University of South Africa MAT1512 Calculus A: Tutorial Letter 001

This document is Tutorial Letter 001 for the MAT1512 Calculus A module at the University of South Africa (Unisa). It serves as an introductory guide for students enrolled in the year module, providing essential information about the course. The letter details the module's purpose, which is to equip...Read More

MAT1512/001/0/2021
of south africa
Tutorial Letter 001/0/2021
CALCULUS A
MAT1512
Year module
Department of Mathematical Sciences
IMPORTANT INFORMATION:
This tutorial letter contains important information about your module.
Please activate your myUNISA and myLife e-mail account(s) and make sure that you have regular
access to the myUNISA module website MAT1512-21-Y1, as well as your group site.
Note: This is a fully online module. It is therefore, only available on myUnisa.
BAR CODE
Define tomorrow. university
Open Rubric

CONTENTS
Page
1 INTRODUCTION.................................................................................................................. 3
1.1 Getting started……………… .............................................................................................. 3
2 OVERVIEW of MAT1512...................................................................................................... 3
2.1 Purpose…............................................................................................................................ 5
2.2 Outcomes………………………………………………………………………………………….. 5
3 LECTURER(S) AND CONTACT DETAILS.......................................................................... 8
3.1 Lecturer(s)............................................................................................................................ 8
3.2 Department........................................................................................................................... 9
3.3 University.............................................................................................................................. 9
4 RESOURCES....................................................................................................................... 9
4.1 Joining myUNISA .................................................................................................................. 9
4.2 Prescribed book(s)…... ......................................................................................................... 10
4.3 Recommended book(s)…………......................................................................................... 10
4.4 Electronic reserves (e-reserves).......................................................................................... 10
4.5 Library services and resources………………………………………………………………… 10
5 STUDENT SUPPORT SERVICES..................................................................................... 11
6 HOW TO STUDY ONLINE ?............................................................................................. 12
6.1 What does it mean to study fully online?.......................................................................... 12
6.2 myUNISA tools…………………………………………………………………………………. 13
7 ASSESSMENT …………………………………………………………………………………. 14
7.1 Assessment plan………………………………………………………………………………… 14
7.2 Year mark and final examination/other options………………………………………………. 16
8 CONCLUSION…………………………………………………………………………………… 17
APPENDIX: GLOSSARY OF TERMS………………………………………………………… 17

MAT1512/001/0/2021
1 INTRODUCTION
Dear Student
Welcome to the MAT1512 module. We trust that you will find the mathematics studied in this module
interesting and useful, and that you will enjoy doing it.
This tutorial letter contains important information about the scheme of work, resources and
assignments for this module as well as exam admission. We urge you to read it carefully before
working through the study material, preparing the assignment(s), preparing for the examination
and addressing questions to your lecturers.
In this tutorial letter, you will find the assignments as well as instructions on the preparation and
submission of the assignments. This tutorial letter also provides all the information you need with
regard to the prescribed study material and other resources. Please study this information
carefully and make sure that you obtain the prescribed material as soon as possible.
You will access all files online, a number of tutorial letters for example, solutions to assignments,
during the semester/ year. These tutorial letters will be uploaded on myUnisa, under Additional
Re-sources and Lessons tools on myUnisa platform. A tutorial letter is our way of communicating
with you about teaching, learning and assessment.
Right from the start we would like to point out that you must read all the tutorial letters you access
from the module site immediately and carefully, as they always contain important and, sometimes
urgent information.
Because this is a fully online module, you will need to use myUnisa to study and complete the
learning activities for this course. Please visit the website for MAT1512 on myUnisa frequently.
The website for your module is MAT1512-21-Y1.
1.1 Getting started
Owing to the nature of this module, you can read about the module and find your study material
online. Go to the website at https://my.unisa.ac.za and log in using your student number and
password. Click on “myModules” at the top of the webpage and then on “Sites” in the top right
corner. In the new window, click on the grey Star icon next to the modules you want displayed on
your navigator bar. Close the window in the right corner. The select the option “Reload to see
your updated favorite sites”. Now go to your navigation bar and click on the module you want to
open.
We wish you every success with your studies!
2 OVERVIEW of MAT1512
2.1 Purpose
This module will be able useful to students interested in developing the basic skills in Calculus
which can be applied in the natural sciences and social sciences. Students who have completed
this module successfully will have an understanding of the basic ideas of Calculus.

MAT1512/001
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2 PURPOSE AND OUTCOMES
2.1 Purpose
This module is useful to students interested in developing the basic skills in differential and integral
calculus. 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 are
already 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 real 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 policy
and guidelines. Recognition takes place, where prior learning corresponds to the re-
quired NQF-HEQF level and in terms of applied competencies relevant to the content
and 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. This
introductory calculus module covers differentiation and integration of functions of one
variable, 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 more
variables.

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Assessment criteria
– A formal definition of the limit with the correct mathematical notation is given which
embraces an understanding of the limit as the y-value of a function.
– A distinction between the limits of a function as x approaches {limx →a f (x ) } and
the 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 instantaneous rate of change of a function.
– The first principle of differentiation is presented using different expressions.
Range: These different expressions include:
     
h
xfhxf
xf h

  0
lim ;      
x
xfxxf
xf x 

  0
lim
  x
y
xf x 

  0
lim ;      
ax
afxf
af ax 

 
lim
– Alternate derivative notations are given. Range: These include:
   xf
dx
d
dx
df
dx
dy
yxf 
– A distinction between continuity and differentiability of a function at point is made
correctly.
– A representation of the first derivative as the slope of the tangent line at the point of
tangency 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:      ][ xgxfxh  ;       xgxfxh .
   
 




  xg
xf
xh .The chain rule is used, together with other rules of differentiation to find
derivatives of composite functions.

MAT1512/001
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2.2.4 Specific outcome 4:
– Use derivatives to solve applied problems.
Assessment criteria
– For the problem solving, the differentiation technique chosen is appropriate to the
problem.
– 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.
2.2.5 Specific outcome 5:
– Demonstrate understanding of basic integration and the Fundamental Theo-
rem of Calculus
Assessment 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 fF  .Anti-differentiation (integration) is recognised as the inverse
of the differentiation process.
_ The Fundamental Theorem of Calculus for a function f on an interval  ba, as:
     aFbFdxxf
b
a
 where  xF is such that    xfxF 
is reproduced and used to:-
∗ explain the way in which differentiation and integration are related.
∗ evaluate given integrals.
– Integral notation is used appropriately.
2.2.6 Specific outcome 6:
– Use integrals of simple functions to solve applied problems
Range: Simple integrals are applied but not limited to problems involving the length
of 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.
2.2.7 Specific outcome 7
– Analyse logarithmic and exponential functions.

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Assessment criteria
– The graphs of the functions y = ex and y = ln x are reproduced.
– The relationship between ex and ln x as inversedifferentiable 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 differentiation 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 differential
equation.
– 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(S) AND CONTACT DETAILS
3.1 Lecturer(s)
The lecturer responsible for this module is Dr. SB Mugisha. You can contact her at:
Dr. SB Mugisha
Tel: [phone]
Room no: C 6-54
GJ Gerwel Building
e-mail: [e-mail]
A notice will be posted on myUnisa if there are any changes and/or an additional lecturer
is appointed to this module.
Please do not hesitate to consult your lecturer whenever you experience difficulties with your stud-
ies. You may contact your lecturer by phone or through correspondence or by making a personal
visit to his/her office. Please arrange an appointment in advance (by telephone or by e-mail)
to ensure that your lecturer will be available when you arrive.

MAT1512/001
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Please come to these appointments well prepared with specific questions that indicated your own
efforts to have understood the basic concepts involved. If these difficulties concern exercises
which you are unable to solve, you must send us your attempts so that we can see where you are
going wrong.
If you should experience any problems with the exercises in the study guide, your lecturer will
gladly help you with them, provided that you send in your bona fide attempts. When sending in
any queries or problems, please do so separately from your assignments and address them
directly to your lecturer.
3.2 Department
You can contact the Department of Mathematical Sciences as follows:
Department of Mathematical Sciences
Fax number: 011 670 9171 (RSA) [phone] (International)
Departmental Secretary: 011 670 9147 (RSA) [phone] (International)
e-mails: [e-mail] or [e-mail]
3.3 University
To contact the University, follow the instructions on the Contact us page on the Unisa website.
Remember to have your student number available whenever you contact the University.
Whenever you contact a lecturer via e-mail, please include your student number in the subject
line to enable the lecturer to help you effectively.
4 RESOURCES
4.1 Joining myUnisa
The myUnisa learning management system is the University’s online campus which will help you
communicate with your lecturers, other students, and the administrative departments within Unisa.
To claim your myUnisa account, Please follow the steps below:
1. Visit the myUnisa website at https://my.unisa.ac.za/portal
2. Click on the “Claim Unisa login” link on the top of the screen under the orange use ID box.
3. A new screen will load, prompting you to enter your student number. Please enter your
student number and click “continue”.
4. Enter your surname, your full name, your date of birth and, finally, your South African ID
number (for South African citizens) OR your passport number (for foreign students). Then
clock “continue”. Remember to enter either an ID number or a passport number, NOT
both.
5. Please read through the guidelines and click all the check boxes to acknowledge that you
have read all the information provided. Once you are done, click the “Acknowledge” button to
redirect you to the final page in the process.
6. The final page will display your myLife e-mail address, and your myLife AND myUnisa
password. This password will also be sent to the cellphone number displayed on the page for
safekeeping.
7. Please note that it can take up to 24 hours for your myLife e-mail account to be created.

10
Remember, the password provided is your myUnisa AND myLife password.
4.2 Prescribed book
The prescribed textbook is
James Stewart
Calculus
Metric version 8E
Early Transcendentals
Cengage Learning
ISBN 13: 978-1-305-27237-8
Please buy the textbook as soon as possible since you have to study from it directly- you cannot
do this module without the prescribed textbook.
Please refer to the list of official booksellers and their addresses in the Study@Unisa brochure.
The prescribed book can be obtained from the University’s official booksellers. If you have difficulty
locating your book at these booksellers, please contact the Prescribed Books Section at (012) 429
4152 or e-mail [e-mail].
4.3 Recommended book(s)
There are no recommended books for this module.
4.4 Electronic Reserves (e-reserves)
E-reserves can be downloaded from the Library catalogue. More information is available at:
https://libguides.unisa.ac.za/request/request
Videos for MAT1512 made by your lecturer and put on You-Tube
We managed to put online (account Youtube) the video from your Lecturer Dr. Mugisha on
Calculus A. As the video is too long we had to cut it into four parts. The videos are all about the
module MAT1512. The videos cover the sections of this module which most student tend to have
difficulties. The videos were made using an old prescribed textbook, but follow the videos with the
new prescribed textbook by James Stewart.
The students can just click on the given list below or copy and paste them on their internet browser
bar.
Video 1 - Limits
http://www.youtube.com/watch?v=GuRGhrt19tM&feature=youtu.be
video 2- Limits and continuity
http://www.youtube.com/watch?v=tEenlPFx6Mk&feature=youtu.be
Video 3-Calculus A-Differentiation
http://www.youtube.com/watch?v=Eyc7C54sPgA
Video 4-Calculus A-Integration
http://www.youtube.com/watch?v=sChEcFeuqT8
4.5 Library services and resources
The Unisa Library offers a range of information services and resources:
 for detailed Library information go to https://unisa.ac.za/library

MAT1512/001
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 for research support and services (e.g. personal librarians and literature search services) go
to http://www.unisa.ac.za/sites/corprate/default/Library/Library-services/R
 The Library has created numerous Library guides:
http://libguides.unisa.ac.za
Recommended guides:
 Request and download recommended material:
http://libguides.unisa.ac.za/request
 Postgraduate information services:
http://lidguides.unisa.ac.za/request/postgrad
 Finding and using library resources and tools:
https://libguides.unisa.az.za/research-support
 Frequently asked questions about the Library:
http://libguides.unisa.ac.za/ask
 Services to students living with disabilities:
http://lidguides.unisa.ac.za/disability
 Assistance with technical problems accessing the Unisa Library or resources:
https://libguides.unisa.ac.za/techsupport
You may also send an e-mail to [e-mail] (please add your student number in
the subject line).
• for detailed Library information go to http://www.unisa.ac.za/sites/corporate/default/Library
• for research support and services (e.g. personal librarians and literature search services) go to
http://www.unisa.ac.za/sites/corporate/default/Library/Library-services/Research-support
The Library has created numerous Library guides:
http://libguides.unisa.ac.za
Recommended guides:
• Request and download recommended material:
http://libguides.unisa.ac.za/request/request
• Postgraduate information services:
http://libguides.unisa.ac.za/request/postgrad
• Finding and using library resources and tools:
http://libguides.unisa.ac.za/Research skills
• Frequently asked questions about the Library:
http://libguides.unisa.ac.za/ask

12
• Services to students living with disabilities:
http://libguides.unisa.ac.za/disability
5 STUDENT SUPPORT SERVICES
The Study@Unisa website is available on myUnisa: www.unisa.ac.za/brochures/studies
This website has all the tips and information you need to succeed at Unisa.
6 HOW TO STUDY ONLINE ?
6.1 What does it mean to study fully online?
Studying fully online modules differs completely from studying some of your other modules at
Unisa.
• All your study material and learning activities for online modules are designed to be
delivered online on myUnisa.
• All your assignments must be submitted online. This means that you will do all your activities
and submit all your assignments on myUnisa. In other words, you may NOT post your assignments
to Unisa using the South African Post Office.
• All communication between you and the University happens online. Lecturers will
communicate with you via e-mail and SMS, and use the Announcements, the Discussion Forums
and the Questions and Answers tools. You can also use all of these platforms to ask questions and
contact your lecturers.
STUDY PLAN
The table in the Tutorial Letter, which gives an Overview of the Module, indicates which sections in
the textbook cover the syllabus of the module and have to be studied.
At the beginning of each assignment there is an indication of the sections in the textbook and study
guide, which have to be studied properly before the assignment is attempted.
It is very important to study each section well at this stage. Make a good start by reading through
the text, studying each and every example and doing the indicated exercises. Study the specific
sections as if the assignment that follows, is a test of your knowledge and understanding of these
sections.

MAT1512/001
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3
The due dates of the assignments set the pace at which you should work through the content.
Month Activities
January
February
Read Tutorial Letter 001 (this letter).
Read pp. vii to ix of the Study Guide and the sections of James
Stewart Calculus to which these pages refer. Make sure you have all
your study material as well as other items such as assignment
covers.
Study Chapters 1 & 2 of James Stewart Calculus as well as
Units 1, 2 & 3 of the Study Guide. Prepare for Assignment 1.
March
Study Chapters 1 & 2 of James Stewart Calculus as
well as Units 1, 2 & 3 of the Study Guide. Prepare for
Assignment 1.
April Submit Assignment 1.
Study Chapters 2 & 3 of James Stewart Calculus as well as
Unit 4 of the Study Guide. Prepare for Assignments 2 & 3.
May Submit Assignments 2 & 3.
Study Chapters 3 & 4 of James Stewart
Calculus as well as Units 4 & 5 of the Study Guide. Prepare
for Assignments 4 & 5.
June Submit Assignment 4 & 5.
Study Chapters 5, 6.1, 7.2 , 14.1 – 14.5 (14.4 read only) of James
Stewart Calculus as well as Units 4 & 5 of the Study Guide.
Prepare for Assignments 6 & 7.
July Submit Assignments 6 & 7.
Study Chapter 9 for Assignment 8 & ALL the above mentioned
Chapters for Assignment 9 ofJames Stewart Calculus as well as Unit
6 for Assignment 8 with Units: 1, 2, 3, 4, 5 & 6 of the Study Guide for
Assignment 9. Prepare for Assignments 8 & 9.
August Submit Assignments 8 & 9.
Study all the Chapters and revise your work.
Prepare for the exam.
September Work through the solutions of
Assignments 1 to 9 and learn from your mistakes.
Prepare for the exam.
October
November
Study for the exam.
Write the exam.
December ENJOY YOUR HOLIDAY!
Draw up your own study schedule and keep to it!
See the brochureStudy @ Unisa for general time management and planning skills.

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6.2 myUnisa tools
The main tool that we will use is the Lessons tool. This tool will provide the content of and the
assessments for your module. At times you will be directed to join discussions with fellow students
and complete activities and assessments before you can continue with the module.
It is very important that you log in to myUnisa regularly. We recommend that you log in at least
once a week to do the following:
• Check for new announcements. You can also set your myLife e-mail account so that you
receive the announcement e-mails on your cellphone.
• Do the Discussion Forum activities. When you do the activities for each learning unit, we
want you to share your answers with the other students in your group. You can read the
instructions and even prepare your answers offline, but you will need to go online to post your
messages.
• Do other online activities. For some of the learning unit activities you might need to post
something on the Blog tool, take a quiz or complete a survey under the Self-Assessment tool. Do
not skip these activities because they will help you complete the assignments and activities for the
module.
We hope that by giving you extra ways to study the material and practice all the activities, this will
help you succeed in the online module. To get the most out of the module, you MUST go online
regularly to complete the activities and assignments on time.
7 ASSESSMENT
Please note that this module has a total of NINE compulsory assignments which contribute 20%
to the final mark and one examination which contributes 80% to the final mark.
Please note that lecturers are not responsible for examination admission, and ALL inquiries
about examination admission should be directed by e-mail to [e-mail]
You will be admitted to the examination if and only if Assignment 01 reaches the University and is
submitted/ uploaded by the stipulated due date.
Note that your marks for the assignments contribute 20% to your final mark (the
remaining 80% is contributed by the examinations).
7.1 ASSESSEMT
Please note that this module has a total of NINE compulsory assignments which contribute 20%
to the final mark.
The questions for the assignments are available online on myUnisa platform. For each
assignment there is a FIXED CLOSING DATE; the date by which the assignment must be

MAT1512/001
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uploaded in the system. Solutions for each assignment will be given as Tutorial Letters 201,
202, etc. will be uploaded on myUnisa under Additional Resources a few days after the closing
date.
Late assignments will be marked, but will be awarded 0%.
Written assignment
Not all the questions in the written assignments will be marked and you will also not be
informed beforehand which questions will be marked. The reason for this is that Mathematics is
learnt by “doing Mathematics”, and it is therefore extremely important to do as many problems
as possible.
You can self-assess the questions that are not marked by comparing your solutions with the
printed solutions that will be sent to you.
Note that Assignment 01 is the compulsory assignment for admission to the examination
and must be uploaded by the due date.
The written/ Online assignments can only be submitted online electronically through myUnisa.
Assignments Feedback as Tutorial Letters
01 201
02, etc, …, 09 202, etc,…, 208. (209 has NO feedback)
Please note that Assignment 01 & Assignment 05 are Multiple-Choice Assignments and are
marked by the system.
The Assignments for the Module are as follows:
Assignment Nr. 01 02 03 04
Unique Nr. 894095 812568 644031 649356
Due date 26 April 2021 14 May 2021 28 May 2021 15 June 2021
Assignment Nr. 05 06 07 08 09
Unique Nr. 852958 784524 677794 703736 643701
Due date 30 June 2021 16 July 2021 30 July
2021
13
August
2021
25 August
2021
Because this is an online module, the assignments contained in this Tutorial Letter are made
available online. You will see the Assignments when you go online. We urge you to tune to the
Website MAT1512-21-Y1 for more updated information.
ASSESSMENT
Assessment criteria
There are nine written assignments and one examination.

16
Assessment plan
Please note that this module has a total of NINE compulsory assignments which contribute 20% to
the final mark.
Eight of the compulsory assignments are relatively short assessments. The ninth assignment
which is submitted at the end of August is a Past Exam Paper. Assignments should be submitted
electronically via myUnisa.
The assessments are as follows: The assignments contribute to the year mark as follows:
Assignment Number Type of assignment Contribution to the final mark (%)
01 Multiple Choice 2
02 Written 2
03 Written 2
04 Written 2
05 Multiple Choice 2
06 Written 2
07 Written 2
08 Written 2
09 Past Exam Paper 4
Total 20
7.2 Year mark and final examination/ other options
The year mark and the examination mark for this module will be divided as follows:
Type of assessment Contribution to the final mark (%)
Formative 20
Summative 80
Final mark 100
Please note that the 20% contribution by the assignments makes it extremely important that
you do all the assignments and score high marks, otherwise it is impossible for you to pass the
module. This also means that if you do all the assignments well, there is less risk for you failing
the module. The final examination is a 2-hours written exam that will be conducted online,
according to the examination calendar, which you can access on the Unisa website.
You only submit your assignments electronically via myUnisa. Assignments may not be
submitted by fax or e-mail.
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.
 Follow the instructions.

MAT1512/001
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The examination
You will write the examination in October/ November 2021 and the supplementary examination will
be written in January/ February 2022.
The Examination Section will provide you with relevant information regarding the examination in
general, examination dates and examination times.
Please note:
 The exam is a two hour examination.
 The use of a pocket calculator is not permitted during the examination.
The examination questions will be similar to the questions asked in the study guide and in the
assignments.
8 CONLUSION
Remember that there are no “short cuts” to studying and understanding mathematics. You need
to be dedicated, work consistently and practice, practice and practice some more. Do not
hesitate to contact us by e-mail if you are experiencing problems with the content of this tutorial
letter or with any academic aspect of the module.
We wish you a fascinating and satisfying journey through the learning material, and trust that
you will complete the module successfully.
Enjoy the journey !
Dr. S.B. Mugisha –Lecturer for MAT1512
Department of Mathematical Sciences.

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ASSIGNMENT 01
MULTIPLE CHOICE ASSIGNMENT
Fixed Closing Date: 26 April 2021
Total Marks:1 5 0
UNIQUE ASSIGNMENT NUMBER: 894095
1. Determine the following limit:
13
26
lim2 

 x
x
x
Then the correct answer is:
(1) 2
(2) 2
(3) 2
1
(4) None of the above. (5)
2. Find the following limit:
76
149
lim 2
2
1 

 xx
xx
x
Then the correct answer is:
(1) 
(2) 
(3) Does not exist
(4) None of the above (5)

MAT1512/001
1
9
3. Determine the following limit:
 xh
x 4
lim

where  












42
4
3
82 2
xifx
xif
x
xh
Then the correct answer is:
(1) 2
(2) 3
8

(3) 8
(4) None of the above. (5)
4. Determine the following limit:
x
xx
x 52
25
lim
2
5
2 


Then the correct answer is:
(1) 0
(2) Does not exist
(3) 5
(4) None of the above. (5)
5. Determine the following limit:
1sin
1sin
lim
2
2 

 x
x
x 
Then the correct answer is:
(1) 2

(2) 2
(3) 1
(4) None of the above. (5)

20
6. Determine the following limit:
 5
25
lim
2


 xx
x
x
Then the correct answer is:
(1) 
(2) 1
(3) Does not exist
(4) None of the above. (5)
7. Use Squeeze Theorem to determine
   
2
lncos1
lim 2
2


 x
xx
x
.
Then the correct answer is:
(1) 2
1

(2) 2
1

(3) 0
(4) None of the above. (5)
8. Let
 
   
 















22cos
2
2
4
4sin
1 2
2
2
xifxa
xifb
xif
x
x
x
xf
Determine the value(s) of a and b that make the function  xf continuous at .2x
Then the correct answer is:
(1) 2a and 5b
(2) 5a and 2b
(3) 5a and 5b
(4) None of the above. (5)

MAT1512/001
2
1
The following Questions from Question 9, below, up to and including Question 20, below, are
about finding Limits from a graph.
Let the graph of the particular function  xg be represented as shown below:
Use the graph of g in the figure above to find the following values, if they exist:
9.  2g
Then the correct answer is:
(1) 1
(2) 3
(3) Undefined.
(4) None of the above. (5)
10.  xg
x 
 2
lim
Then the correct answer is:
(1) 2
(2) 1
(3) Undefined.
(4) None of the above. (5)

22
11.  xg
x 
 2
lim
Then the correct answer is:
(1) 2
(2) 3
(3) 1
(4) None of the above. (5)
12.  xg
x 2
lim

Then the correct answer is:
(1) 1
(2) 2
(3) 3
(4) None of the above. (5)
13.  xg
x 
 2
lim
Then the correct answer is:
(1) 2
(2) 1
(3) 5
(4) None of the above. (5)
14.  xg
x 
 2
lim
Then the correct answer is:
(1) 2
(2) 1
(3) 2
(4) None of the above. (5)

MAT1512/001
2
3
15.  xg
x 2
lim

Then the correct answer is:
(1) 5
(2) 1
(3) Does not exist.
(4) None of the above. (5)
16.  3g
Then the correct answer is:
(1) 3
(2) 2
3
(3) Undefined.
(4) None of the above. (5)
17.  xg
x 
 3
lim
Then the correct answer is:
(1) 4
(2) 2
3
(3) Undefined.
(4) None of the above. (5)
18.  xg
x 
 3
lim
Then the correct answer is:
(1) 2
(2) 1
(3) 2
3
(4) None of the above. (5)

24
19.  xg
x 3
lim

Then the correct answer is:
(1) 2
3
(2) 2
(3) 2
(4) None of the above. (5)
20. Identify the discontinuities in the function  xg graphed above.
Then the correct answer is:
(1) 2x , removable discontinuity and 2x , Jump discontinuity
(2) 3x , essential discontinuity and 2x , Jump discontinuity
(3) 2x , Jump discontinuity and 2x , removable discontinuity.
(4) None of the above. (5)
The following Questions from Question 21, below, to Question 27, below, refer to the
function h below:
Let h be a function defined as:
 











2
2
1
20
08
2 xifx
xifx
xifx
xh
21. Determine the following limit:
 xh
x 
 0
lim
Then the correct answer is:
(1) 0
(2) 8
(3) 2
(4) None of the above. (5)

MAT1512/001
2
5
22. Determine the following limit:
 xh
x 
 0
lim
Then the correct answer is:
(1) 0
(2) 8
(3) 2
(4) None of the above. (5)
23. Determine the following limit:
 xh
x 0
lim

Then the correct answer is:
(1) 0
(2) 8
(3) Does not exist.
(4) None of the above. (5)
24. Determine the following limit:
 xh
x 
 2
lim
Then the correct answer is:
(1) 2
(2) 8
(3) 6
(4) None of the above. (5)
25. Determine the following limit:
 xh
x 
 2
lim
Then the correct answer is:
(1) 2
(2) 8
(3) 6
(4) None of the above. (5)

26
26. Determine the following limit:
 xh
x 2
lim

Then the correct answer is:
(1) 8
(2) 2
(3) 6
(4) None of the above. (5)
27. Determine if the function h is continuous at x = 0 and x = 2.
Then the correct answer is:
(1) Yes, the function h is continuous at both 0x and 2x .
(2) No, the function h is NOT continuous at both 0x and 2x .
(3) The function h is NOT continuous at 0x but is \continuous at 2x .
(4) None of the above. (5)
28. Sketch the graph of h .
Then the correct answer is:
(1)

MAT1512/001
2
7
(2)
(3)
(4) None of the above. (5)

28
29. Use the Squeeze Theorem to determine
 
2
sin3
lim 2 

 x
ex
x
Then the correct answer is:
(1) 0
(2) 2
4
(3) 2
2
(4) None of the above. (5)
30. Let the functionf be defined by:
 













22
2
2
2
xifcx
xif
x
x
xf
wherec is a constant. Thenf is continuous at 2x for:
(1) No value ofc .
(2) 0c
(3) 4c
(4) 2c (5)
Total: [150]

MAT1512/001
21
ASSIGNMENT 02
Fixed Closing Date: 14 May 2021
Total Marks: 30
UNIQUE ASSIGNMENT NUMBER: 812568
1. Use the first principles of differentiation to determine  xf  for the following functions:
(a)   143 2  xxxf (2)
(b)   3
12


 x
x
xf (4)
(c)   x
xf 
 1
4 (4)
[10]
2. Use appropriate differentiation techniques to determine the first derivatives of the
following functions (simply your answers as far as possible).
(a)   v
vev
vf
v
23 
 (4)
(b)   t
ttc 7
50  (4)
(c)   1
1
3
2


 x
x
xf (4)
(d)    xxf tansincos (4)
(e)   x
x
xf sec
1tan 
 (4)
[20]
Total: [30]
(4)

ASSIGNMENT
Date: May
UNIQUE ASSIGNMENT NUMBER:
22
1. Determine the derivatives of the following functions:
(a)    xxxf lnln  (2)
(b)   1
1
4 

 x
x
xg (3)
(c)    3
2
1
2
  xexxh xx (3)
(d)     xxxy lnsin (2)
[10]
2. Determine the derivatives of the following inverse trigonometric functions:
(a)   xxf 1
tan 
 (5)
(b)   








1
cot
ln
12
x
xx
xy (5)
(c)     





 
2
cos3sin 11 x
xxg (5)
(d)    1tan 21   xxxh (5)
[20]
Total: [30]

23
MAT1512/001
ASSIGNMENT
Date: June
UNIQUE ASSIGNMENT NUMBER:
1. Use the method of implicit differentiation to determine the derivatives of the following
functions:
(a) 1sinsin  xyyx (3)
(b)   2
1
tan x
y
yx 
 (3)
(c) 44 yxyx  (3)
(d) yxyxy 2
cos  (3)
2. Find the number “ c ” that satisfy the Mean Value Theorem (M.V.T) on the given
intervals.
(a)   x
exf 
 , ]2,0[ (3)
(b)   2
 x
x
xf ,  ,1 (3)
3. Determine the equation of the tangent and normal at the given points:
(a) yxyxy 2
cos  , 



2
,1  . (4)
(b)   1
2
2 
 x
xh , at 1x . (4)
[8]
4. Find the derivative of    

x
x
dvvxf 22 [4]
Total: [30]

24
ASSIGNMENT
MULTIPLE CHOICE ASSIGNMENT
Date:
UNIQUE ASSIGNMENT NUMBER:
1. The equation of the ellipse is given as
1
22












b
y
a
x
Use implicit differentiation to determine the derivative of the equation of the ellipse given above.(5)
Then the correct answer is:
(1) y
xb
dx
dy 2

(2) ya
xb
dx
dy 2
2

(3) xb
ya
dx
dy 2
2

(4) None of the above.
2. Determine the slope of the equation in Question 1, above, at 00 , yx . (5)
Then the correct answer is:
(1)
0
0
2
y
xb
dx
dy  at  00 , yx .
(2)
0
2
0
2
ya
xb
dx
dy  at  00 , yx
(3)
0
2
0
2
xb
ya
dx
dy  at  00 , yx
(4) None of the above.

25
MAT1512/001
3. Hence or otherwise find the equation of the tangent at 00 , yx .
The equation referred to is in Question 1, above. (10)
Then the correct answer is:
(1) The equation of the tangent is:
1 ByAx
Where
4
0
a
x
A  and 2
0
b
y
B 
(2) The equation of the tangent is:
1 ByAx
Where
4
0
a
x
A  and 2
0
b
y
B 
(3) The equation of the tangent is:
1 ByAx
Where
4
0
a
x
A  and 2
0
b
y
B 
(4) None of the above.
[20]
4. Let 322  yxyx be the equation of an ellipse. By implicit differentiation determine the
equation of the normal of the equation given above at 1,1 . [10]
Then the correct answer is:
(1) 12  xy
(2) 1 xy
(3) xy 
(4) None of the above.

26
ASSIGNMENT 06
Fixed Closing Date: 16 July 2021
Total Marks: 30
UNIQUE ASSIGNMENT NUMBER: 784524
1. Determine the first order partial derivative of the following functions:
(a)  2
ln txz  (3)
(b)    dteyxF
x
y
t
 cos, (3)
(c)   xz
exyzyxf 
 2
,, (5)
[11]
2. Clairaut’s Theorem holds that yxxy UU  , show that the following equations obey
Clairaut’s Theorem.
(a)  yxu 2ln  (5)
(b) yeu xy sin (5)
[10]
3. Laplace’s equation holds that 0 yyxx UU , verify that the second derivative of the
following equations are Laplace’s equation.
(a) 22
ln yxu  (5)
(b) 22 yxu  (4)
[9]
Total: [30]

27
MAT1512/001
ASSIGNMENT
Date:
UNIQUE ASSIGNMENT NUMBER: 677794
1. Determine the following integrals:
(a) dUUU 




  7
2
2 56 (2)
(b) dx
x
xx
 






 1 (2)
(c)  




 
4
1
64 du
u
u (2)
(d)  
2
0
12 dxx (4)
[10]
2. Deter mine the following integrals:
(a)  dtt
t
0
3sin  (2)
(b) dx
x
x

1
0
2
cos
2sin (2)
(c) dx
x
xx
 






 1 (2)
(d) dxx 
4
0
1 (2)
3. Use substitution method to determine the following integrals:
(a) dx
x
x
  4
3
1 (3)
(b)  dsincos 4
 (4)

28
(c)  dxxx 122 (5)
[12]
Total:[30]

29
MAT1512/001
ASSIGNMENT
Date:
UNIQUE ASSIGNMENT NUMBER: 703736
1. Show whether or not the following differential equations are separable:
1.1 1
1


 y
x
dx
dy (3)
1.2 22 


x
ye
dx
dy yx
(3)
1.3    22 8ln tSt
dx
dy t  (3)
2. Solve the following differential equation by using separation of variables method:
2.1 y
dx
dy
x 4 (3)
2.2    
 tp
tp
dt
dp
sin
cos1 2

 (3)
3. Solve the following differential equation subject to the given initial conditions
3.1 
 siny
d
dy  ;   3y . (4)
3.2 xyy
dx
dy
x 2 ;   11 y . (4)
4. The population of a certain community is known to increase at a rate proportional to the
number of people present any time. If the population had double in 5 years, how long will
it take to triple ? [7]
Total: [30]
(3)
1 (3)

30
ASSIGNMENT 09
Fixed Closing Date: 25 August 2021
Total Marks: 100
UNIQUE ASSIGNMENT NUMBER: 643701
PAST EXAM PAPER

31
MAT1512/001

32

33
MAT1512/001