4x [3] 7x (1)

QUESTION 4: The middle term in the expansion of 41 x is

[1] 4x [2] 26x [3] 34x (1)

QUESTION 5: If

2 3 2

2 3 0

3 4

x y z

x y z

x y z

then y =

[1]

2 3 1

1 2 3

3 1 1

2 2 1

1 0 3

3 4 1

[2]

2 2 1

1 0 3

3 4 1

2 3 1

1 2 3

3 1 1

[3]

2 2 1

0 1 3

4 3 1

2 3 1

1 2 3

3 1 1

(1)

QUESTION 6: cos3 cos7

[1] 2sin 2 sin5 [2] 2sin 2 sin5 [3] 2sin 2 cos5 (1)

QUESTION 7: The equation 2 225x y represents

[1] circle [2] hyperbola [3] ellipse (1)

MAT1581/101

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QUESTION 8: The equation 2 2 16y x represents


QUESTION 9: If 43y x , then dy

dx =

[1] 1 [2] 43 [3] 0 (1)

QUESTION 10: If 24 5 33 x xf x e , then 'f x

[1] 24 5 3 123 4 5 3x x

x x e

[2] 24 5 33 8 5 x xx e [3] 8 5xe (1)

QUESTION 11: If 232 3x x

f xx

, then 'f x

[1] 8

31 5x

[2]

5 32 3 2

2

3 2 3x x x

x

[3] 1

2

2 2x x

x

(1)

QUESTION 12: If 1

2 35 3y x

, then dy

dx =

[1]

423

10

3 5 3

xy

x

[2]

2

423

10

3 5 3

xy

x

[3]

1

10 3x (1)

QUESTION 13: If 104 2 1g x x x , then ' 1g

[1] 1 [2] 220 [3] 24 (1)

QUESTION 14: If 5 28 2 1y u u u and 10u x , find dy

dx when x = 9.

[1] 0 [2] 9 [3] 9/2 (1)

QUESTION 15: If n 2 7y x , then 'y

[1] 1

2 7x [2]

1

2 7x [3]

31

2 7x (1)

QUESTION 16: If 2 25 1 xy x x e , then 'y

[1] 2 22 12 3 xx x e [2] 22 5 xx e [3] 2 12 5

2

xx e

x

(1)

QUESTION 17: If sin 3 2cos 2y x x , then dy

dx =

[1] 3cos3 4sin 2x x [2] cos3 2sin 2x x [3] cos3 4sin 2x x (1)

18

QUESTION 18: If 3 2 43x x xy e a , then dy

dx =

[1 anane xxx 423 33 [2] anane xxx 423 .433.23 3)

[3] anane xxx 423 333 (1)

QUESTION 19: If 2 2sin 1 4f x x , then 'f x

[1] 241cos2 x [2] xx 8.41cos2 2

[3] xxx 8.41cos.41sin2 22 (1)

QUESTION 20: If 3 .sinxy n e x , then dy

dx =

[1] 3

3

3 .cos

.sin

x

x

e x

e x [2] xcot3 [3]

xe

xex

x

sin.

cos.3

3

(1)

QUESTION 21: 34

12x dx

x

[1] 23

13

4x C

x [2]

4

3

32

4

xx C

x [3]

4

3

12

4 3

xx C

x (1)

QUESTION 22: 22

1

2

xdx

x x

[1] 2

1

2 2C

x x

[2]

32 2

3

x xC

[3] 12 2x x C

(1)

QUESTION 23: 1

02 dx

[1] 2

2 [2]

1

2 2 [3] 2 (1)

QUESTION 24: 14x dx

[1] 114

4x C

n

[2] 24 4xn C [3]

2

4

4x

n

(1)

QUESTION 25: 2 3 4

2

x xdx

x

[1] 2

6 22

xx n x C [2]

2

6 22

xx n x C

[3] 1

22

n x C (1)

MAT1581/101

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Assignment 04 (Optional) Last Date assignment will be accepted: 25 April


This assignment is a written assignment based on Study Guide 1 and 3

Source: Paper October/November 2012


QUESTION 1 1.1 Find the turning points on the curve 3 23 6 3 1y x x x and determine

their nature. (7)

1.2 Determine the following limits:

1.2.1 0

lim 5x

x (1)

1.2.2 1

1lim

1t

t

t

(3)

1.3 Differentiate the following with regard to x and simplify if possible:

1.3.1 3 4

cos2

n xy

x

(4)

1.3.2 24 sin 3xy e x (2)

1.3.3 2 3 5y x x (3)

[20]

QUESTION 2 2.1 Determine the following integrals:

2.1.1 22 3 1x x dx

(3)

2.1.2 22 1x

dxx

(2)

2.1.3 55cos sint t dt (3)

2.1.4 2

11

2

xdx

x x

[Use partial fractions] (7)

2.2 The distance, s, covered by a train on a straight track in the first two seconds

is given by 2

020 1 ts e dt . Find s. (5)

[20]

20



Assignment 1 Due Date: 30 July

Unique number: 890695 This assignment contributes 10% to your year mark.


Evaluate the expressions, correct to 2 decimal places where necessary:

(Decimal comma used as in study guides)

QUESTION 1: 27 5 3

[1] 42 [2] 6 [3] 96 (1)

QUESTION 2: 9,2 6,8 3,4 3 2 1,7

[1] 6,3 [2] 9,34 [3] 8,78 (1)

QUESTION 3: 2 1

3 6

[1] 3

9 [2]

5

6 [3]

3

18 (1)

QUESTION 4: 2 3

115 5

[1] 12 [2] 16

25 [3]

1

12 (1)

QUESTION 5: 3

25,68

[1] 614,4 [2] 1,56 [3] 9,6 (1)

QUESTION 6: 16 1

[1] 4,12 [2] 5 [3] 289 (1)

QUESTION 7: 22x

[1] 4x [2] 2 9x [3] 4x (1)

QUESTION 8: 22x

[1] 2 4x [2] 2 4 4x x [3] 2 4 4x x (1)

MAT1581/101

21

QUESTION 9: Solve for x : 22 3 1 0x x

[1] 3 17

6

[2]

3 17

2

[3]

3 17

4

(1)

QUESTION 10: True or false: The equations 5 0x and 5 0x x are equivalent.


QUESTION 11: True or false: The equations 5 0x and 7 5 0x are equivalent.


QUESTION 12: 3 3x x

[1] 2 3x [2] 2 3x [3] 2 9x (1)

QUESTION 13: 3 2

2

4 12

4

x x x

x

[1] 24 16 48x x [2] 3

14

x

x [3] 24 12

4

xx x (1)

QUESTION 14: 7 6

3 21

x

x x

[1] 2

x [2] 3 [3] x (1)

QUESTION 15: 2 1

4 4

x x

x x

[1] 1

4

x

x

[2] 1

4

x

x

[3] 2

1

16

x

x

(1)

QUESTION 16: One factor of 3 64x is

[1] 2 16x [2] 2 16x [3] 2 4 16x x (1)

QUESTION 17: Factoring 4 3 3 48 2 8 2x x x x gives

[1] 3 36 8 2x x [2] 3 3

2 8 2 5x x x [3] 3 310 8 2x x (1)

QUESTION 18: Solve for x: 4 5 3 4

9 6 2 3

x x x

[1] 5

6 [2]

5

3 [3]

6

5 (1)


1

x

x

is

[1] undefined [2] 0 [3] (1)

22


3

x

x

is

[1] undefined [2] 0 [3] (1)

MAXIMUM: 20



Assignment 02(Compulsory) Due Date: 9 September Unique number:878379



QUESTION 1

Use the binomial theorem to expand 321 x to four terms,

Simplify your answer as far as possible. [8]

QUESTION 2 If three cables are joined at a point and three forces are applied so the system is in equilibrium,

the following system of equations results:

1 2 3

1 2 3

1 2 3

2 4 1

3 5 5

2 3 2 3

F F F

F F F

F F F

Use Cramer’s rule to determine the three forces , and A B CF F F , measured in newton.

[12]

QUESTION 3

Resolve into partial fractions: 2

2

4 3

1 1

x x

x x x. [12]

QUESTION 4

4.1 Write the polar coordinates 5;2

in Cartesian form. (3)

4.2 Solve for x and y if 2 1 4 1yj j j x yj (12)

[15] Overall presentation and logic [2]

MAXIMUM: 50

MAT1581/101

23



Assignment 03(Compulsory) Due Date: 8 October

Unique number:817985

This assignment is a written assignment based on all Study Guides

This assignment contributes 40% to your yearmark.


QUESTION 1: 300

log100

[1] 3 [2] log3 [3] log 200 (1)

QUESTION 2: 5

5 664

[1] 2 [2] 32 [3] 5 2 (1)

QUESTION 3: Solve for x if log 1 log log 2x x

[1] 1x [2] 4

3

nx

[3] 3

4x

n

(1)

QUESTION 4: Solve for x if 32 5 3xe

[1] 4

3 [2] 9x [3] 9x (1)

QUESTION 5: If

2 3 2

2 3 0

3 4

x y z

x y z

x y z

then x =

[1]

2 3 1

1 2 3

3 1 1

2 2 1

0 1 3

4 3 1

[2]

2 2 1

1 0 3

3 4 1

2 3 1

1 2 3

3 1 1

[3]

2 3 1

0 2 3

4 1 1

2 3 1

1 2 3

3 1 1

(1)

QUESTION 6: sin3 sin5

[1] 2cos4 sin [2] 2sin 4 cos [3] 2cos4 sin (1)

24

QUESTION 7: The equation 2 23 27x y represents


QUESTION 8: The equation 2 2

19 16

y x represents


QUESTION 9: If 22y x , then dy

dx =

[1] 1 [2] 22 [3] 0 (1)

QUESTION 10: If 24 1xf x e , then 'f x

[1] 24 1 14 1 xx e [2] 24 18 4 1 xx e [3] 8 4 1xe (1)

QUESTION 11: If 2

2

4

2

xf x

x

, then 'f x

[1] 22

12

2

x

x [2]

22

12

2

x

x

[3]

2

22

4 1

2

x x

x

(1)

QUESTION 12: If 3 2

1

5 3y

x

, then

dy

dx =

[1]

423

10

3 5 3

x

x

[2]

2

423

10

3 5 3

x

x [3]

1

10 3x (1)

QUESTION 13: If n 2 7y x , then 'y

[1] 1

2 7x [2]

1

2 7x [3]

31

2 7x (1)

QUESTION 14: If 2 25 1 xy x x e , then 'y

[1] 2 22 12 3 xx x e [2] 22 5 xx e [3] 2 12 5

2

xx e

x

(1)

QUESTION 15: If cosec 3 sec 2y x x , then dy

dx =

[1] 3cosec3 cot 3 2sec 2 tan 2x x x x [2] 3cosec3 cot 3 2sec 2 tan 2x x x x

[3] cosec3 ot 3 sec 2 tan 2xc x x x (1)

MAT1581/101

25

QUESTION 16: If 22y n , then

dy

dx

[1] 0 [2] 2 2n [3] 22 ne (1)

QUESTION 17: If 3 28 6 7 2f x x x x , then '''f x

[1] 24x [2] 24 [3] 48 (1)

QUESTION 18: If 3 2 43x x xy e a , then dy

dx =

[1 anane xxx 423 33 [2] anane xxx 423 .433.23

[3] anane xxx 423 333 (1)

QUESTION 19: If 2 2sin 1 4f x x , then 'f x

[1] 241cos2 x [2] xx 8.41cos2 2

[3] xxx 8.41cos.41sin2 22 (1)

QUESTION 20: If 3 .sinxy n e x , then dy

dx =

[1] 3

3

3 .cos

.sin

x

x

e x

e x [2] xcot3 [3]

xe

xex

x

sin.

cos.3

3

(1)

QUESTION 21: 2 1

3

xdx

[1] 2

3C [2]

22 1

6

xC

[3] 21

3x x C (1)

QUESTION 22: 2

33 dx

[1] 2

33 x C [2] 5

33

35

C

[3] 0 C (1)

QUESTION 23: 3 4xe dx

[1] 3 33 4 xx e C [2] 3 413

xe C [3] 3 43 xe C (1)

QUESTION 24: 2

4

1

xdx

x

[1] 2

Cx [2] 22 1n x C [3] 2 1n x C (1)

26

QUESTION 25: 1 2

05x x dx

[1] 11

2 [2]

11

4 [3] 11 (1)



Assignment 04(Optional) Last Date on which assignment will be accepted: 10 October

Unique number842734:


Source: Paper May/June 2013

This assignment contributes 0% to your yearmark.

QUESTION 1

1.1 Determine the following integrals:

1.1.1 3 4( 2) 8x x x dx

(3)

1.1.2 tan(2 ) 2sec (2 )xe x dx

(2)

1.1.3 2 1xn e dx (3)

1.1.4 2

2

2 5 3

4 4 1

x xdx

x x x


1.2 A force F stretches a spring through a distance x. The spring obeys Hooke’s Law, namely F kx , where k is the spring constant. The work done, W, is given by

W F dx . Determine the work, W, if a spring, with spring constant 30 N/m is

stretched by 40 cm. (5)

[20]

MAT1581/101

27

QUESTION 2

2.1 Determine the following limit: 2

23

6lim

4 3t

t t

t t

(3)


2.2.1 2sin

1 cos

xy

x

(4)

2.2.2 2( 2)xy x e (3)

2.2.3 1

1

xy n

x

(4)

2.3 The distance,s, in meter moved by a body in t seconds is given by

3 2

6 83 2

t ts t

Find:

2.3.1. the velocity of the body when t = 5 seconds (3)

2.3.2 the value of t when the acceleration is 11 m/s2 (3)

[20]

28

8.5 Marking and Feedback on Assignments

After you hand in your assignment it gets recorded by the department of student assessment. The assignments are ordered by date and will be sent to external markers. It is impossible for the lecturer to mark all assignments. In theory the earlier you submit your assignment, the earlier you will receive your marked assignment.

A selection of assignments will be checked by the lecturer to make sure that the memorandum is followed and that all answers are marked consistently. After the marks are recorded the assignment is returned to you.

When you receive the marked assignment please check that the marks are added correctly and contact us as soon as possible if you find any mistakes. For written assignments, markers will comment constructively on your work. Solutions to all questions in the written assignments will be available on myUNISA two weeks after the due date. You may request a copy of the assignment solutions by e-mail from the lecturer two weeks after the due date. E-mails without a student number and module code will be deleted. As soon as you have received the commentaries and solutions, please check your answers. The assignments and the commentaries on these assignments constitute an important part of your learning and should help you to be better prepared for the next assignment and the examination.

9 OTHER ASSESSMENT METHODS

There are no other assessment methods for this module.

10 EXAMINATION

Particulars about the examination will be sent to you by the examinations division during the year. 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 a month before the examination 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 without memorandum. Results can be viewed on myUnisa and will be posted to you. Included are preparation examination papers. These are old papers showing the kind of questions as well as the topics covered. The paper must be done after you have submitted your second compulsory assignment. The solutions of these papers will be posted on MyUnisa with other important information regarding your examination paper. When attempting this preparation paper, work under examination conditions. Your paper will be two hours and 80 marks. Divide the paper into two sections of one hour each. Sit down and attempt to do all the questions in that section without referring to your notes. Use the memorandum to mark your work. Determine which areas you need to concentrate on before the examination.

MAT1581/101

29

10.1 Preparation Paper I (May/June 2010) Question 1

1.1 Solve the following system of equations for t only using Cramèr's rule:

3 4 7

5 6

2 3 4 9

x y

x t

x y t

(6)

1.2 Expand 4

22 yx using the binomial theorem. (5)

1.3 Solve for x: 1.3.1 2 2log log ( 1) 5x x (4)

1.3.2 2 3 2 0x xe e (3) 1.4 A pie-shaped piece (sector) is going to be cut from a circular piece of metal.

The radius of the circle is 50,0 mm and the central angle of the sector is 75°.

The cost of the metal is 20 cents per square millimeter.

What is the cost of the pie-shaped piece? (4)

[22]

Question 2

2.1 Given the curve 3 2

2 13 2

x xy x

2.1.1 determine the maximum and or minimum turning points of the curve.(6)

2.1.2 sketch the curve. (3)

2.1.3 calculate the area between the curve and the lines 2x and 0x .

(6)

2.2 Determine

2.2.1 2

1lim

1y

y y

y

(2)

2.2.2

5lim

3x

x

x (3)

[20]

30

Question 3


3.1.1 7

3 5

4

xy

x

(4)

3.1.2 33 sin y x x (3)

3.1.3 xy n e tan (2)


3.2.1 cot nx

dxx

(2)

3.2.2 3 2 1x x dx (3)

3.2.3 2

7

6

xdx

x x

[Hint: Use partial fractions] (9)

[23]

Question 4

4.1 Express (1+ i) in polar form. (3)

4.2 Find 101 i using DeMoivre’s theorem. (6)

4.3 Solve for x if 4 4sin sin 1 ; 0 2x x x (6)

Give your answer in terms of .

[15]

TOTAL :80

10.2 Preparation Paper II (October 2010) Question 1

1.1 The currents in an electrical network are given by the solution of the system:

1 2 3

1 3

1 2

0

8 10 8

6 3 12

I I I

I I

I I

where 1 2 3, and I I I are measured in amperes.

Solve for 2I only using Cramèr's rule. (6)

MAT1581/101

31

1.2 Expand 1711 y to four terms using the binomial theorem. (4)

1.3 Solve for x:

1.3.1 25 6n x nx n x (5)

1.3.2 22 64 log 64x (3)

[18]

Question 2

2.1 Given the curves

2 2

2

2 2

2 2

11

4 16

3 12 1

3 21

9 16

1 3 16

x yA

B y x

x yC

D x y

( ).

. ( )

.

.

2.1.1 Identify the circle and write down the coordinates of the centre. (1)

2.1.2 Identify the ellipse and write down the coordinates of the centre. (1)

2.1.3 Identify the hyperbola and write down the coordinates of the centre. (1)

2.2 Resolve into partial fractions: 2

3 2

6 2 2

2

x x

x x x

(7)

2.3 Represent the complex number 8 cos sin2 2

i

graphically and find the

rectangular form of the number. (3)

2.4 Find all the solutions for x if 3 27 0x j (6)

2.5 Find the angular speed, in radians per second, of a wheel running at 1700

revolutions per minute. (3)

[22]

Question 3

3.1 Differentiate the following with regard to x :

3.1.1 532 5y x x (2)

32

3.1.2 32 cos 3y x x (3)

3.1.3 2 37 .

tan

xx ey n

x

(4)


3.2.1 42 36 5 2 5x x x dx

(2)

3.2.2 2tan3 .sec 3x x dx (3)

3.2.3

42

1

2 4

xdx

x x

(3)

3.2.4 sin

1 cos

xdx

x

(2)

[19]

Question 4

4.1 The concentration of a drug in a patient’s bloodstream h hours after it was

injected is given by 2

0,17

2

hC h

h

. Determine

4.1.1 1

limh

C h

(1)

4.1.2 limh

C h

(3)

4.1.3 'C h the rate at which the concentration changes. (4)

4.2. Given the curve 3 22 15 6y x x x

4.2.1 determine the maximum and or minimum turning points of the curve. (6)

4.2.2 sketch the curve. (3)

4.2.3 calculate the area between the curve, the x -axis and the lines

0x and 1x . (6)

[23]

TOTAL: 82 (FULL MARKS: 80)

MAT1581/101

33

10.3 Preparation Paper III (May 2011) Question 1

1.1 Differentiate the following with regard to x :

1.1.1 3 3 7y x (2)

1.1.2 5 24 .cos(3 )y x x (3)

1.1.3 2 xy n x e ( . ) (2)

1.1.4 1

sec2

xy

x

(3)

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

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

or minimum. (4)

1.2.2 Find the current at these times. (2)

1.3 Determine

1.3.1 0

limcosx

x

x (1)

1.3.2 2

2

3 2lim

2 3x

x x

x x

(3)

[20]

Question 2

2.1 Resolve into partial fractions:

2

2

4

1 1

x

x x (5)


2.2.1

2

2

4

1 1

xdx

x x

(3)

2.2.2 2

3

3 sin

cos

x xdx

x x

(3)

2.2.3 2

1

2

xdx

x x

(3)

34

2.2.4 55 xe dxx

(2)

2.2.5 2 32 3

12 4x x dx

(4)

[20]

Question 3

3.1 Represent the complex number 2 2 i graphically and

find the polar form of the number. (3)

3.2 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)

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

3.4. Find the radius of a circle given that the area of a segment subtended

by an angle of 120° is 9,4 m2. (4)

3.5 Identify and sketch the following curve:

2 2

116 25

x y (4)

[20]

Question 4

4.1 Find the area bounded by the parabola 2 2 8,y x x the x –axis

and the lines 1x and 3x . (6)

4.2. 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). (6) 4.3 Solve for x:

4.3.1 3 27 7 7log log 2 logx x x (5)

4.3.2 2

12 16

x (3)

[20]

TOTAL: 80

MAT1581/101

35

10.4 Preparation Paper IV (October 2011)

Question 1

1.1 Expand 21 x to three terms using the binomial theorem. (4)

1.2 Solve for x:

1.2.1 2log 1 3 4 2x (3)

1.2.2 3 12xe (3)

1.2.3 321 27x (2)

1.2.4 33 tan tanx x for 0 2x .

Give your answer in terms of . (5)

1.3 Identify the following curves and rewrite the equations in standard form:

1.3.1 2 24 16x y (2)

1.3.2 2 2

2 11

9 5

x y

(1)

[20]

Question 2


2.1.1 2

20lim

1

x

x

e

x (2)

2.1.2 0

cos 1lim

2sinx

x

x

(3)


2.2.1 3 tan2y x x (2)

2.2.2 1

1

x

x

ey

e

(4)

2.2.3 22 1y n x (2)

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

Tg

,

36

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

dl (3)

2.4 Find the turning points of the function 3 22 5 4 1y x x x and determine

their nature. Find the y-intercept. Sketch the function. (10)

[26]

Question 3


3.1.1 (cos 2 sin )x x dx (2)

3.1.2 1

x

x

edx

e

(3)

3.1.3 2

1

2 1

xdx

x x

(4)

3.2 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)

[14]

Question 4

4.1 Resolve into partial fractions: 2

3

3 4 4

4

x x

x x

(7)

4.2 The relationship between the displacement, s, velocity, v, and acceleration,

MAT1581/101

37

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

(4)

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

1 2 2

3 1 4 41.

3 2 1

4.3 Rewrite 3z j in polar form and find 7z ,giving the answer in polar form.

(5)

4.4 The total impedance of an AC circuit containing two impedances Z1

and Z2 in parallel is given by 1 2

1 2

Z ZZ

Z Z

.

Find Z when Z1 = 2 j and Z2 = 3 + j (4)

[20]

10.5 Preparation Paper V (May/June 2012)

Question 1

1.1 Find the turning points on the curve 3 23 6 3 1y x x x and determine

their nature. (7)


1.2.1 0

lim 5x

x (1)

1.2.2 1

1lim

1t

t

t

(3)


1.3.1 3 4

cos2

n xy

x

(4)

1.3.2 24 sin 3xy e x (2)

1.3.3 2 3 5y x x (3)

[20]

38

Question 2

2.1 Expand 1

1 x

to three terms using the binomial theorem. (3)

2.2 Solve for x:

2.2.1 2log 1 log4 log 1x x (3)

2.2.2 1 3 42x xe e (4)

2.2.3 32 cos cos 0x x for 0 360x . (6)

2.3 A football stadium floodlight can spread its illumination over an angle

of 45° to a distance of 55m.

Determine to the nearest square meter the maximum area that is floodlit.(4) [20]

Question 3


3.1.1 22 3 1x x dx

(3)

3.1.2 22 1x

dxx

(2)

3.1.3 55cos sint t dt (3)

3.1.4 2

11

2

xdx

x x


3.2 The distance, s, covered by a train on a straight track in the first two seconds

is given by 2

020 1 ts e dt . Find s. (5)

[20]

Question 4

4.1 The simultaneous equations representing the currents flowing in an unbalanced,

three-phase, star-connected electrical network are as follows:

MAT1581/101

39

1 2 3

1 2 3

1 2 3

2,4 3,6 4,8 1,2

3,9 1,3 6,5 2,6

1,7 11,9 8,5 0

I I I

I I I

I I I

Find the I 3 using Cramèr's rule (8)

4.2 Solve for x if 2 2 5 0x x . Show the roots on an Argand diagram. (5)

4.3 Find the resultant force, 1 2f f , on a body when forces 1 10 45f and

2 8120f act on the body. Give the answer in rectangular form. (4)

4.4 Rewrite 2 3z j in exponential form. (3)

[20]

40

11 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,all papers are in this letter or on MyUnisa.

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

Answer: In some modules all follow-up letters are only posted on myUnisa, so no post is sent to students. If a tutorial letter is send by post and you owe money on your student account it will not be posted to you however you will still be able to access the letter on myUNISA.

12 SOURCES CONSULTED

Study Guides and past papers for MAT1581

13 CONCLUSION

The semester system have some disadvantages but it also 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.

We wish you success with your examination and future studies.

14 ADDENDUM

14.1 Errata Study guides

Page Correction 7 Activity 6

2.2 6213

29 d) second to last line of solution 24 = 6x 45 c) answer 32

3 2 or x x

47 Activity 2

(b) 22 4 3 0x x 48 a) answer 2

3y

69 (j) answer

6

2

xy

z

77 Activity 2

e) 42 2 123 10 2 10 3,6 10

MAT1581/101

41

42 12

4 8 12

0

5 10 3,6 10

5 10 3,6 10

4,4 10

100 Activity 1

c) 2 33 17x answer x = 2,79

105 Response 1

c) 2 33 17x 108 b) The solution is x = 316,7 116 f) The solution is 2,67 123 Example 2 4 4 2 2 3 46 4a b a a b ab b 3- 4a b

128 Response 3 T4 can simplify further

3 3

415 5

384 128

x xT

149

Similarly we can show that

a h

c kak chy

a bad cb

c d

175 Line above Activity 2.

Remove double brackets:

61 18

31 5 9 31 3 7x x

177 Replace line 8 and 9 from the top of the page: that is , 6 = 4A + C (e) Subtract equations (d) and (e): 0 = 3A

182 Last line: Answer:

2

11 3 11 3 2 5

1 3 1 32 3

x x

x x x xx x

206 f) 5th line ; sign error : 20 4 3x x 226

Conversion Factor: radians to degrees multiply by 180

Degrees to radians multiply by 180

243 Table b) Last entry under 360° must be 0

250 Spelling: Block middle of page NOTE: We do ……

254 Response 3 a) answer 203,58 or 336,42°

254 Response 3 b) Third quadrant 243,43° 262 Double angles

2 2cos2 cos sinA A A 264 Solution example 7. First figure x and y values must be swapped.

Thus y = 3 and x = 4

266 Example 10 Remove x = 0° as a solution.

269 3. 4th line must read: 2tan tan 2 0

271 7. Draw the graph of 2sin 30 0 360 .y

42

274 4a) 3rd line:

22 225 60h h

4b)

60 mmˆsin ½ 0,7184,5 mm

ˆ½ 0,789 radians

ˆ 1,579 radians

=90,47°

=90°28'

O

O

O

276 8.3 sin 0

0 ; 180°

x

x

8.4 Sign error in 6th line: 8.4 sin 2x

277 8.6 Method 1: Line 7 sin 2 0

2 0 or 2 180°

0 90

278 8. Line 1 and 2 of the solution should be swapped.

5 812 15

5 812 15

tan tan9.2 tan

1 tan tan

1

7 11

60 921

220

x yx y

x y

288 Solution: Point (2; - 1,75) not placed in correct position. 289 4th line from bottom of page

degree sign omitted

2 sin 225 290 Spelling: Activity 1: 1……. co-ordinates 291 Solution: Let Ship A be “Always Leaks” and ship B be “Mayday, Mayday”. 303 Figure 2 : Add A on left side of line, that is line AB 306 Activity 3 c) …parallel to the line 2 3 4 0y x

346 2c) answer

1 12 2

2 2 2

c Given 7;3 ; ' 3;3 and 6;3 thus the major axis is along 3.

The centre is at the midpoint of ' that is at (2;3)

half the length of ' 7 3 10 5

length from centre to 6;3 = 6 2 4

16 2

V V F y

VV

a VV

c F

c a b

2

2 2

5

3

2 31

25 9

b

b

x y

326 and 328 The shading for the greater and smaller than graphs is lost in print production. 345 1(d) foci (- 1 ; 2 7 )

MAT1581/101

43

366 Activity 4

C (1 – j 6) 381 Last line 4 3z j

391 Redo example 1 on your own to find

Thus 2 30 5 45 4 120 7, 268 6 9, 42 39,54°j

405 3. 4 2 , 3 3 and 2 - 5A j B j C j

407 2b)

316 9 tan4

5180 36,9

216,05 cis 143,1

r

z

420 and 421 Activity 1 number 5

Calculate the value of 22f x h x 2

454 b) 3 1

'3

y tt t

464 Answer example 7

2

1

1x x

466 Response 4(b)

cos' .sin

2

tf t t t

t

472 Example 2

23dy

xdx

478 Answer b) 2

2 2

2 1

1

x

x x

481 (c ) n ny x

485 Response 3 (e)

4

6

5

1

dy x

dx x

487 Response 5:

122 2½( j) ' ½ 1 5 3y x x x

488 Response 6

e) 1

'4 2

f

Response 8

2( ) 9sin 2 3 cos 2 3b x x

497 Response 1 (i) Delete second answer.

504 Activity 2 : 2. Delete question. 508 Response 2

(i). Do not use implicit differentiation:

44

12

12

2 2

2

25 25

12 25

24

At 4 : Gradient =3

y x x

dyx x

dxdy

xdx

(Implicit differentiation is in Mathematics II.)

515 Top of the page

The graph of the second derivative ''y f x shows clearly that when:

2

2

2

2

2

2

is a maximum, '' 0;

is a minimum, '' 0;

has an inflection point, '' 0.

d yy f x f x

dx

d yy f x f x

dx

d yy f x f x

dx

517 Replace line 6 from bottom of the page:

2 22 23 3

2223

6 4 3 4 1 2 1''

x x x x xf x

x

Replace line 5 from the bottom of the page:

21

341

3

2 0At 1: '' 1 0 positivex f

528 Response 1 f)

3 2

For 1,36 '' 6 1,36 4 0

1,36 is a point of minimum value.

Value of 1,36 2 1.36 11 1,36 12

2,52 3,70 14,96 12

20,74

x f x

x

y

530 3 002 1Length 600

300

2

536 Solution Example 2: Line 4 41y x

547

2

2

2

2

2

2 3

56If 2, then

8

0 if 2. Therefore the curve has a minimum.

3 6 28If 2, then

27 6

0

The curve has a maximum value at 2.

d yx

dx

d yx

dx

d yx

dx

x

MAT1581/101

45

564

1 12 2

1 12 2

1 1 1 12 2 2 2

12

12

1 12 2

1 11 12 1 2 1

1 11 1 1 11 1 1 1 11 2 1 2 1 1 12 1 2 1 1 1

1 11 11 1 1

1 1 1 1

1)

1

x xx x

x xx x x xx x x x xx x x x

x xx x x

x x x x

x

dre

dx

x

597 Last line: Add + C to answer. 609

1 1) .

1 1

12

'Now it is possible to recognise that the integrand is of the form

2

2 1

dxb dx

xx x x

duu

f x

f x

n u C

n x C

666 g)

22 9 35

1 2 3 1 2 3

2 3 1 3 1 2

x x A B C

x x x x x x

A x x B x x C x x

x +1 x - 2 x + 3

668 f) Let 5 2sec3 , then 6sec tanu x du x x dx 3 3 669

2

2

coscos cos

1 0

1

)

1

b

a

b

a

x

e

c I du

u

e e e

e e

46

14.2 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

47

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

jrnren

bjbee

rrerre

jrre

nkn

krz

nz

njnrnrr

j

ajba

jj

j

nn

n

nnn

.11

sincos.10

sinandcos

sincos.9

1,,2,1,0with360

:rootsdistinct has.8

sincos

Theorem sMoivre' De.7

`

11

1

48

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

49

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

50

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

51

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

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