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Tutorial letter 101/3/2018

Calculus B

MAT1613

Semesters 1 & 2

Department of Mathematical Sciences

IMPORTANT INFORMATION:

This tutorial letter contains important information about yourmodule.

MAT1613/101/3/2018

CONTENTS

Page

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

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

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

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

3 LECTURER AND CONTACT DETAILS ..................................................................5

3.1 Lecturer ...............................................................................................................................5

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

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

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

4.1 Prescribed book .................................................................................................................6

4.2 Recommended books ........................................................................................................7

4.3 e-Reserves ..........................................................................................................................7

4.4 Library services and resources information .........................................................................7

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

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

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

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

8.1 Assessment plan ................................................................................................................9

8.2 Assignment numbers............................................................................................................9

8.2.1 General assignment numbers ...........................................................................................9

8.2.2 Unique assignment numbers ............................................................................................9

8.2.3 Assignment due dates .........................................................................................................9

8.3 Submission of assignments............................................................................................10

8.4 The assignments ..............................................................................................................11

8.5 Other assessment methods ...............................................................................................33

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MAT1613/101/3/2018

9 The examination ...............................................................................................................33

10 FREQUENTLY ASKED QUESTIONS ....................................................................33

11 SOURCES CONSULTED ..........................................................................................33

12 IN CLOSING ..................................................................................................................33

3

1 INTRODUCTION

Welcome to module MAT1613 on Calculus. I hope you will find it both interesting and rewarding.This module is offered as a semester module. You will be well on your way to success if you startstudying early in the semester and resolve to do the assignments properly.

I hope you will enjoy this module, and wish you success with your studies.

Tutorial matterYou will find all the study material online. The topics and outcomes for this module are specified

in your study guide and the topics must be studied in both the textbook and study guide.A list of topics with refernce to page numbers in the textbook will be posted on myUnisa.Tutorial Letter 101 and the study guide which will be available at the time of registration all othertutorial letters and some extra material will be uploaded later in the semester. The solutions tothe assignments will also be uploaded online about a week after the closing date.Please access the myUnisa website at http://my.unisa.ac.za

Tutorial Letter 101 contains important information about the scheme of work, resources and assign-ments for this module. I urge you to read it carefully and to keep it at hand when working throughthe study material, preparing the assignments, preparing for the examination and addressing ques-tions to your lecturers.

In this tutorial letter you will find the assignments as well as instructions on the preparation andsubmission of the assignments. This tutorial letter also provides information with regard to otherresources and where to obtain them. Please study this information carefully.

Certain general and administrative information about this module has also been included. Pleasestudy this section of the tutorial letter carefully.

You must read all the tutorial letters carefully, as they always contain important and, sometimes,urgent information.

2 PURPOSE AND OUTCOMES

2.1 Purpose

This module will be useful to students interested in developing those skills in integral and differentialcalculus which can be used in the natural economic, social and mathematical sciences. Studentscredited with this module will have the knowledge of those basic techniques in differential andintegral calculus which are used in related rates problems, graph sketching, evaluating integrals,calculating volumes and areas, and in maximum and minimum problems.

4

MAT1613/101/3/2018

2.2 Outcomes

2.2.1 Calculate and use the derivatives of a function to sketch a graph of the function.

2.2.2 The first derivative is used to determine the relationship between the rates of change ofvarious quantities in the rates-of-change word problem.

2.2.3 The student is able to solve maximum or minimum word problems using the theory of deriva-tives.

2.2.4 Ability to use L’Hopital’s rule to determine limits of indeterminate forms.

2.2.5 Calculation of the volumes of solids of revolution.

2.2.6 An improper integral is tested for convergence or divergence and evaluated if convergent.

2.2.7 The student is able to use various integration techniques to evaluate integrals.

2.2.8 The student is able to calculate the Taylor polynomial of any order at a given point.

3 LECTURER AND CONTACT DETAILS

3.1 Lecturer

The lecturer responsible for this module is:

Dr L LindeboomRoom C6-53E-mail address: [email protected] Florida Campus

All queries that are not of a purely administrative nature but are about the content of this moduleshould be directed to me. Email is the preferred form of communication to use. Contact me [email protected]. If you phone me please have your study material with you when you contactme. If you cannot get hold of me, leave a message with the Departmental Secretary. Please clearlystate your name, time of call and how I can get back to you. You are always welcome to come anddiscuss your work with me, but please make an appointment before coming to see me. Pleasecome to these appointments well prepared with specific questions that indicate your own efforts tohave understood the basic concepts involved.

You are also free to write to me about any of the difficulties you encounter with your work for thismodule. If these difficulties concern exercises which you are unable to solve, you must send your

5

attempts so I can see where you are going wrong, or what concepts you do not understand. Mailshould be sent to:

Dr L LindeboomDepartment of Mathematical SciencesPO Box 392UNISA0003

.

3.2 Department

The number of the secretary of the Department Mathematical Sciences at the Unisa Florida cam-pus is: 011 670 9147The fax number of the Department Mathematical Sciences at the Unisa Florida campus is: 011670 9171

3.3 University

If you need to contact the University about matters not related to the content of this module,please consult My studies @ Unisa.It contains information on how to contact the University (e.g. to whom you canwrite for different queries, important telephone and fax numbers, addresses and details ofthe times certain facilities are open).

Always have your student number at hand when you contact the University.

4 RESOURCES

4.1 Prescribed book

The prescribed textbook is

James StewartCALCULUS Early Transcendentals(International Metric version)Cengage Learning8th EditionISBN 978-1-305-27237-8

Make sure that you obtain the correct book (NB make very sure that the ISBN number is identi-cal to the above.) Please refer to the list of official booksellers and their addresses in the publication

6

MAT1613/101/3/2018

my Studies @ Unisa . Prescribed books can be obtained from the University’s official booksellers.If you have difficulty in locating your book(s) at these booksellers, please contact the PrescribedBook Section at Tel: 012 429-4152 or e-mail [email protected].

4.2 Recommended books

There are no recommended books for this module.However any book on Calculus which contains

the relevant study topics can be used to obtain more exercises. Remember however that thenotation might not be the same.

4.3 e-Reserves

The links to some e-reserves will be uploaded on myUnisa.

4.4 Library services and resources information

For brief information, go to www.unisa.ac.za/brochures/studies

For detailed information, go to http://www.unisa.ac.za/library. For research support and services ofpersonal librarians, click on ”Research support”.

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

For information on the various student support systems and services available at Unisa (e.g. stu-dent counselling, tutorial classes, language support), please consult My studies @ Unisa .Study groups

It is advisable to have contact with fellow students. On the MAT1613 website you can do so underdiscussion forums.

7

Another way to do this is to form study groups. The addresses of students in your area may beobtained from the following department:

Directorate: Student Administration and RegistrationPO Box 392UNISA0003

myUnisa

All resources can be accessed through the myUnisa learning management system. You need a@mylife account for this.To go to the myUnisa website, start at the main Unisa website, www.unisa.ac.za, and then clickon the “myUnisa” link below the orange tab labelled “Current students”. This should take you tothe myUnisa website. You can also go there directly by typing my.unisa.ac.za in the address bar ofyour browser. Please consult My studies @Unisa.

e-Tutoring and Tutorial classesUnisa offers online tutorials (e-tutoring) to students registered for modules at NQF level 5 and 6,this means qualifying first year and second year modules. Please communicate with your moduleleader to find out if any of the modules that you have registered for falls in this category.Once you have been registered for a qualifying module, you will be allocated to a group of studentswith whom you will be interacting during the tuition period as well as an e-tutor who will be yourtutorial facilitator. Thereafter you will receive an sms informing you about your group, the name ofyour e-tutor and instructions on how to log onto MyUnisa in order to receive further information onthe e-tutoring process.Online tutorials are conducted by qualified E-Tutors who are appointed by Unisa and are offeredfree of charge. All you need to be able to participate in e-tutoring is a computer with internetconnection. 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 MyUnisawhere you are expected to connect with other students in your allocated group. It is the role of thee-tutor to guide you through your study material during this interaction process. For your to get themost out of online tutoring, you need to participate in the online discussions that the e-tutor will befacilitating.There are modules which students have been found to repeatedly fail, these modules are allocatedface-to-face tutors and tutorials for these modules take place at the Unisa regional centres. Thesetutorials are also offered free of charge, however, it is important for you to register at your nearestUnisa Regional Centre to secure attendance of these classes.

6 STUDY PLAN

Study plan Semester 1 Semester 2Outcomes 2.2.1 to 2.2.4 to be achieved by 12 March 22 AugustOutcomes 2.2.5 to 2.2.8 to be achieved by 10 April 12 SeptemberWork through previous exam paper 20 April 30 SeptemberRevision 30 April 15 October

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MAT1613/101/3/2018

7 PRACTICAL WORK AND WORK INTEGRATED LEARNING

There are no practicals for this module.

8 ASSESSMENT

8.1 Assessment plan

In each semester there are three assignments for MAT1613. Assignment 01 is a written assign-ment and Assignment 02 and 03 are multiple choice assignments. The solutions to 01, 02 and03 will be available on myUnisa. The questions for the assignments for both semesters are givenin 8.4. All assignments count towards your semester mark. Please make sure that you answerthe questions for the semester for which you are registered. When marking the assignments,constructive comments will be made on your work (onsceen marking), which will then be returnedto you. The assignments and the comments on these assignments constitute an important part ofyour learning and should help you to be better prepared for the next assignment and the examina-tion. Some general comments and examples will also be available on myUnisa. Please do not waituntil you receive Assignment 01 back before you start working on Assignments 02 and 03.

To be admitted to the examination you need to submit the first assignment before the compulsorydate.

Your semester mark for MAT1613 counts 20% and your exam mark 80% of your final mark. Thefirst assignment counts 20% the second assignment counts 40% and the third assignment counts40% towards the year mark.

8.2 Assignment numbers

8.2.1 General assignment numbers

The assignments are numbered as 01, 02 and 03 for each semester.

8.2.2 Unique assignment numbers

Please note that each assignment also has their own unique assignment number which needs toincluded in your assignment upon submission.

8.2.3 Assignment due dates

The due dates for the submission of the assignments in 2018 are:

Assignment number Semester 1 Semester 201 06 April 31 August02 13 April 19 September03 26 April 05 October

9

8.3 Submission of assignments

You must submit written assignments and multiple choice assignments (following the correct proce-dure) electronically via myUnisa. Assignments may not be submitted by fax or e-mail. For detailedinformation on assignments, please refer to myStudies @ Unisa.

To submit an assignment via myUnisa:

• Go to myUnisa.• Log in with your student number and password.• Select the module.• Click on assignments in the menu on the left-hand side of the screen.• Click on the assignment number you wish to submit.• Follow the instructions.

PLEASE NOTE: Although students may work together when preparing assignments, each studentmust write and submit his or her own individual assignment. In other words, you must submit yourown calculations in your own words. It is unacceptable for students to submit identical assign-ments on the basis that they worked together. That is copying (a form of plagiarism) and none ofthese assignments will be marked. Furthermore, you may be penalised or subjected to disciplinaryproceedings by the University.

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MAT1613/101/3/2018

8.4 The assignments

SEMESTER 1ASSIGNMENT 01

Graphs of functions, rates of change, optimization, L’Hopital’s ruleFIXED CLOSING DATE: 06 April 2018Unique assignment number: 689189

1. How fast is the perimeter of a square increasing when the area is 9 cm2 and the area isincreasing at 18 cm2/sec?

[10]

2. Let f be the function defined byf (x) =

x

(2x− 1)2.

(a) Determine the y–intercept. (1)

(b) Determine the horizontal and vertical asymptotes. (3)

(c) Use the sign pattern for f ′ (x) to determine

(i) the interval(s) over which f rises and where it falls; (5)(ii) the local extrema. (2)

(d) Use the sign pattern for f ′′ (x) to determine

(i) where the graph of f is concave up and where it is concave down; (5)(ii) the inflection point(s) (if any). (4)

[20]

3. Suppose that the area of a rectangle is 16 cm2. What is the smallest length of the diagonalof such a rectangle? [10]

4. Use L’Hopital’s rule to obtain the following limits:

(a) limx→0+

√x ln 2x (5)

(b) limx→∞

(lnx)1/2x (5)[10]

TOTAL: [50]

11

ONLY FOR SEMESTER 1 STUDENTSASSIGNMENT 02

Substitution and integration by partsFIXED CLOSING DATE: 13 April 2018

Unique Number: 720829

This is a multiple-choice assignment, so you must submit your answers either via myUnisaor on a mark reading sheet. Please consult the booklet My studies @ Unisa before usingmyUnisa or completing the mark reading sheet.

Determine the following integrals and choose the corrrect answer.

1. Calculate the following limit:

L = limx→0

e2x − ex − xx2

.

1. L = 0

2. L = −32

3. L = 32

4. L = −15. None of the above.

2. Calculate the following limit:

L = limx→π

4

cos 2x1√2− sin x

The correct answer is

1. L =√2

2. L = 2√2

3. L =√22

4. L = 1√2

5. None of the above.

12

MAT1613/101/3/2018

3. I =

∫((ln 3x) + 2)−2

xdx, x > 1


1. I = −13((ln 3x) + 2)−3 + c

2. I = 3((ln 3x) + 2)−3 + c

3. I = −((ln 3x) + 2)−1 + c

4. I = ((ln 3x) + 2)−1 + c


4. I =∫(4 + sin θ

2)−

12 cos θ

2dθ


1. I = 12(4 + cos θ

2)12 + c

2. I = −14(4 + cos θ

2)12 + c

3. I = −2(4 + cos θ2)−

12 + c

4. I = 4(4 + cos θ2)12 + c


5. I =∫ cos(

√x+ π

6)

√x

dx


1. I = −2√x sin(

√x+ π

6) + c

2. I = −32sin(√x+ π

6)

3. I = 23sin(√x+ π

6)

4. I = 12sin(√x+ π

6)


6. I =

∫[1− (lnx)]

32dx

x2

, x > 1


1. I = 2(1− lnx)52 + c

13

2. I = −45(1− lnx)

52 + c

3. I = 2(1− lnx)12 + c

4. I = −35(1− lnx)

12 + c


7. I =

∫tan−1 3xdx


1. I = 12x2 tan−1 3x− 1

6tan−1(1 + 9x2) + c

2. I = 12x2 tan−1 3x− 1

18tan−1(1 + 9x2) + c

3. I = x tan−1 3x− 16ln(1 + 9x2) + c

4. I = x tan−1 3x− 118ln(1 + 9x2) + c


8. I =

∫x3 ln

√xdx where x > 1


1. I = 3x2 ln√x− 6

5x

52 + c

2. I = 32x2 lnx− 2x

12 + c

3. I = 18x4(lnx− 1

2) + c

4. I = 18x4(lnx− 1

4) + c


9. I =

∫x3e− xdx

The correct answer is:

1. I = − 13e−x(x− 1) + c

2. I = −e−x(13x− 1) + c

3. I = −13e−x(x+ 1)

4. I = −e−x(13x+ 1) + c


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MAT1613/101/3/2018

10. I =

∫x sin(

x

3)dx


1. I = 3x cos(x

3)− 3 sin(x

3) + c

2. I = −3x cos(x3) + 9 sin(x

3) + c

3. I = − x cos(x

3) + 3 sin(x

3) + c

4. I = 3x cos(x

3)− 9 sin(x

3) + c


11. I =

∫x2 sin(

x

3) dx

The correct answer is:

1. I = (27− 3x2) cos x3+ 9x sin x

3+ c

2. I = (3x2 − 27) sin x3+ 9x cos x

3+ c

3. I = (54− 3x2) cos x3+ 18x sin x

3+ c

4. I = (3x2 − 54) sin x3+ 18x cos x

3+ c


12. I =∫ 2 cos(ex)

sin2(ex)exdx


1. I = ln(sin(ex) + c

2. I = −2 ln(sin(ex) + c

3. I = −1sin(ex)

+ c

4. I = −2sin(ex)

+ c


13. I =∫ sin 1

x

x2dx


1. I = 2x−1 cos 1x+ c

15

2. I = −12x−1 sin 1

x+ x−2 cos 1

x+ c

3. I = cos 1x+ c

4. I = 1xsin 1

x+ c


14. I =

∫ 1

0

x2 cos πx dt


1. I = −1

2. I = − 2π2

3. I = 2π

4. I = 2

π3


15. I =∫x3√x2 − 1dx


1. I = 13(x2 − 1)

32 − 1

5(x2 − 1)

52 + c

2. I = 23(x2 − 1)

32 − 1

3(x2 − 1)

52 + c

3. I = 15(x2 − 1)

52 − 1

3(x2 − 1)

32 + c

4. I = 23(x2 − 1)

52 − 1

3(x2 − 1)

32 + c


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MAT1613/101/3/2018


Integration, Taylor seriesFIXED CLOSING DATE: 26 April 2018


This is a multiple-choice assignment, so you must submit your answers either via myUnisaor on a mark reading sheet. Please consult the booklet My studies @ Unisa before usingmyUnisa or completing the mark reading sheet.


1. I =

∫−3x2

x9 + 1dx


1. I = − tan−1(x3) + c

2. I = −3 tan−1(x3) + c

3. I = −2 tan−1(x4) + c

4. I = 2 tan−1(x4) + c


2. I =

∫3x√1−9x2

dx


1. I = −13

√1− 9x2 + c

2. I = −12

√1− 9x2 + c

3. I = −13sin−1 3x+ c

4. I = −12sin−1(3x) + c


17

3. I =

∫1− 9x2

3xdx , x > 1

The correct alternative is

1. I = ln(sin 3x) + c

2. I = 3 ln(sin 3x) + c

3. I = 13lnx− 3

2x2 + c

4. I = ln 3x− sin−1 3x+ c


4. I =

∫ √1− 9x2

3xdx

With trigonometric substitution this integral reduces to the following:


1. I = 3[∫cosec θ cot θdθ] where we used the substituion with 3x = sin θ etc.

2. I = −3 [∫cosec θ cot θdθ] where we used the substituion with 3x = sin θ etc.

3. I = 13[∫cosec θdθ −

∫sin θdθ] where we used the substituion with 3x = sin θ etc.

4. I =∫

cos2θsinθ

dθ


5. I =

∫dx

1 + 3ex − 2e−x


1. I = 35ln(

3ex + 2

ex − 1) + c

2. I =3

5ln(3ex + 2 ) + 1

5ln(ex − 1) + c

3. I =3

5ln(3ex − 2 ) + 1

5ln(ex + 1) + c

4. I = 15ln(

3ex − 2

ex + 1) + c

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MAT1613/101/3/2018


6. I =

∫cosx

1− cosxdx

With z-subsitution we obtain the following form which needs to be solved with partial frac-tions:The correct answer is

1. I =∫ 2(1+z2)

z2(1−z2)dz

2. I =∫ 2(1+z2)

z(1−z2)dz

3. I =∫ (1−z2)

z(1+z2)dz z

4. I =∫ (1−z2)

z2(1+z2)dz


7. I =

∫2x3 − 3x2 − 3

(x2 + 1)2dx

With partial fractions this integral reduces to the following integral:


1. I = 2∫

1x2+1

dx− 3∫

xx2+1

dx− 2∫

x(x2+1)2

dx

2. I = 2∫

1x2+1

dx− 3∫

xx2+1

dx+ 2∫

x+1(x2+1)2

dx

3. I = 2∫

xx2+1

dx− 3∫

1x2+1

dx− 2∫

x(x2+1)2

dx

4. I =∫

xx2+1

dx− 2∫

1x2+1

dx− 3∫

x(x2+1)2

dx


8. I =

∫ 0

1

−3(2 + u)(1 + 2u)

du

The integral reduces with partial fractions to:


1. I = ln |(2 + u)(1 + 2u)| |01

2. I = ln |(2 + u)(1 + 2u)| |10

3. I = ln∣∣ u+21+2u

∣∣ |0119

4. I = ln∣∣ u+21+2u

∣∣ |105. None of the above.

9. I =

∫2dx√4− 4x2


1. I = 2sin−1(2x) + c

2. I = sin−1(x) + c

3. I = 2 sin−1(x) + c

4. I = sin−1(2x) + c


10. I =

∫ 2

−∞

dx

(3− 5x)3


1. I = 172

2. I = 110.72

3. I = 170

4. I =1

5.72


11. I =

∫ 1

0

2 ln 3xdx

Why is this an improper integral? What is correct form of the limit of the answer?

The correct answer is to the above questions are:

1. limx→1 2 ln 3x = 0 and limt→1[2x(ln 3x) − 2x] | t02. limx→1 ln 3x = 0 and limt→1[2x(ln 3x) − 2x] | t03. limx→0+ ln 3x = −∞ and limt→0[2x(ln 3x) − 2x] |1t4. limx→0+ ln 3x = ∞ and limt→0[ x(ln 3x) − 2x] |1t5. None of the above.

20

MAT1613/101/3/2018

12. The area bounded by the curves x = y2 + 2 and the line y = 3x + 2 and the y − axisfrom intersection point to intersection point is rotated about the x− axis. The volume ofresulting solid of revolution will be given by the formula?The correct answer is

1. V (x) =

∫ 11

2

[π(y − 2)− π(y−2

2)2]dy.

2. V (x) =

∫ 11

2

[π(y − 2)2 − π(y−2

2)2]dy.

3. V (x) =

∫ 3

0

[π(3x+ 2)2 − π(x2 + 2)2] dx.

4. V (y) =

∫ 3

0

[π(x2 + 2)2 − π(3x+ 2)2] dx.


13. The area bounded by the curves x = y2 + 2 and the line y = 3x + 2 from intersectionpoint to intersection point is rotated about the line x = 3. The volume of resulting solidof revolution will be given by the formula?The correct answer is

1.

V (y) =

∫ 3

0

[π(3− (3x+ 2))2 − π(3− (x2 + 2))2

]dx.

2.

V (y) =

∫ 3

0

[π(3− (x2 + 2))2 − π(3− (3x+ 2))2

]dx.

3.

V (y) =

∫ 11

2

[π(3− (

√y − 2))2 − π(3− (

y − 2

3) )2]dy.

4.

V (y) =

∫ 11

2

[π(3− (

y − 2

3) )2 − π(3− (

√y − 2))2

]dy


21

14. Find the Taylor polynomial of order n for f(x) = e−2x in a = 1?


1. Pn,1(f(x)) =∑n

k=1(−2)k−1 (x−1)

k−1

k!e2k

2. Pn,1(f(x)) =∑n

k=0(−2)k−1 (x−1)

k−1

k!e2k

3. Pn,1(f(x)) = 1e2

∑n

k=0

(−2)k(x− 1)k

k!

4. Pn,1(f(x)) = 1e2

∑n

k=1(−2)k−1(x− 1)k−1


15. Determine the derivative of order n for the function f(x) = 1√x


1. fn(x) = (−1)n+1 1.3.5....(2n−1)2n

x−(n+12)

2. fn(x) = (−1)n 1.3.5....(2n−1)2n

x−(n+12)

3. fn(x) = (−1)n 1.3.5....(2n+1)2n+1 x−(n+

12)

4. fn(x) = (−1)n+1 1.3.5....(2n−1)2n+1 x−(n+

12)


22

MAT1613/101/3/2018


Graphs of functions, rates of change, optimization, L’Hopital’s ruleFIXED CLOSING DATE: 31 August 2018


1. The volume of a cube is increasing at a rate of 1200 cm3/min at the moment when thelengths of the sides are 20cm. How fast are the lengths of the sides increasing at thatmoment? [10]

2. Let f be the function defined by

f (x) =9 + x2

3− x2.

(a) Determine the y–intercept. (1)(b) Determine the horizontal and vertical asymptotes. (3)(c) Use the sign pattern for f ′ (x) to determine

(i) the interval(s) over which f rises and where it falls; (5)(ii) the local extrema. (2)

(d) Use the sign pattern for f ′′ (x) to determine(i) where the graph of f is concave up and where it is concave down; (5)(ii) the inflection point(s) (if any). (4)

[20]

3. The sum of two non-negative numbers is 36. Determine the numbers if the differencebetween between their square roots is a maximum. [10]

4. Use L’Hopital’s rule to obtain the following limits:

(a)

limx→∞

ln(lnx)

x

(5)(b)

limx→0

(ex + x)1/x

(5)[10]

TOTAL: [50]

23


Substitution and integration by partsFIXED CLOSING DATE: 19 SEPTEMBER 2018


This is a multiple-choice assignment, so you must submit your answers either viamyUnisa or on a mark reading sheet. Please consult the booklet My studies @ Unisabefore using myUnisa or completing the mark reading sheet.

Determine the following limits and integrals and choose the corrrect answer.

1.L = lim

x→π4

1− sin 2x

1 + cos 4x


1. L = 14

2. L = − 14

3. L = 18

4. L = −18


2.L = lim

x→0+[1

x− 2

ln(1 + 2x)]


1. L = 0

2. L = −23. L = 2

4. L = 4

5. None of the above

3. I =∫ x− 2√

5 + 4x− x2dx


1. I = − 2(5 + 4x− x2 ) 12 + c

24

MAT1613/101/3/2018

2. I = (5 + 4x− x2 ) 32 + c

3. I = − (5 + 4x− x2 ) 12 + c

4. I = −14(5 + 4x− x2 ) 1

2 + c


4. I =

∫1

2(1 +√x)√xdx


1. I = (1 +√x)−

32 + c

2. I = 2 ln(1 +√x) + c

3. I = ln(1 +√x) + c

4. I =√x√

1 +√x+ c


5. I =

∫sin θ√

1 + 3 cos θdθ


1. I = −13(1 + 3 cos θ)

12 + c

2. I = 13(1 + 3 cos θ)

12 + c

3. I = −23(1 + 3 cos θ)

12 + c

4. I = 23(1 + 3 cos θ)

12 + c


6. I =

∫ √2

2

0

sin−1 x√1− x2

dx


1. I = π2

18− 1

2, I = π2

18− π2

8

3. I = π2

32− π2

18

4. I = π2

32


7. I = I =

∫ 16

4

dx

x−√x


1. I = ln 4

2. I = ln 3

25

3. I = ln 9

4. I = ln 6


8. I =∫x−

12 ln 2xdx


1. I =√x(ln 2x− 2) + c

2. I = 2√x(ln 2x− 2) + c

3. I =√x(ln 2x− 4) +c

4. I = 2√x(ln 2x− 1) + c


9. I =

∫(tan 2x)−3/2 sec2 2x dx


1. I = −1√tan 2x

+ c

2. I = 1√tan 2x

+ c

3. I = 2√tan 2x

+ c

4. I = −2√tan 2x

+ c


10. I =

∫e−x sin 2xdx


1. I = −25e−x cos 2x− 1

5e−x sin 2x+ c

2. I = −15e−x cos 2x− 2

5e−x sin 2x+ c

3. I = 25e−x cos 2x− 1

5e−x sin 2x+ c

4. I = −25e−x cos 2x+ 1

5e−x sin 2x+ c


11. I =∫x lnx2dx


1. I = 2x lnx− 14x4 + c

2. I = 12x2[2 lnx− 1] + c

3. I = x2[lnx− 14] + c

4. I = x2[2 lnx− 12] + c

26

MAT1613/101/3/2018


12. I =∫ 9

1

1√x(1 +

√x)2

dx


1. I = −12

2. I = 12

3. I = −32

4. I = 32


13. I =

∫tan−1 3x dx


1. I = ln 3x− 13tan−1 3x+ c

2. I = x − 16x tan−1 3x + c

3. I = x tan−1 3x− 16ln(1 + 9x2) + c

4. I = x tan−1 3x− 13ln(1 + 9x2) + c


14. I =

∫ π2

4

0

1√xcos(2π +

√x)dx


1. I = 1

2. I = −13. I = 2

4. I = −25. None of the above.

15. I =

∫ 2

0

xex2dx


1. I = 14(e4 − e)

2. I = 12(e4 − 1)

3. I = 12(e4 − e)

4. I = (e4 − 1)


27


Integration, Taylor seriesFIXED CLOSING DATE: 5 OCTOBER 2018


This is a multiple-choice assignment, so you must submit your answers either viamyUnisa or on a mark reading sheet. Please consult the booklet My studies @ Unisabefore using myUnisa or completing the mark reading sheet.


1. I =

∫x3√

1− 3x2dx


1. I = −x2

3(1− 3x2)

12 − 2

9(1− 3x2)

32 + c

2. I = x2

3(1− 3x2)

12 + 2

9(1− 3x2)

32 + c

3. I = −x2

2(1− 3x2)

12 + 2

27(1− 3x2)

32 + c

4. I = −x2

3(1− 3x2)

12 − 2

27(1− 3x2)

32 + c


2. I =

∫ √x2−4x

dx


1. I =√x2 − 4 + 2 sec−1(x

2) + c

2. I =√x2 − 4− 2 sec−1(x

2) + c

3. I =√x2 − 4− 2 tan−1(x

2) + c

4. I =√x2 − 4 + 2 tan−1(x

2) + c


3. I =

∫ √1−16x2x2

dx

The correct alternative is

1. I =−√

1−16x24x

− sin−1 4x+ c

2. I =−√

1−16x24x

− 4 sin−1 4x+ c

3. I =

√1−16x24x

− sin−1 4x+ c

28

MAT1613/101/3/2018

4. I =−√

1−16x2x

− 4 sin−1 4x+ c


4. I = 3

∫ 4

1

dx

(x− 1)2 + 9The correct answer is

1. I =π

2

2. I =π

4

3. I =π

6

4. I =π


5. I =

∫ 3

2

dx√−x2 + 4x− 3


1. I =π

3

2. I = −π2

3. I =π

24. I = 0


6. I =

∫dx

1 + sin x+ cosx

After you have used z-substitution (z = tan x2etc.) the above integral becomes:


1. I =∫

11+z2

dz

2. I =∫

11+z

dz

3. I =∫

2(1+z)2

dz

4. I =∫

22z+1

dz


7. I =

∫10x2 + 4x+ 2

(1 + 4x2)(x+ 2)dx

We use the method of partial fractions and the above integral becomes:The correct answer is

29

1. I =∫

4x−11+4x2

dx+∫ 1

(x+ 2)dx

2. I = −∫

6x1+4x2

dx+∫ 4

(x+ 2)dx

3. I =∫

2x1+4x2

dx+∫ 2

(x+ 2)dx

4. I =∫

2x−31+4x2

dx+∫ 2

(x+ 2)dx


8. I =

∫ ∞1

u−1(u+1)(u2+1)

du

First applying the technique of integration by parts and then for the improper integralthe limit, the integral becomes:


1. I = limt→∞ ln u2+1|u+1| |

t1

2. I = limt→∞ ln√u2+1

(u+1)2|t1

3. I = limt→∞ ln |u+1|√u2+1|t1

4. I = limt→∞ ln√u2+1|u+1| |

t1


9. I =

∫5dx

4 sinx+ 3 cosxThe correct answer is

1. I = 3 ln

∣∣∣∣tan x2+ 1

tan x2− 3

∣∣∣∣+ c

2. I = ln

∣∣∣∣ tan x2− 3

3 tan x2+ 1

∣∣∣∣+ c

3. I = ln

∣∣∣∣3 tan x2+ 1

tan x2− 3

∣∣∣∣+ c

4. I = 13ln

∣∣∣∣tan x2− 3

tan x2+ 1

∣∣∣∣+ c


10. I ==

∫ 1

-∞

1

(3− x)3dx


1. I = − 14

2. I = 18

30

MAT1613/101/3/2018

3. I = −2

4. I =1

2


11. I =

∫ ∞0

1

e2xdx


1. I =1

22. I = 2

3. I = ln 2

4. I =∞5. None of the above.

12. The area bounded by the curves x = y2

2and the line y = 2 in the first quadrant is rotated

around the line x = 2.

The volume of the solid of revolution is given by:

1. V (y) = π

∫ 2

0

(2y2 − y4

4)dy

2. V (y) = π

∫ 2

0

[(2− y2

2)2 − (

√2)2]dy

3. V (x) = π

∫ 4

0

[(4)2 − (

√2x− 2)2

]dx

4. V (x) = π

∫ 4

0

[(2x− 2) − (4)2] dx


13. The area bounded by the curves y = x2 + 1 and y = x + 3 is rotated around the liney = 1.

The volume of the solid of revolution obtained is given by

1. V (x) =

∫ 2

−1[ π(1− (x2 + 1))2 − π(1− (x+ 3))2] dx

2. V (x) =

∫ 2

−1[ π(x+ 2)2 − π(x2)2] dx

3. V (y) =

∫ 5

2

[ π(y − 2) − π( y − 4)2] dy

4. V (y) =

∫ 5

2

[π(√y − 1− 1)2 − π(y − 3− 1 )2

]dy


31

14. Determine the derivative of order n for the function f(x) = (2x− 1)−1


1. fn(x) = (−1)n+1(n− 1)!2n+1(2x− 1)−n

2. fn(x) = (−1)n+1n!2n+1(2x− 1)−n

3. fn(x) = (−1)n(n− 1)!2n (2x− 1)−(n+1)

4. fn(x) = (−1)nn!2n (2x− 1)−(n+1)


15. Find the Taylor polynomial of order n for f(x) = ln x2

in a = 2?


1. Pn,2f(x) =∑n

k=0(−1)k(x−2)k2kk!

2. Pn,2f(x) =∑n

k=1(−1)k(x−2)k2kk!

3. Pn,2f(x) =∑n

k=1(−1)k−1(x−2)k2kk

4. Pn,2f(x) =∑n

k=0(−1)k(x−2)k2kk


32

MAT1613/101/3/2018

8.5 Other assessment methods

There are no other assessment methods for this module.

9 The examination

Examination admissionTo be admitted to the examination you must submit the compulsory assignment, i.e. Assignment01, by the due date (06 April 2018 for Semester 1, and 31 August 2018 for Semester 2).Examination period

This module is offered in a semester period of fifteen weeks. This means that if you are registeredfor the first semester, you will write the examination in May/June 2018 and the supplementary ex-amination will be written in October/November 2018. If you are registered for the second semesteryou will write the examination in October/November 2018 and the supplementary examination willbe written in May/June 2019.

During the semester, the Examination Section will provide you with information regarding the ex-amination in general, examination venues, examination dates and examination times.

Examination paper

The textbook forms the basis of this course. The study outcomes are listed under 2.2 of this tutorialletter. The examination will be a single written paper of two hours duration.Refer to myStudies @ Unisa for general examination guidelines and examination preparationguidelines.

You are not allowed to use a calculator in the exam. Previous examination paper(s) will beavailable online and memorandums to some of the papers will also be availble online.

10 FREQUENTLY ASKED QUESTIONS

For any other study information see my Studies@Unisa.

11 SOURCES CONSULTED

No other books except the textbook and study guide were used in this module.

12 IN CLOSING

Read your tutorial letter carefully, follow the study guide reference and outcomes and do as manyexercises as possible.

33

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