Unisa Study Notes - MAT1512 · Assessment criteria – A formal deﬁnition of the limit with the correct mathematical notation is given which embraces an understanding of the limit

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Tutorial letter 101/3/2018

Calculus A

MAT1512

Semesters 1 & 2

Department of Mathematical Sciences

IMPORTANT INFORMATION:

This tutorial letter contains important information about yourmodule.

MAT1512/101/3/2018

CONTENTS

Page

1 INTRODUCTION AND WELCOME ..........................................................................4

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

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

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

3 LECTURER AND CONTACT DETAILS ..................................................................8

3.1 Lecturer ...............................................................................................................................8

3.2 Department .........................................................................................................................9

3.3 University ............................................................................................................................9

4 RESOURCES ................................................................................................................10

4.1 Prescribed book ...............................................................................................................10

4.2 Recommended books ......................................................................................................10

4.3 Electronic reserves (e-Reserves) ......................................................................................10

4.4 Library services and resources information .......................................................................11

5 STUDENT SUPPORT SERVICES ..........................................................................11

6 STUDY PLAN ................................................................................................................12

7 PRACTICAL WORK AND WORK INTEGRATED LEARNING ......................12

8 ASSESSMENT..............................................................................................................13

8.1 Assessment plan ..............................................................................................................13

8.2 Assignment numbers..........................................................................................................13

8.2.1 General assignment numbers .........................................................................................13

8.2.2 Unique assignment numbers ..........................................................................................13

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8.3 Assignment due dates .....................................................................................................13

8.4 Submission of assignments............................................................................................14

8.5 Other assessment methods.......................................................................................14

8.6 The examination............................................................................................................15

8.7 Examination paper ...........................................................................................................15

9 FREQUENTLY ASKED QUESTIONS ....................................................................15

10 SOURCES CONSULTED ..........................................................................................15

11 IN CLOSING ..................................................................................................................15

12 ADDENDUM ......................................................................................................................16

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1 INTRODUCTION AND WELCOME

Welcome to Calculus A module with the module code MAT1512. I hope you will find it both inter-esting and rewarding. This module is offered as a semester module. You will be well on your wayto success if you start studying 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 Matter

The Department of Despatch should supply you with the Study Guide and Tutorial Letter 101 atregistration and others later.

You will receive an inventory letter that will tell you what you have received in your study packageand also show items that are still outstanding. Also see the booklet entitled Study @Unisa. Checkthe study material that you have received against the inventory letter.

PLEASE NOTE: Your lecturers cannot help you with missing study material.

Apart from Tutorial Letter 101 , you will also receive other tutorial letters during the semester.These tutorial letters will not necessarily be available at the time of registration. Tutorial letters willbe despatched to you as soon as they are available or needed.

If you have access to the Internet, you can view the study guide and tutorial letters for the modulesfor which you are registered on the University’s online campus, myUnisa, 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 you receive during the semester immediately and care-fully, as they always contain important and, sometimes, urgent information.

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MAT1512/101/3/2018

2 PURPOSE AND OUTCOMES OF THE MODULE

2.1 Purpose

This module is useful to students interested in developing the basic skills in differential and integralcalculus. Differential and integral calculus are essential for physical, life and economic sciences.Students credited with this module will have a firm understanding of the limit, continuity at a point,differentiation and integration, together with a background in the basic techniques and some appli-cations of Calculus.

2.1.1 Learning Assumptions: The learning is based on the assumption that students arealready competent in terms of the following outcomes or areas of learning and must:

– Have a Senior Certificate or equivalent qualification (as required) for further study.– Have obtained an NQF/HEQF Level equivalent to 4 with the ability to:– Be able to learn from predominantly written material in the language of tuition– Take responsibility for their own progress and independently adjust to the learning

environment– Have basic computer skills like using a mouse, keyboard and windows features– Demonstrate an understanding of the most current topics in mathematics including∗ Functions∗ The ability to algebraically manipulate reall numbers and solve equations.∗ An ability to sketch graphs and find equations from these graphs.∗ Substantive knowledge about basic trigonometry∗ Knowledge about the following mathematical concepts: absolute values, partial

fractions and inequalities.

Recognition of prior learning will take place in accordance with the institution’s policyand guidelines. Recognition takes place, where prior learning corresponds to the re-quired NQF-HEQF level and in terms of applied competencies relevant to the contentand outcomes of the qualification, at the discretion of the department.

2.1.2 Range statement for the module: The techniques selected involve polynomial, ratio-nal, trigonometric, exponential and logarithmic functions and their composites. Thisintroductory calculus module covers differentiation and integration of functions of onevariable, with applications.

2.2 Outcomes

2.2.1 Specific outcome 1:Demonstrate knowledge of the concept of a limit of a function and its application.Range:The knowledge includes limits of one variable and an introduction to limits of two or morevariables.

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Assessment criteria

– A formal definition of the limit with the correct mathematical notation is given whichembraces an understanding of the limit as the y-value of a function.

– A distinction between the limits of a function as x approaches a {limx→a f (x)} andthe value of the function at x = a is made correctly.

– Laws governing limits are stated and used to determine and evaluate limits of sums,products, quotients and composition of functions.

– The limits of functions are evaluated graphically and numerically.– The limit definition of continuity is used to determine whether a function is continuous

or discontinuous at a point.– The Squeeze Theorem is used to determine certain undefined limits.

2.2.2 Specific outcome 2:Demonstrate an understanding of differentiation.Assessment criteria

– The derivative is defined as an instataneous rate of change of a function.– The fist principle of differentiaion is presented using different expressions.

Range: These different expressions include:

f ′ (x) = limh→0f (x+ h)− f (x)

h; f ′ (x) = lim∆x→0

f (x+ ∆x)− f (x)

∆x

f ′ (x) = lim∆x→0∆y

∆x; f ′ (a) = limMx→a

f (x)− f (a)

x− a– Alternate derivative notations are given. Range: These include:

f ′ (x) = y′ =dy

dx=df

dx=

d

dxf (x)

– A distinction between continuity and differentiability of a function at point is madecorrectly.

– A representation of the first derivative as the slope of the tangent line at the point oftangency is given.

2.2.3 Specific outcome 3:

– Calculate derivatives.Assessment criteria

– The derivative of a function is computed from the first principle of differentiation.– The basic rules of differentiation such as the power rule, product and quotient rules

are used to compute derivatives of different functions.– Range: The functions are in the form:

[h (x) = f (x)± g (x)] ; [h (x) = f (x) · g (x)] ;[h (x) =

f (x)

g (x)

].

– The chain rule is used, together with other rules of differentiation to find derivativesof composite functions.

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MAT1512/101/3/2018


– Use derivatives to solve applied problems.Assessment criteria

– For the problem solving , the differentiation technique chosen is appropriate to theproblem.

– Mathematical notations and language are used appropriately.– The derivative is used to find equations of tangent and normal lines of different

curves.– Where appropriate, the Mean Value Theorem is applied.


– Demonstrate understanding of basic integration and the Fundamental Theo-rem of CalculusAssessment criteria

– The definite integral is defined and interpreted using :∗ the concept of definite integral to obtain areas under the curve.∗ as the net change in a quantity from x = a to x = b if f(x) is the rate of change

of the quantity with respect to x.– A function F is defined as an anti-derivative (indefinite integral) of the function f if

the derivative F ′ = f. Anti-differentiation (integration) is recognised as the inverse ofthe differentiation process.

– The Fundamental Theorem of Calculus for a continuous function fon an interval[a; b] as:b∫a

f (x) dx = F (b)− F (a) where F (x) is such that F ′ (x) = f (x)

is reproduced and used to:-∗ explain the way in which differentiation and integration are related.∗ evaluate given integrals.

– Integral notation is used appropriately.


– Use integrals of simple functions to solve applied problemsRange: Simple integrals are applied but not limited to problems involving the lengthof a curve, area between curves, velocity and acceleration.Assessment criteria

– Substitution or term by term integration techniques are used appropriately.– The anti-derivatives of basic algebraic and trigonometric functions are determined

correctly.– For the problem solving process:-∗ The estimations of the definite integrals of the functions are correct.∗ The solution is consistent with the problem.

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2.2.7 Specific outcome 7

– Analyse logarithmic and exponential functions.Assessment criteria

– The graphs of the functions y = ex and y = lnx are reproduced.– The relationship between ex and lnx as inverse differentiable functions is recognised

and used as a device for simplifying calculations.– Rules of differentiation and integration are applied to functions involving logarithmic

and exponential functions.– Logarithmic diffefentiation is used correctly.– Exponentials and logarithmic models for solving applied problems are identified.

2.2.8 Specific outcome 8

– Solve exponential growth and decay problems using elementary differential equa-tions.Range: The solutions are limited to first-order, separable, constant coefficient initial-value problems, with contextual situations involving exponential growth and decay.Assessment criteria

– The contextual situation (problem) is analysed and represented with a differentialequation.

– A suitable method for determining the solution is chosen.– Initial or boundary conditions are identified and used to determine the constant of

integration.– The differential equation is solved correctly.– Partial derivatives are computed where necessary.– Mathematical notation is used to communicate the results clearly

3 LECTURER AND CONTACT DETAILS

3.1 Lecturer

The lecturer responsible for this module MAT1512 is:

Mrs S.B. MugishaUNISA Florida Campus,GJ Gerwel, Room 6-54Tel: 011 670 9154E-mail address: [email protected]

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. If you phone meplease have your study material with you when you contact me. If you cannot get hold of me, leavea message with the Departmental Secretary. Please clearly state your name, time of call and howI can get back to you. You are always welcome to come and discuss your work with me, but

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MAT1512/101/3/2018

please make an appointment before coming to see me. Please come to these appointmentswell prepared with specific questions that indicate your own efforts to have understood the basicconcepts 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 yourattempts so I can see where you are going wrong, or what concepts you do not understand. Mailshould be sent to:

Mrs S.B. MugishaDepartment of Mathematical SciencesPO Box 392UNISA0003

PLEASE NOTE: Letters to lecturers may not be enclosed with or inserted into assignments.

3.2 Department

Departmental SecretaryTel: 011 670 9147Mathematical SciencesUNISA Florida Campus,GJ Gerwel, 6th floor

3.3 University

If you need to contact the University about matters not related to the content of this module,please consult the publication Study @Unisa that you received with your study material.This brochure 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.

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4 RESOURCES

4.1 Prescribed book

The prescribed textbook is

James StewartCalculusMetric version 8EEarly TranscendentalsCengage Learning

ISBN 13: 978-1-305-27237-8

Please refer to the list of official booksellers and their addresses in the Study @Unisa brochure.Prescribed books can be obtained from the University’s official booksellers. If you have difficulty inlocating your book(s) at these booksellers, please contact the Prescribed Book Section at Tel: 012429-4152 or e-mail [email protected].

4.2 Recommended books

There are no recommended books for this module.

4.3 Electronic reserves (e-Reserves)

We managed to put online in (account Youtube) the video from your Lecturer Mrs Mugisha onCalculus A. As the video is too long we had to cut it into four parts. The videos are all about themodule MAT1512. The videos cover the sections of this module which most student tend to havedifficulties. The videos wer made using an old prescribed textbook, but follow the videos with thenew prescribed textbook by Stewart.The students can just click on the given list below or copy and paste them on their internet browserbar.Video 1 - Limitshttp://www.youtube.com/watch?v=GuRGhrt19tM&feature=youtu.bevideo 2- Limits and continuityhttp://www.youtube.com/watch?v=tEenlPFx6Mk&feature=youtu.beVideo 3-Calculus A-Differentiationhttp://www.youtube.com/watch?v=Eyc7C54sPgAVideo 4-Calculus A-Integrationhttp://www.youtube.com/watch?v=sChEcFeuqT8

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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 the publication Study @Unisathat you received with your study material.

Study groups

It is advisable to have contact with fellow students. One way to do this is to form study groups. Theaddresses of students in your area may be obtained from the following department:

Directorate: Student Administration and RegistrationPO Box 392UNISA0003

myUnisa

If you have access to a computer that is linked to the internet, you can quickly access resourcesand information at the University. The myUnisa learning management system is Unisa’s onlinecampus that will help you to communicate with your lecturers, with other students and with theadministrative departments of Unisa all through the computer and the internet.

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 the publication Study @Unisa which you received with your study material for moreinformation on myUnisa.

11

Tutorial Classes

Tutorial classes might be given by tutors at some learning centres. This will depend on how manystudents are registered at that learning centre. You have to pay for these classes. You may contactthe learning centre in connection with tutorial classes. The information and telephone numbers arein the Study @Unisa brochure.

Information on e-tutoring

Please be informed that, with effect from 2013, Unisa offers online tutorials (e-tutoring) to studentsregistered for modules at NQF level 5 and 6, this means qualifying first year and second yearmodules. You are lucky since this module falls in this category.Once you have been registered for this module, you will be allocated to a group of students withwhom you will be interacting during the tuition period as well as an e-tutor who will be your tutorialfacilitator. Thereafter you will receive an sms informing you about your group, the name of youre-tutor and instructions on how to log onto MyUnisa in order to receive further information on thee-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 you to get themost out of online tutoring, you need to participate in the online discussions that the e-tutor will befacilitating.

6 STUDY PLAN

The outcomes of the Study Plan below refers to pages 5 to 8 of this tutorial letter 101.

Study Plan Semester 1 Semester 2Outcomes 2.2.1 to 2.2.4 to be achieved by 20 March 14 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

7 PRACTICAL WORK AND WORK INTEGRATED LEARNING

There are no practicals for this module.

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MAT1512/101/3/2018

8 ASSESSMENT

8.1 Assessment plan

In each semester there are two assignments for MAT1512. Assignment 01 is a written assignmentand Assignment 02 is a multiple choice assignment. The solutions to both 01 and 02 will be sent toALL registered students after the due date. The questions for the assignments for semester 1 andsemester 2 are found under Addendum, section 12 of this tutorial letter 101. Both assignmentscount towards your semester mark. Please make sure that you answer the questions for thesemester for which you are registered. After marking the assignments, they will then be re-turned to you. The assignments and the comments in the form of solutions on these assignmentsconstitute an important part of your learning and should help you to be better prepared for the nextassignment and the examination. Please do not wait until you receive Assignment 01 back beforeyou start working on Assignment 02. The solutions to each assignment will be sent to you orsolutions would be posted on myUnisa at least two weeks after each assignment’s due date.

To be admitted to the examination you need to submit the first assignment due before the compul-sory date.

Your semester mark for MAT1512 counts 20% and your exam mark 80% of your final mark. Thefirst assignment counts 50% and the second assignment 50% towards the year mark.

8.2 Assignment numbers

8.2.1 General assignment numbers

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

8.2.2 Unique assignment numbers

Please note that each assignment also has their own unique assignment number which has to bewritten on the cover of your assignment upon submission.

8.3 Assignment due dates

The closing dates for submission of the assignments are:Semester 1

Assignment Nr. 01 02Type Written Multiple choiceUnique Nr. 888348 882589Due date 23 March 2018 20 April 2018

13

Semester 2Assignment Nr. 01 02Type Written Multiple choiceUnique Nr. 667972 686451Due date 17 August 2018 21 September 2018

8.4 Submission of assignments

VERY IMPORTANT NOTICE TO STUDENTS!!

Due to a very important rule of the university, a student must submitat least one assignment to reach Unisa BEFORE the specific date given below,

to be considered as an active student and admitted to the examination.

You must submit written assignments and multiple choice assignments (following the correct pro-cedure) electronically via myUnisa. Assignments should not be submitted by fax or e-mail. Fordetailed information on assignments, please refer to Study @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.

8.5 Other assessment methodsThere are no other assessment methods for this module.

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8.6 The examination

Examination admissionTo be admitted to the examination you must submit the compulsory assignment, i.e. Assignment01, by the due date (23 March 2018 for Semester 1, and 17 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 supplementaryexamination 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.

8.7 Examination paper

The textbook and the study guide form the basis of this course. The study outcomes are listedunder 2.2 of this tutorial letter. The examination will be a single written paper of two hours duration.Refer to the Study @Unisa brochure for general examination guidelines and examination prepara-tion guidelines.You are not allowed to use a calculator in the exam. Previous examination paper(s) will beavailable to students without memorandum on myUnisa under Official Study Material.

9 FREQUENTLY ASKED QUESTIONS

For any other study information see the brochure Study @Unisa.

10 SOURCES CONSULTED

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

11 IN CLOSING

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

Your MAT1512 lecturers.

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12 ADDENDUM

SEMESTER 1

ASSIGNMENT 01Closing date: 23 March 2018

Unique assignment number: 888348

1. Finding Limits from a graph.

Let the graph of the particular function g(x) be represented as shown below

(a) Use the graph of g in the figure to find the following values, if they exist, If a limit doesnot exist explain why.

(i) g(−2) (vi) limx→2+

g(x)

(ii) limx→−2−

g(x) (vii) limx→2

g(x)

(iii) limx→−2+

g(x) (viii) g(3)

(iv) limx→−2

g(x) (ix) limx→3−

g(x)

(v) limx→2−

g(x) (x) limx→3+

g(x)

(xi) limx→3

g(x)

(b) Identify the discontinuities in the function g(x) graphed above.

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2. (a) Let h be a function defined as

h(x) =

8− xx12x2

ififif

x < 00 < x ≤ 2x > 2

(i) limx→0−

h(x)

(ii) limx→0+

h(x)

(iii) limx→0

h(x)

(iv) limx→2−

h(x)

(v) limx→2+

h(x)

(vi) limx→2

h(x)

(b) Determine if function h is continuous at x = 0 and x = 2.

(c) Sketch the graph of h

3. Determine the following limits

(a) limx→1

x2 − 9x+ 14

x2 + 6x− 7

(b) limx→4

h(x), where h(x) =

2x2 − 8

3if x < 4

2x if x ≥ 4

(c) limx→−2

5

−5x2 − 2x

|2 + 5x|

(d) limx→π

2

sin2 x− 1

sinx− 1

(e) limx→−∞

|25− x2|x(x+ 5)

4. Use Squeeze Theorem to determine

limx→0

(x+ 1) cos(lnx2)√x2 + 2

17

5. Let

f(x) =

(x2 + 1)

sin(x2 − 4)

x2 − 4if x < 2

b if x = 2a cos(x− 2) if x > 2

Determine the value of a and b that make the function f(x) continuous at x = 2

6. Use the first principle of differentiation to find the derivative of g(x) = 3x2 − 4x+ 2

7. Find the derivative of the given function by using the appropriate rule of differentiation:

(a) h(x) = x2 sin 5− 7x2 + x√

2 + ln 2

(b) h(y) =ey − 9y2 + 1

(y − 1)2

(c) h(t) = sin t(t3 + 5)2

(d) h(x) = cos(sin(tan x))

(e) h(θ) = cot√θ.

(f) h(t) = 3√et + 1

8. Use implicit differetiable to finddy

dxin the followimg

(a) y2 = lnx

(b) cos (cos(x3y2))− x cot y = −2y

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MAT1512/101/3/2018

SEMESTER 1ASSIGNMENT 02

Closing date: 20 April 2018Unique assignment number: 88289

MULTIPLE CHOICE QUESTIONS. ANSWER ON A MARK READING SHEET.THERE IS ONLY ONE CORRECT ALTERNATIVE FOR EACH QUESTION.MARKS WILL BE DEDUCTED FOR INCORRECT ANSWERS.

1. Let the function f be defined by:

f (x) =

x+ 2√x+ 2

if x > −2

2x+ c if x ≤ −2

where c is a constant. Then f is continuous at −2 for:

(1) no value of c

(2) c = 0

(3) c = 4

(4) c = −2

2. Let the function f be defined by

f (x) =

4x3 + 5

−6x2 + 7xif x > 2

1

2x+ 3if |x| < 2

6x3 + 3x2 − 8

7x4 + 16x2 + 2if x < −2

Which one of the following statements is true?

(1) limx→∞

f (x) = −∞

(2) limx→−∞

f (x) =∞

(3) limx→∞

f (x) =2

3

(4) None of the above.

19

3. Let

f (x) =x2 + 1

(2x− 1)2.

Then the derivative of f is

(1) f ′ (x) =2 (2x− 1) · 2x

2x

(2) f ′ (x) =−2x− 4

(2x− 1)3

(3) f ′ (x) =2x (2x− 1)2 − 2 (2x− 1) · 2x (x2 + 1)

(x2 + 1)2

(4) f ′ (x) =2 (2x− 1) (x+ 2)

(x2 + 1)2 .

4. Let

f (x) =x+ 2

x2 + 1.

Then the second derivative of f is:

(1) f” (x) =(3x+ 1) (x+ 1)

(x2 + 1)2

(2) f” (x) =2x3 + 12x2 − 6x− 4

(x2 + 1)3

(3) f” (x) =x+ 2

2x (x2 + 1)

(4) f” (x) =2x3 + 9x2 − 2x− 3

(x2 + 1)3

5. Let g (θ) = sin4(θ3 − 3

). Then the derivative of g is

(1) g′ (θ) = 12θ2 cos(θ3 − 3

)sin3

(θ3 − 3

)(2) g′ (θ) = −12θ2 cos

(θ3 − 3

)sin3

(θ3 − 3

)(3) g′ (θ) = 3 sin2

(θ3 − 3

)(4) None of the above.

6. Let f (θ) = sin 3θ cos 4θ. Then the derivative of f is

(1) f ′ (θ) = 3 sin θ + 4 cos θ

(2) f ′ (θ) = 3 cos 3θ cos 4θ + 4 sin 4θ sin 3θ

(3) f ′ (θ) = 3 cos 3θ cos 4θ − 4 sin 4θ sin 3θ

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MAT1512/101/3/2018

(4) f ′ (θ) = −3 cos 3θ cos 4θ − 4 sin 4θ sin 3θ

7. The derivative of the functionh (z) =

√tan (1− 2z)

is:

(1) h′ (z) =− sec2 (1− 2z)√

tan (1− 2z)

(2) h′ (z) =sec2 (1− 2z)

2√

tan (1− 2z)

(3) h′ (z) =sec (1− 2z)√tan (1− 2z)


8. Let

f (x) =

x

1− 2 |x|if − 1

2< x < 1

2

1 if x ≤ −12

or x ≥ 12

Which one of the following statements is false?

(1) limx→−∞

f (x) = 1

(2) f is continuous at every real number except −12

and 12

(3) limx→0

f (x) does not exist


9. Suppose the equation2x2y − x3 = 3y + x

defines y implicitly as a differentiable function of x. Then

(1)dy

dx=

3− 2y2

4xy − 3y2 − 1

(2)dy

dx=

1 + 3x2 − 4xy

2x2 − 3

(3)dy

dx=

2xy2 − y3

3x+ y

(4)dy

dxdoes not exist.

21

10. Let

f (x) =

sec2 x if 0 ≤ x <

π

4

sin 2x if π4< x ≤ π

2

Then the value of the integral ∫ π2

0

f (x) dx

is

(1)3

2

(2)1

2

(3) 2

(4)π

2

11. Determine ∫x√

2x+ 1dx

The correct alternative is

(1)1

10(2x+ 1)

52 + 1

2(2x+ 1)−

12 + c

(2)1

10(2x+ 1)

52 +

1

4(2x+ 1)−

12 + c

(3)1

4

[2

5(2x+ 1)

32 − 2

3(2x+ 1)

32

]+ c

(4)1

10(2x+ 1)

52 − 1

6(2x+ 1)

32 + c

12. Determine the integral ∫4x2 − 2x+ 2

4x2 − 4x+ 3dx


(1)1

2ln 2− 1

2ln 3

(2) x+1

4ln |4x2 − 4x+ 3|+ c

(3) ln |4x2 − 4x+ 3|+ c

22

MAT1512/101/3/2018


13. Letf (x) = tan x x ∈

[−π

4,π

6

]The area between the graph of f and the x–axis is given by

(1)1

2ln 2− 1

2ln 3

(2)3

2ln 2− 1

2ln 3

(3)1

2ln 3

(4) ln(π

4

)− ln

(π6

).

14. Letf (x) = x2

and

g (x) =

{x+ 2 for x > 02− x for x < 0

The area enclosed by the curves f and g is given by

(1)10

6(units)2

(2)20

3(units)2

(3)48

3(units)2

(4) Not one of the above.

15. Determine the integral ∫ π3

π4

tan2 x

x− tanxdx.


(1) − ln

∣∣∣∣π3 − 1√3

∣∣∣∣+ ln∣∣∣π4− 1∣∣∣

(2) ln

∣∣∣∣∣π − 3√

3

3

∣∣∣∣∣+ ln

∣∣∣∣π − 4

4

∣∣∣∣

23

(3) ln

∣∣∣∣π − 4

4

∣∣∣∣− ln

∣∣∣∣∣π − 3√

3

3

∣∣∣∣∣(4) ln

∣∣∣∣π3 − 1√3

∣∣∣∣− ln

∣∣∣∣∣π − 3√

3

3

∣∣∣∣∣16. Determine the integral ∫ ln 3

1

ex − e2x

1 + exdx.


(1) ln(1 + eln 3

)− ln (1 + e) + ex

(2) 2 ln(1 + eln 3

)− 3 ln (1 + e) + e

(3) ln

(4

1 + e

)+ 1 + e

(4) 2 ln

(4

1 + e

)+ e− 3

17. Let

w = 2 cotx+ y2z2

x = uv

y = sin (uv)

z = eu

Find∂w

∂ufor u =

1

4and v = π.


(1) −4 + e2π

(2) π(−1 + 11

4e2π)

(3) −1

2+

2√2e2π


18. Write a Chain Rule formula for dwdx

where

w = f (x, y) ,

y = g (x) .

24

MAT1512/101/3/2018


(1)dw

dx=dw

dg

dg

dx+dw

dx

(2)dw

dx=∂f

∂x+∂f

∂y

dy

dx

(3)dw

dx=∂w

∂y· dydx

+∂g

∂x· dydx

(4)dw

dx=dw

dg

dg

dx

19. Let

T (x) =

∫ 4

4r3t sin t3dt

Find T ′ (r) .


(1) −48r5 sin(4r3)3

(2) 3x4 sin 3x8 + 3 sin (3)4

(3) 3x4 sin (3x4)4 · 12

(4) 2r3 sin r6

20. Letxy2 + x2 + y + sin

(x2y)

= 0

Finddy

dxby implicit differentiation.


(1)xy − y2 cosxy2

x2 + 2y − 2xy cos (xy2)

(2)−2xy − y2 cosxy2

x2 + 2y + 2xy cos (xy2)

(3)−(2x+ y2 + 2xy cos(x2y))

1 + 2xy + x2 cos (x2y)


25

SEMESTER 2

ASSIGNMENT 01CLOSING DATE: 17 August 2018

UNIQUE NUMBER: 667972

1. Finding limits from a graph:

Let the graph of the particular function f (x) be represented as shown below

2 11

65421 3

1

2

3

4

5

x0

y

(a) Use the graph of f in the figure to find the following values, if they exist. If a limit doesnot exist explain why (12)

(i) f (1) (vii) limx→3+

f (x)

(ii) limx→1−

f (x) (viii) limx→3

f (x)

(iii) limx→1+

f (x) (ix) f (2)

(iv) limx→1

f (x) (x) limx→2−

f (x)

(v) f (3) (xi) limx→2+

f (x)

(vi) limx→3−

f (x) (xii) limx→2

f (x)

(b) Identify the discontinuities in the function f (x) graphed above. (2)

2. Let g be a function defined as:

g (x) =

x if x < 0x2 if 0 < x ≤ 2

8− x if x > 2

26

MAT1512/101/3/2018

(a) Evaluate each of the following limits, if it exists (6)

(i) limx→0−

g (x) (iv) limx→2−

g (x)

(ii) limx→0+

g (x) (v) limx→2+

g (x)

(iii) limx→0

g (x) (vi) limx→2

g (x)

(b) Determine if the function g is continuous at x = 0 and x = 2. (2)

(c) Sketch the graph of g. (4)

3. Determine the following limits.

(a) limx→2

f (x) , where f (x) =

{3x− 1 if x < 2x2 + 1 if x ≥ 2

(3)

(b) limx→1

x2 + 6x− 7

x2 + 5x− 6(3)

(c) limx→ 3

2

2x2 − 3x

|2x− 3|(3)

(d) limx→2

sin (x2 − 4)

x− 2(3)

(e) limx→∞

2− 3x2

−x |x− 2|(3)

4. Use Squeeze Theorem to detemine

limx→0

3− sin (ex)√x2 + 2

(5)

5. Let h (x) =

2 sinxx

if x < 0a if x = 0b cosx if x > 0

Determine the values of a and b that make the function h (x) contnuous at x = 0. (6)

6. Use the first principle of differentiation to find the derivative of f (x) = x2 − 2x− 15. (4)

27

7. Find the derivative of the given function by using the appropriate rules of differentiation:

(a) f (x) = x4 − 3x3 + 2x− 1 (3)

(b) f (t) = t2 (t+ 2)3 (3)

(c) f (x) = 6x2

(x−1)2(3)

(d) f (θ) = tan√θ (3)

(e) f (x) =√

lnx+ 1 (3)

(f) f (θ) = sin (cos (tan θ)) (3)

8 (a) Find the equation of the tangent to the graph of f (x) = x2 + 3x− 1 which makes an angleof 45◦ with the x−axis. Assume that the scales along the x−axis and y−axis are the same.Angles are measured counterclockwise from the positive x−axis. (5)

(b) What is the equation of the normal line to the curve of f at the point of tangency of thetangent considered in part (a). (2)

9. Use implicit differentiation to find dydx

in the following:

(a) x2 − xy + y2 = 1 (3)

(b) sin (x2y3) + 2x = y tanx (3)

10 Derivatieves of logarithmic functions: Use the properties of logarithms to simplify the followingfunction before computing f ′ (x)

(a) f (x) = ln (3x+ 1)4 (3)

(b) f (x) = ln(2x− 1) (x+ 2)3

(1− 4x)2 (3)

11. Logarithm differentiation: Use logarithmic differentiation to evaluate f ′ (x) of the following:

(a) f (x) = xlnx (3)

(b) f (x) =(x3 − 1)

4√3x− 1

x2 + 4(4)

[100]

28

MAT1512/101/3/2018

SEMESTER 2

ASSIGNMENT 02CLOSING DATE: 21 September 2018

UNIQUE NUMBER: 686451

1. The value of definite intergral

∫ 1

0

x2 (1 + 2x3)5dx is: (5)

The correct answer is:

(1)9

182

(2) 182

(3)182

9

(4) None of the above

2. Determine the indefinite integral

∫x√

2x+ 1dx (5)


(1)1

10(2x+ 1)

52 + 1

2(2x+ 1)−

12 + k

(2)1

4

[25

(2x+ 1)32 − 2

3(2x+ 1)

32

]+ k

(3)1

10(2x+ 1)

52 − 1

6(2x+ 1)

32 + k


3. Determiine the possible value of the definite integral

∫ e3

e2

ln (lnw)

w log3wdw (5)


(1)ln 2

3

[(ln 3)2 − (ln 2)2]

(2)ln 3

2

[(ln 3)2 − (ln 2)2]

(3)(

ln 32

)2[ln 3− ln 2]


29

4. Determine the indefinite integral

∫ex

1 + e2xdx (5)


(1) arctan ex + c

(2) arctan x+ ex + c

(3) arctan(xe

)+ c


5. Determine the integral

∫1√x+ x

dx (5)


(1) (√x+ x)

3+ 3 + c

(2) ln (1 + x) +√x+ c

(3) 2 ln (1 +√x) + c



∫4x2 − 2x+ 2

3− 4x+ 4x2dx (5)


(1) x+ 14

ln |4x2 − 4x+ 3|+ c

(2) x2

+ 12

ln |4x2 − 4x+ 3|+ c

(3) ln |4x2 − 4x+ 3|+ c



∫ ln 3

1

ex − e3x

1 + exdx (5)


(1) −1

3e3 ln 3 + eln 3 − e+

e3

3

(2)e2

2− e− 3

2

(3)e2

2− e

(4)1

2eln 2 − e+

e2

2

30

MAT1512/101/3/2018


∫1

sec2 x cos ec5xdx (5)


(1)3 sec2 x

2+ 4 cos ec3x+ c

(2)2 cos5 x

5− cos3 x

3− cos7 x

7+ c

(3) 2cos5 x

5+

3 sin7 x

7+ c


9. The value of the integral

∫ 4

−3

||x| − 4| dx is: (5)


(1)31

2

(2)2

31

(3) 62


10. Let G (x) =∫ 3x4

3t sin t4dt, for all x. Find G′ (x) . (5)


(1) 3x4 sin (3x4)4 · 12

(2) 3x4 sin 3x8 + 3 sin (3)4

(3) 36x7 sin (3x4)4

(4) 2x3 sinx6

11. Let k (x) =

∫ h(x2)

cosα

1

3r + r2dr where h (x) = sinx4 and α is constant. Use the Fundamental

Theorem of Calculus to finddk

dx. (5)


(1)2 cosx4 + 3 sinx4

2x sin2 x4

31

(2)8x7 cosx8

3 sinx8 + sin2 x8

(3)4x4 cosx4

3 sinx4 + sin2 x4


12. Let f (x) = x2 and g (x) = |x| . The area enclosed by the curves f and g is: (5)


(1)1

6

(2)4

3

(3)1

3


13. Find the area of the region R between the graphs of the given functions f and g (5)

where f (x) = x2 + 1g (x) = 3− x2

x = −2 and x = 2

(Hint: Sketch the graphs and show the area(s) involved).


(1) 8

(2)8

3

(3)28

3


14. Solve for y in the following initial value problem, where y is a function of x. (5)

dy

dx=

1 + x

xy, x > 0 and y satisfies the initial condition y(1) = −4.


(1) y2 = 2 ln x+ 2x+ 14

(2) y2 = x2 + 2x+ 14

(3) y = lnx+ 2x+ 7

32

MAT1512/101/3/2018


15. Solve for y as a function of x in the following initial value problem (5)

dy

dx=

sin(x2

)[1 + cos

(x2

)]3 ; y (0) = π


(1) y = 1

2(1+cos(x2 ))3 + π − 1

4

(2) y =sin(x2

)1 + cos

(x2

)2 + π − 1

4

(3) y = x

2(1+cos(x2 ))2 + π − 1

4

(4) y = 1

(1+cos(x2 ))2 + π − 1

4

16. Write a Chain Rule formula for∂w

∂tif w = f (x, y, z) , and x, y and z are functions of s and t


(1)∂w

∂t=∂f

∂x+∂f

∂y+∂f

∂z

(2)∂w

∂t=∂x

∂t+∂y

∂t+∂z

∂t

(3)∂w

∂t=∂f

∂x· ∂x∂t

+∂f

∂y· ∂y∂t

+∂f

∂z· ∂z∂t

(4) None of the above. (5)

17. If u = x4y + y2z3

where x = rset

y = rs2e−t

z = r2s sin t

Find the value of∂u

∂twhen r = 2, s = 1, t = 0. (5)


(1) 96

33

(2) 69

(3) −96


18. Let w = 2 cot x+ y2z2

where x = uvy = sin (uv)z = ev

Find∂w

∂xfor u = 1

4and v = π. (5)


(1) −4 + e2π

(2) −4

(3) 4

(4) 4− e2π

19. You have deposited A0 rands in a bank account that pays 13% interest, compounded continu-ously. How long will it take your money to grow to 4 times the original amount? (5)


(1) t = 13 ln 4

(2) t = 100 ln 4

(3) t =13

100ln 4

(4) t = 10013

ln 4

20. A culture initially has P0 number of bacteria. At t = 1 hour, the number of bacteria is measuredto be 3

2P0. If the rate of growth is proportional to the number of bacteria P (t) present at time

t. Determine the time necessary for the number of bacteria to triple. (5)


(1) t = 2

(2) t =ln 3

ln(

32

)(3) t =

ln 3

ln 2


[100]

34

Documents

Unisa Study Notes - MAT1512 · Assessment criteria – A formal deﬁnition of the limit with the correct mathematical notation is given which embraces an understanding of the limit