Study Guide
Mathematics and Applied Mathematics
Calculus
WTW 114
© 2025 University of Pretoria
1
Contents
Module Calendar: Important Dates & Overview
4
Course Details
A
Welcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
7
Welcome
7
B
Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
C
Assessment and related matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
D
Contact time and study hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
E
Disciplinary issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1
Functions
1.1 Logic/Proofs (2 lecture) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Numbers, intervals and inequalities (1 lecture) . . . . . . . . . . . . . . . . . . . . . . .
1.3 The absolute value (2 lectures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Trigonometry (1 lecture) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Functions and their graphs (2 lectures) . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 New functions from old functions (2 lectures) . . . . . . . . . . . . . . . . . . . . . . .
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2
Limits and Continuity
2.1 The tangent and velocity problems (1 lecture) . . . . . . . . . . . . . . . . . . . . . . .
2.2 The limit of a function (1 lecture) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Limit laws (1 lecture) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Precise definition of a limit (1 lecture) . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Continuity (3 lectures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Limits at infinity and horizontal asymptotes (1 lecture) . . . . . . . . . . . . . . . . . .
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3
Differentiation
3.1 Derivatives and rates of change (1 lecture) . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The derivative as a function (1 lecture) . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Differentiation rules (2 lectures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Derivatives of trigonometric functions (1 lecture) . . . . . . . . . . . . . . . . . . . . .
3.5 The chain rule (2 lectures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Exponential functions and the number e (1 lecture) . . . . . . . . . . . . . . . . . . . .
3.7 The derivatives of inverse functions (2 lectures) . . . . . . . . . . . . . . . . . . . . . .
3.8 Implicit differentiation (1 lecture) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Logarithmic functions and their derivatives (2 lectures) . . . . . . . . . . . . . . . . . .
3.10 Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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49
2
4
Applications of Differentiation
4.1 Maximum and minimum values (2 lectures) . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The Mean Value Theorem (2 lectures) . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 How the derivative affects the shape of the graph (2 lectures) . . . . . . . . . . . . . . .
4.4 Indeterminate forms and L’Hospital’s rule (2 lectures) . . . . . . . . . . . . . . . . . . .
4.5 Curve sketching (2 lectures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Optimization problems (2 lectures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Antiderivatives (1 lecture) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5
Integration
5.1 The area problem (1.5 lectures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The definite integral (2.5 lectures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 The Fundamental Theorem of Calculus (2 lectures) . . . . . . . . . . . . . . . . . . . .
5.4 Indefinite integrals (1 lecture) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Transformation of the integral (3 lectures) . . . . . . . . . . . . . . . . . . . . . . . . .
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Appendices
I
Differentiation formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II
Differentiation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
III List of standard indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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FLY@UP Support
62
Department’s Vision & Mission
64
3
Module Calendar: Important Dates & Overview
Below “WS” stands for “Worksheet” and “TT” stands for “timetable” and “A” stands for “Appendix”
(located at the back of our textbook).
Week
1
1
1
1
2
2
2
2
3
3
3
3
4
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4
4
5
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5
∗6
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Lecture date
Unit
Homework
Mon, Feb 10, 2025
1.1(a)
WS1
Tues, Feb 11, 2025
1.1(b)
WS1
Wed, Feb 12, 2025
1.2
(p. A9) no. 17, 19, 27, 35, 39, 57, 59, WS1
Fri, Feb 14, 2025
1.3
(p. A9) no. 43, 45, 47, 49, 51, 53, 55, 65, WS1
Mon, Feb 17, 2025
1.4
(p. A33) no. 23, 27, 29, 31, 45, 61, 65, 67, WS2
Tues, Feb 18, 2025
1.5
(p. 17) no. 1, 35 - 45 odd, WS2
Wed, Feb 19, 2025
1.6
(p. 42) no. 33, 35, 37, 45, 47, 49, 51, WS2
Fri, Feb 21, 2025
2.1
WS2
Mon, Feb 24, 2025
2.2
(p. 92) no. 5, 7, 9, 19, 21, 29, 31, 35, 37, WS3
Tues, Feb 25, 2025
2.3
(p. 102) no. 1, 11-33 odd, 37, 41, 49, WS3
Wed, Feb 26, 2025
2.4
(p. 113) no. 1, 19, 23, WS3
Fri, Feb 28, 2025
2.5(a)
(p. 124) no. 3, 5, 9, 19, 21, 23, 33, 35, 43, 49, WS3
Mon, Mar 3, 2025
2.5(b)
continue above problems, WS4
Tues, Mar 4, 2025
2.6
(p. 137) no. 1, 3, 7, 9, 15-31 odd, WS4
Wed, Mar 5, 2025
3.1
(p. 149) no. 5, 7, 19, 21, 23, 25, 33, WS4
Fri, Mar 7, 2025
3.2
(p. 161) no. 1, 3, 5, 7, 21-31 odd, 41, WS4
Mon, Mar 10, 2025
3.3(a)
(p. 181) no. 3-33 odd, 37, 39, 49, (p. 189) 1-29 odd, 33, 35, WS5
Tues, Mar 11, 2025
3.3(b)
continue above problems, WS5
Wed, Mar 12, 2025
3.4
(p. 197) no. 1-27 odd, 35, 39, WS5
Fri, Mar 14, 2025
3.5(a)
(p. 206) no. 7-55 odd and not exponential, WS5
Mon, Mar 17, 2025 3.5(b), Fri TT
continue above problems, WS6
Tues, Mar 18, 2025
3.6
(p. 52) no. 3, 15, 17, 21, (p. 206) 7-55 odd exponential, WS6
Wed, Mar 19, 2025
3.7
(p. 64) no. 5, 7, 13, 19, 69-71 odd, 75, 77 (p. 224) 63-75 odd, WS6
Fri, Mar 21, 2025 Public Holiday
Mon, Mar 24, 2025 UP Test Week
Tues, Mar 25, 2025 UP Test Week
Wed, Mar 26, 2025 UP Test Week
Fri, Mar 28, 2025 UP Test Week
continues on next page...
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Week
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Lecture date
Unit
Homework
Mon, Mar 31, 2025
3.8
(p. 214) no. 5-21 odd, 27-33 odd, WS7
Tues, Apr 1, 2025
3.9(a)
(p. 224) 3-39 odd, 45-55 odd, WS7
Wed, Apr 2, 2025
3.9(b)
continue above problems, WS7
∗Lecture Videos∗
3.10
(p. 266) 1, 3, 5, 11-15 odd, 19-23 odd, 35-43 odd, WS7
Fri, Apr 4, 2025
4.1(a)
(p. 286) 1-13 odd, 29-47 odd, 51-65 odd, WS7
Mon, Apr 7, 2025
4.1(b)
continue above problems, WS8
Tues, Apr 8, 2025
4.2(a)
(p. 295) no. 9-17 odd, WS8
Wed, Apr 9, 2025
4.2(b)
continue above problems, WS8
Fri, Apr 11, 2025
4.3(a)
(p. 305) no. 1, 5, 9-29 odd 29, WS8
Mon, Apr 14, 2025
UP Recess
Tues, Apr 15, 2025
UP Recess
Wed, Apr 16, 2025
UP Recess
Fri, Apr 18, 2025
Public Holiday
Mon, Apr 21, 2025 Public Holiday
Tues, Apr 22, 2025 4.3(b), Mon TT
continue above problems, WS9
Wed, Apr 23, 2025
4.4
(p. 316) 9-69 odd, WS9
Fri, Apr 25, 2025
4.5
(p. 327) no. 1-19 odd, WS9
Mon, Apr 28, 2025 Public Holiday
Tues, Apr 29, 2025
4.6(a)
(p. 342) no. 3, 7, 9, 13, 15, 19, 21, 25, WS10
Wed, Apr 30, 2025 4.6(b), Mon TT
continue above problems, WS10
Fri, May 2, 2025
4.7
(p. 361) 1-25 odd, 29-53 odd, WS10
Mon, May 5, 2025
UP Test Week
Tues, May 6, 2025
UP Test Week
Wed, May 7, 2025
UP Test Week
Fri, May 9, 2025
UP Test Week
Mon, May 12, 2025
5.1
(p. 381) no. 1(a), 5, WS11
Tues, May 13, 2025
5.2(a)
(p. 394) no. 1, 5, 7, WS11
Wed, May 14, 2025
5.2(b)
continue above problems, WS11
Fri, May 16, 2025
5.2(c)
continue above problems, WS11
Mon, May 19, 2025
5.3(a)
(p. 406) 25-53 odd, WS12
Tues, May 20, 2025
5.3(b)
continue above problems, WS12
Wed, May 21, 2025
5.4
(p. 415) 1-23 odd, 27-53 odd, WS12
Fri, May 23, 2025
5.5(a)
(p. 425) no. 1-53 odd, 59-79 odd, WS12
Mon, May 26, 2025
5.5(b)
continue above problems, WS13
Tues, May 27, 2025
5.5(c)
continue above problems, WS13
Wed, May 28, 2025
5.5(d)
continue above problems, WS13
Fri, May 29, 2025
UP Recess
tutorials week by week schedule on next page...
5
Tutorials Week by Week
Week 1. (Feb 10 - 14) No tutorials.
Week 2. (Feb 17 - 21) Tutorial Activity 1 - Attendance Required
Week 3. (Feb 24 - 28) Tutorial Activity 2 - Attendance Required
Week 4. (Mar 3 - 7) Tutorial Activity 3 - Attendance Required
Week 5. (Mar 10 - 14) Tutorial Activity 4 - Attendance Required
Week 6. (Mar 17 - 21) Practice Problems Set 1 - Attendance Not Required
Week 7. (Mar 24 - 28) Semester Test Week. No tutorials.
Week 8. (Mar 31 - Apr 4) Tutorial Activity 5 - Attendance Required
Week 9. (Apr 7 - 11) Tutorial Activity 6 - Attendance Required
Week 10. (Apr 14 - 18) UP Recess. No tutorials.
Week 11. (Apr 21 - 25) Practice Problems Set 2 - Attendance Not Required
Week 12. (Apr 28 - May 2) Practice Problems Set 3 - Attendance Not Required
Week 13. (May 5 - 9) Semester Test Week. No tutorials.
Week 14. (May 12 - 16) Tutorial Activity 7 - Attendance Required
Week 15. (May 19 - 23) Tutorial Activity 8 - Attendance Required
Week 16. (May 26 - 30) Practice Problems Set 4 - Attendance Not Required
The General Rule: If we have a normal, full week (no holidays, etc.) then tutorial attendance is required,
and you will work on a tutorial activity. If we don’t have a normal, full week, then tutorial attendance is
not required, and you will work on a practice problem set.
Attendance Required Tutorials: You must attend the tutorial group assigned to you in your timetable.
Late registration students (after Feb 6th): You must register for a tutorial by emailing wtw114@up.ac.za
(attach your late registration papers). You must attend your registered tutorial group.
Attendance Not Required Tutorials: You do not have to attend your assigned/registered tutorial group.
You can attend any tutorial group you want.
6
Course details
A
Welcome
Welcome to WTW114 Calculus 2025!
We are happy that you are joining us and want to encourage everyone to fully engage with the module—
take it upon yourself to seize upon (and to create!) opportunities to interact with fellow students and
staff so that we may grow a vibrant learning environment.
The theme for this class is Work hard on maths every day, then success will come your way.
We wish you all the best with WTW114 as well as all your other modules.
The WTW114 Team
Aim of the module
Calculus provides us with a powerful tool to solve mathematical problems as well as real world problems
that typically arise in Physics, Biology, Engineering and Economics.
The emphasis in WTW114 is twofold. To equip students with the analytical skills required to solve
such mathematical and real world problems, and to develop the methodological techniques necessary
for mathematical proof.
This module prepares students for courses in Advanced Calculus and Analysis that they will meet in
their second and third years of study.
B
Organization
B.1
Textbook
James Stewart, Single variable calculus: Early transcendentals, International Metric Ed, 9th ed.1
1
You are encouraged to read the few pages on the “Principles of Problem Solving” (p. 71).
7
B.2
Staff
(a) Dr Christopher Michael Schwanke (course coordinator)
Vetman Building 1-8, christopher.schwanke@up.ac.za
(b) Prof John van den Berg
Mathematics Building 1-24.1, john.vandenberg@up.ac.za
(c) Dr Ruaan Kellerman
Mathematics Building 2-28 ruaan.kellerman@up.ac.za
(d) Miss Baphumelele Nxala (module administrator)
Mathematics Building 2-35, baphumelele.nxala@up.ac.za
(e) All Admin Queries wtw114@up.ac.za
B.3
Communication
All admin related module correspondence should be sent to wtw114@up.ac.za.
If you have a mathematical content question however, you may email any of the staff members listed
above directly. Whilst every effort will be made to respond to your query quickly, please be aware that
with a class of several hundred students, this will not always be possible.
Announcements will be made via clickUP.
B.4
Consulting hours
Details on how to book a consultation with a staff member are available on clickUP.
B.5
Lectures
All lectures will be held face-to-face on campus. Any changes will be communicated via clickUP. You
should have four scheduled time slots for lectures per week.
Due to the substantially large size of this class, we offer three lectures per day on the days we do have
lectures. As such, our lectures are organized in three groups. You choose which group you want
to join and then just show up. No admin needed from you! We highly recommend that you, if
possible, stick with one lecture group for the duration of the semester! The three lecture groups are
taught by three different lecturers, whom each have different teaching styles, and go at slightly different
paces.
Caution! Each lecture group can hold at most 260 students, which is much smaller than the number
of students in this class. If you attend a lecture group that is full, you must, if at all possible, switch to
another group.
The lecture schedule is below. (HB is the Humanities Building, and “vd” is short for “van der”.)
Monday
–Group G01 - 10:30 - 11:20 - Muller Hall
–Group G02 - 7:30 - 8:20 - Centenary 1
–Group G03 - 11:30 - 12:20 - vd Bijl Hall
Tuesday
–Group G01 - 7:30 - 8:20 - Te Water Hall
–Group G02 - 8:30 - 9:20 - Centenary 4
–Group G03 - 9:30 - 10:20 - Centenary 4
8
Friday
–Group G01 - 8:30 - 9:20 - Muller Hall
–Group G02 - 9:30 - 10:20 - vd Bijl Hall
–Group G03 - 10:30 - 11:20 - HB 4-2
Wednesday
–Group G01 - 8:30 - 9:20 - Muller Hall
–Group G02 - 9:30 - 10:20 - vd Bijl Hall
–Group G03 - 7:30 - 8:20 - vd Bijl Hall
• Lecture Group G01 will be taught by Dr. Schwanke.
• Lecture Group G02 will be taught by Dr. Kellerman.
• Lecture Group G03 will be taught by Prof. van den Berg.
Note that there will also be short “mini lectures” in video format posted on clickUP throughout the
semester. The content of these videos is examinable! The reason we need to make these videos is
because we lost two weeks of lectures this year due to the implementation of the two semester test
weeks.
B.6
Tutorials (also called Practicals)
All tutorials will be held face-to-face on campus in two-hour sessions. Any changes will be communicated via clickUP. We have six timeslots for WTW 114 tutorials. You should have one scheduled
time slot for tutorials/practicals per week. Officially, UP allocates three hours per tutorial session per
week, but again, face-to-face tutorials will be held for only two hours per week. For the third hour, we
will present solutions to the tutorial exercises on ClickUP.
All first-year NAS students registered on or before Feb 6: Your tutorial timetable is set centrally
by the faculty. You must attend the tutorial session allocated to you on your timetable! Switching
tutorial sessions is not allowed for any reason.
All other students: You must send an email to wtw114@up.ac.za to register for a tutorial group. Please
note that we may not be able to accommodate your first choice, as some of the venues may be full.
You must attach a copy of your timetable, and, if applicable, your late registration papers to your
email.
Group T01
–Tues - 14:30 - 16:30 - EMS 4-151
Group T02
–Wed - 10:30 - 12:30 - North Hall
Group T03
–Thurs - 10:30 - 12:30 - EMS 4-150 and EMS 4-151 (go to either one of these venues, but if one of the
venues is full, go to the other)
Group T04
–Tues - 10:30 - 12:30 - South Hall and Theology 1-9 (go to either one of these venues, but if one of the
venues is full, go to the other)
Group T05
–Wed - 14:30 - 16:30 - HB 4-1 and HB 4-3 (go to either one of these venues, but if one of the venues is
full, go to the other)
Group T06
–Mon - 13:30 - 15:30 - Centenary 6
9
During eight of the tutorial sessions, you will be working on tutorial activities for a grade in teams of
four. Students will form the teams themselves, so you may want to liaise with anyone you study with before you choose a tutorial team. See Section B.5 for more information on these tutorial activities.
(Note that in WTW114 a “practical” is the same thing as a “tutorial.”) You should keep these time slots
open for course activities. You should complete the exercises in this study guide of all the units that
were covered in lectures during the week.
B.7
Code of conduct
We are not only facilitating learning in a module, we are also preparing you for the world of work. We
expect you to adhere to the code of conduct as spelled out in the Escalation policy of UP.
Communication via email
When you email your lecturer, you have to use a respectful tone and include all the following aspects:
• A clear and explanatory subject line;
• Your full name and surname at the end of the mail;
• Your student number;
• The module involved; and
• Short and clear message.
Compliments and complaints
You are more than welcome to express your appreciation to your lecturer or tutor and supply feedback
about aspects of the course that you enjoy and find valuable.
If you have a query or complaint, you have to submit it in writing with specifics of the issue or the nature
of the complaint. It is imperative that you follow the procedure outlined below in order to resolve your
issues:
1. Consult the lecturer concerned about your complaint/concerns.
If the matter has not been resolved,
2. consult the class representative (the primary function of the Class Representative is to serve as a
two-way communication channel between the class and the lecturer).
If the matter has not been resolved,
3. consult the module coordinator (large modules with multiple lecturers).
If the matter has not been resolved,
4. consult the Head of Department.
If the matter has still not been resolved,
5. consult with the Dean of the Faculty.
C
Assessment and related matters
All class tests, semester tests and the exam will be conducted on campus. The following serves as a
guideline.
10
C.1
Material for tests and the Examination
Material for tests and the Examination will be posted on clickUP. Dates for the two Semester Tests are
given on the UP Website. Please note, however, that these dates are provisional and might change. In
the event that they do change, you will be given advance warning.
Tests and exams in this course aim primarily to test your understanding of the relevant mathematical
concepts, as opposed to the rote application of recipes and memorization of proofs. This may be done
using a variety of questions which test your ability to perform certain calculations, your understanding
of definitions, theorems and their proofs, your ability to solve mathematical problems, and, your ability
to give mathematical formulations of real-world problems (so called word sums).
Also take note of the following:
1. Student cards must be presented on request during tests and examinations.
2. The removal of examination papers from the examination venue is a transgression of the examination rules and is regarded as a serious offence.
3. All test queries must be made within 3 days after tests are handed back to students.
C.2
Absence from assessments
If you are absent from a Tutorial Activity or Semester Test 2, you must submit to a sick note via
clickUP within 3 working days of the test being written. You must present convincing proof of the
reason for your absence, for example a medical certificate (see below). In the case of a normal invigilated
exam, the relevant faculty office should be informed of the absence.
There will be no “make ups” for tutorial activities. If you miss a tutorial activity for a valid reason (with
proof), then that those marks will be excluded when your total mark is calculated.
If you missed a tutorial activity due to illness, you must submit a doctor’s note on clickUP. If you missed
a tutorial activity due to late registration, you must submit your late registration papers on clickUP.
Missed worksheet submissions are not excluded for any reason, but we offer bonus marks for those
instead. Thus you can miss a few worksheets and still get full marks for them overall. If you miss
Semester Test 2 (and you have acceptable excuse), then you may write a Sick Test at a time that will
be announced. It will cover the work of both semester tests. If you miss Semester Test 1 for whatever
reason, your Semester Test 2 score will simply count as your Semester Test 1 score as well.
• Valid original sick notes are accepted if issued by a medical doctor registered at the Health Professions Council of South Africa (HPCSA). The only other type of sick note that is accepted are
those issued by an Advanced Practice Nurse (a registered nurse with a postgraduate qualification)
as determined by the South African Nursing Council who has a BHCF practice number, provided
that the diagnosis falls only within their specific field of specialisation.
• An affidavit will only be accepted if supported by substantiating documentation, e.g. case report
or criminal charge with case number obtained from a police station, valid medical certificate
for injuries, a death certificate for a funeral, etc. Please note that submission of fraudulent sick
notes and affidavits is a criminal offense, which will lead to disciplinary action and may result in
dismissal.
11
C.3
Practice Problems
Practice problems are given for each and every unit in this study guide. It is extremely important that
you attempt to complete as many as these practice problems as you have time for before starting the
weekly worksheet! However, these practice problems are not to be submitted for a grade. Thus these are
a great, low-stress way to practice calculus!
C.4
Weekly Worksheets
Most weeks you will be given a weekly worksheet consisting a set of problems to complete before your
next week’s tutorial session. These problems will be more difficult than the practice problems you were
given. That’s one reason it’s critical to start with the practice problems! Although the worksheets are not
marked, you will submit them and receive a small amount of marks for completing the worksheet. In
order to receive marks for worksheet submissions, you must attempt all of the problems and show
sufficient effort for each one! Marks for worksheet submissions are given on an “all or nothing” basis.
Worksheets are submitted online. Details on how to submit can be found on ClickUP. Unless otherwise
specified on ClickUP, all worksheets will be due on Mondays at 7:00am.
No late submissions of worksheets are accepted for any reason, so be sure to not wait until the
last minute! However, you can miss three worksheets and still receive full marks in total for the worksheets.
Although these worksheets are not marked, they play a fundamental role in the tutorial activities, which
are marked, see next section.
C.5
Tutorial Activities
Provisionally, eight weeks of tutorials are reserved for “tutorial activities”. Students will work on these
tutorial activities in teams of four, with no more than two teams of less than four. There will be no
teams more than four in size.
These tutorial activities consist of three questions to be worked on, and turned in, by each team.
C.6
Practice Problem Sets
Provisionally, four weeks of tutorials are reserved for “practice problem sets”. These are similar to
tutorial activities, except they are not mandatory and not submitted for a mark. Nonetheless, they prepare
you for the semester tests and exam, as so they are just as important as tutorial activities!
C.7
Semester Tests
There will be two semester tests given in this module. Times and venues are available on the university’s
website, but those could potentially change. The course coordinator will clearly communicate with you
on ClickUP regarding the times, venues, and content of the semester tests.
C.8
How Will Everything Be Marked?
With the exception of both semester tests, every single question written for a grade will be graded out
of five marks. That includes all tutorial activities. The rules for the marking tutorial activities are as
follows:
–5. Perfect answer!
–4. Only very minor error(s) present, solid understanding of the material clearly demonstrated.
12
–3. There exist more than just very minor errors, a passworthy level of understanding has been demonstrated.
–2. There exist more than just very minor errors, an unpassworthy level of understanding has been
demonstrated, but the work is close to passworthy.
–1. The work is not close to passworthy, but at least some meaningful progress toward a correct answer
has been shown (e.g. at least writing down a relevant definition).
–0. Not close to passworthy, no meaningful progress toward a correct answer shown.
–No half marks will be given.
For Semester Tests 1 and 2, the marking is almost the same, except each question will be worth 15
marks:
–15. Perfect answer!
–12. Only very minor error(s) present, solid understanding of the material clearly demonstrated.
–9. There exist more than just very minor errors, a passworthy level of understanding has been demonstrated.
–6. There exist more than just very minor errors, an unpassworthy level of understanding has been
demonstrated, but the work is close to passworthy.
–3. The work is not close to passworthy, but at least some meaningful progress toward a correct answer
has been shown (e.g. at least writing down a relevant definition).
–0. Not close to passworthy, no meaningful progress toward a correct answer shown.
–No in-between marks will be given.
C.9
Semester Mark
There are 380 total marks available prior to the semester exam. The breakdown of these marks is as
follows:
2 marks each × 10 worksheets = 20 marks
Worksheet Submissions:
Tutorial Activities:
15 marks each × 8 tutorial activities = 120 marks
Semester Test 1:
120 marks
Semester Test 2:
120 marks
Total:
380 marks
Your semester mark will be given as a percentage out of 100, rounded to the nearest integer.
So for example:
285/380 marks = 75%
190/380 marks = 50%
152/380 marks = 40% =⇒ exam admission
Note: There will 13 worksheets to be submitted for a mark. Thus there will be the opportunity to earn
six bonus marks by submitting all 13 worksheets!
13
C.10
Cumulative Semester Test 2
Semester Test 2 will be cumulative, meaning it will cover everything from the first week of class up
to where we are in the class at the time Semester Test 2 is written. Your Semester Test 2 score will
replace your Semester Test 1 score provided you score better on Semester Test 2 than Semester
Test 1.
For example, if you score 54 on Semester Test 1 and 114 on Semester Test 2, then your official grade
book will show a score of 114 for Semester Test 1 and a score of 114 for Semester Test 2. Your Semester
Test 1 score got replaced by your improved Semester Test 2 score! However, for example, if you score
90 on Semester Test 1 and 50 on Semester Test 2, then your official grade book will show a score of 90
for Semester Test 1 and 50 for Semester Test 2. Your Semester Test 1 score did not get replaced since
your Semester Test 2 score was lower.
Again, Semester Test 2 will only replace your Semester Test 1 score if you do better on Semester Test
2. Semester Test 1 cannot affect your Semester Test 2 score in any way.
Two key takeaways from this policy are the following.
(i) Your learning in WTW 114 should be permanent.
(ii) You can always keep improving in WTW 114, and you get rewarded for doing so.
C.11
Missing Tutorial Activities and Semester Tests
Each Tutorial Activity that is missed with a valid excuse (e.g. sick with a doctor’s note) cannot be made
up but can be excluded from the calculation of your marks.
If you miss Semester Test 1 for any reason, you will score a zero but your Semester Test 2 score will
later replace it.
If you miss Semester Test 2 with valid excuse with proof (e.g. sick with doctor’s note), your mark for the
Special Test/Sick Test will be used in its place in the above calculation for the Semester Mark. However,
the Special Test/Sick Test will not replace your Semester Test 1 score, even if you score better on
the Special Test.
If you miss Semester Test 1 and write Semester Test 2, then you do not need to take the Special
Test/Sick Test. In fact, you do not qualify to write the Special Test/Sick Test in this case.
C.12
Examination admission
A minimum Semester Mark of 40% is required to be admitted into the examination.
C.13
Semester Examination
Only those who are admitted into the examination will receive a semester exam mark.
The semester exam will be graded out of 120 marks, just like the two semester tests.
Your Semester Exam mark will be given as a percentage out of 100, rounded to the nearest integer.
C.14
Final Mark
Your Final Mark will be computed as follows:
14
• Semester Mark: 60%
• Examination Mark: 40%
As such, your final mark is calculated using the following formula:
Final Mark = 0.6 × (Semester Mark) + 0.4 × (Examination Mark)
To pass the module, a Final Mark of at least 50% is required together with a sub-minimum of 40% for
the Examination.
C.15
Supplementary Examination
A student who has a final mark between 40% and 49%; will not have passed the module, but will have
qualified for a Supplementary Examination.
A student who is admitted to the Supplementary Examination must obtain a minimum mark of 50% for
this exam in order to pass the course, regardless of their Semester Mark.
D
Contact time and study hours
This module carries a weight of 16 credits, indicating that, on average, a student should spend some 160
hours to master the required skills (including time for preparation for tests and the examination). This
means that, on average, you should devote some 11.5 hours of study time per week to this module.
E
Disciplinary issues
The policy of the Department is to refer all incidents in which there is a suspicion of dishonesty or other
irregularity to the Disciplinary Committee of the University.
Plagiarism is a serious form of academic misconduct. It involves both appropriating someone else’s work
and passing it off as one’s own work afterwards. Thus, you commit plagiarism when you present someone else’s written or creative work (words, images, ideas, opinions, discoveries, artwork, music, recordings, computer-generated work, etc.) as your own. Only hand in your own original work. Indicate precisely and accurately when you have used information provided by someone else. Referencing must be
done in accordance with a recognised system. Indicate whether you have downloaded information from
the Internet. For more details, visit the library’s website: http://www.library.up.ac.za/plagiarism/index.htm.
15
Theme 1: Functions
Unit 1.1: Logic/Proofs
2 lecture
Note: This unit was added in order to help students with their proofwriting. Although proofwriting
has always been expected of students in this course, there was little time given in class to help students
prepare for this responsibility. We have added this unit to address this issue.
Textbook: None
Learning outcomes: On completion of this unit you should be able to
1. understand basic symbolic logic
2. be able to fill in truth tables
3. be able to use mathematical qualifiers
4. be able to write direct mathematical proofs
5. be able to write mathematical proofs by contradiction
6. be able to write mathematical proofs by contraposition
7. be able to write mathematical proofs via mathematical induction
8. to be able to disprove false mathematical statements using a counterexample
Problems: None from the textbook, but there will be some problems in the weekly worksheets to help
you practice these important concepts.
Unit 1.2: Numbers, intervals and inequalities
1 lecture
Textbook: Appendix A, pp. A2–A6.
Learning outcomes: On completion of this unit you should
1. know what the symbols N, Z, Q, R and ∅ (see additional notes) stand for.
2. be able to use set-builder notation to describe sets.
3. be able to use the union and the intersection of two sets, as well as the compliment of one set with
another (see additional notes), to describe a set.
4. be able to use interval notation and set-builder notation to describe intervals.
5. know the rules for inequalities and be able to use them to solve inequalities.
16
6. be able to use standard bracket notation for finite sets.
Additional notes:
1. Let S be a set. We use the notation x ∈ S to denote that x is an element of S . A set S is contained
in a set T , denoted by S ⊂ T , if x ∈ S implies that x ∈ T .
2. On page A3, study the paragraph explaining the following ideas: set; element of a set; the union
S ∪ T of sets S and T ; the intersection S ∩ T of sets S and T ; set-builder notation.
3. The empty set, written as ∅, is the set that contains no elements.
4. Let S and T be sets. Then S compliment T , written as S \ T , is the set which consists of all
elements in S that is not in T .
5. You must remember the following special symbols that will be used in all the mathematics modules:
R is the set of all the real numbers.
Q is the set of all the rational numbers.
Z is the set of all the integers.
N is the set of all the natural numbers.
6. In set-builder
n notation,
o
Q = mn | m, n ∈ Z and n , 0
Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}
N = {1, 2, 3, . . .}
7. In list 1 on page A4, study the set-builder notation in which the different types of intervals are
expressed. Here are some examples:
(a) Describe the set {x ∈ Z | −4 ≤ x < 1} in your own words and give all its elements.
Solution. It is the set of all integers that are greater or equal to −4 and less than 1. Therefore
{x ∈ Z | −4 ≤ x < 1} = {−4, −3, −2, −1, 0}.
(b) Write the following sets as intervals or as the union or intersection of intervals:
i. {x ∈ R | 1 ≤ x < 3} = [1, 3)
ii. {x ∈ R | 0 ≤ x ≤ 3 and 1 < x < 4} = [0, 3] ∩ (1, 4) = (1, 3]
iii. {x ∈ R | 0 ≤ x ≤ 3 or 1 < x < 4} = [0, 3] ∪ (1, 4) = [0, 4)
iv. {x ∈ R | −3 ≤ x < −1 or 0 ≤ x ≤ 2} = [−3, −1) ∪ [0, 2].
8. Simplify the following expressions:
i. (−∞, 3] \ (0, 4) = (−∞, 0]
ii. [3, 4] \ ∅ = [3, 4]
iii. ([3, 4] ∪ (4, 5)) ∩ ∅ = ∅
9. The Rules for Inequalities (list 2 on page A4) is a useful summary of the rules for solving inequalities. Study Examples 1, 2, 3, and 4.
10. In Example 3 (page A5) the inequality x2 − 5x + 6 ≤ 0 is solved.
Get used to making use of what is called the “visual method” and draw the parabola
y = x2 − 5x + 6, showing its intersection with the x-axis.
17
Then the solution is all x for which y ≤ 0, that is, 2 ≤ x ≤ 3. The solution set is the interval [2, 3].
11. In Example 4 the inequality x3 + 3x2 − 4x > 0 is solved.
First factorise y = x3 + 3x2 − 4x = x(x2 + 3x − 4) = x(x + 4)(x − 1).
Then draw the 3rd degree polynomial by observing for example that if x = −1 then y = 6.
Solution: −4 < x < 0 or x > 1, that is the set (−4, 0) ∪ (1, ∞).
18
Unit 1.3: The absolute value
2 lectures
Textbook: Appendix A, pp. A6–A9.
Learning outcomes: On completion of this unit you should be able to
1. write down and use the definition of the absolute value of a number.
2. solve absolute value inequalities.
Additional notes:
1. One of the most important applications of the absolute value will be the type discussed in Example
5 on page A7. Study it well and rewrite the following expressions without the absolute value
symbol:
(a) |x + 1|
(b) |x2 − 1|
(c) |x2 + 1|
(d) |2x − 1|
2. On page A7 in list 6, there are three rules that are helpful for solving absolute value equations and
inequalities. (One can also solve these equations by making use of the definition of the absolute
value.)
In Example 7 we must solve |x − 5| < 2. That means the number x − 5 is distance less than 2 units
away from 0, that is, −2 < x − 5 < 2.
So 3 < x < 7 and the solution set is the open interval (3, 7).
3. The properties in block 5 (p. A7) can be proved by using property 4, |a| =
√
4. Property 4, |a| = a2 , also implies that |a|2 = a2 .
19
√
a2 .
Unit 1.4: Trigonometry
1 lecture
Textbook: Appendix D, pp. A24–A33.
Learning outcomes: On completion of this unit you should be able to
1. find the sin, cos, tan, cosec, sec and cot of angles in radians.
2. solve simple trig equations.
3. write down and use the trigonometric identities.
Additional notes:
1. An angle has a vertex and two sides (Figure 1, p. A24) and can be measured in degrees or radians.
Formula 1 on page A24 gives the conversion rule:
π rad = 1800 .
In Calculus, we use radians to measure angles.
2. Figure 1 on page A24 shows a sector OAB of a circle with an angle θ with vertex the center of the
circle and θ subtending the arc AB “of length a”. The circle has radius r.
We know the definition of the length of a straight line. Without properly defining what the length
of an arc is, we will work from the starting point that the circumference of a circle is 2π times
the radius of the circle. We cannot prove this here, since then we would need the definition of the
length of an arc. By using the circumference of a circle, we can prove (see p. A24) that θ = ar . We
can use this formula to determine the length of an arc of a circle.
With this formula one can now see that in a circle with radius 1, the circumference of 2π is
subtended by an angle of 2π radians.
3. Study the definitions of the standard position of an angle and what is meant by a positive angle
and a negative angle on page A25.
4. Study and memorize the formulas in lists 4 and 5 on page A26 very well—especially the new
ones, namely
• the reciprocal of sin θ, namely csc θ;
• the reciprocal of cos θ, namely sec θ;
• the reciprocal of tan θ, namely cot θ.
5. Study and memorize the special triangles at the bottom of page A26 and/or the unit circle.
6. Study Example 3 (p. A27) and use the technique in the example to do Exercise D (p. A32) no. 23.
Exercises: Find sin 0, sin π4 , sin π6 , sin π, sin 1, sin 2 and sin 3 by using a calculator.
7. Example: Find all values of x in the interval [0, 2π] that satisfy the equation 2 cos x − 1 = 0.
Solution: From 2 cos x − 1 = 0 follows that cos x = 12 .
Using one of the special triangles gives one solution x = π3 . Since y = cos x is also positive in the
fourth quadrant, x = 2π − π3 = 5π
3 is another solution.
Exercises: Solve the following equations in [0, 2π] without a calculator:
(a) 2 sin x − 1 = 0
20
(b) 2 sin2 x = 1
(c) tan x = 1
8. Example: Solve 3 sin x + 1 = 0 in [0, 2π]. (Use a calculator.)
Solution: It follows that sin x = − 13 .
Because x is in radians, switch the calculator from degrees to radians. Enter − 13 and find the
inverse function of sin of − 31 , namely −0.34.
There are solutions in the 3rd and the 4th quadrants, namely
x = π + 0, 34 or x = 2π − 0.34
= 3.48
= 5.94.
Exercise: Find all the solutions of the equation tan2 x − 9 = 0 in [0, 2π]. (Use a calculator.)
Trigonometric Identities (p. A28)
9. Memorize list 7 and the basic Identity 8
sin2 θ + cos2 θ = 1.
10. Identity 9, tan2 θ + 1 = sec2 θ, follows from Identity 7 by dividing both sides by cos2 θ :
1
1
(sin2 θ + cos2 θ) =
,
2
cos θ
cos2 θ
sin θ
cos θ
2
that is, cos
2 θ + cos2 θ = sec θ, and so
2
2
sin θ 2
+ 1 = sec2 θ. Thus tan2 θ + 1 = sec2 θ.
cos θ
11. Identity 10, 1 + cot2 θ = cosec2 θ, follows by dividing both sides of Identity 7 by sin2 θ. Do it as an
exercise.
21
12. The following identities are well-known from school and must be memorized:
(10a) sin(−θ) = − sin θ
(10b) cos(−θ) = cos θ
(12a) sin(x + y) = sin x cos y + cos x sin y
(12b) cos(x + y) = cos x cos y − sin x sin y
13. The identities (13a), (13b), (16a), (16b), (17a), (17b), (18a) and (18b) on page A29 and A30
can be proved using the previous ones.
Now prove:
(16a) sin 2x = sin(x + x) = . . .
(16b) cos 2x = cos(x + x) = . . .
Also prove (17a), (17b), (18a) and (18b).
Graphs of the Trigonometric Functions
14. In Grade 11 you used to draw the function f (x) = sin x or y = sin x in the xy-system where the
y-axis represented the real numbers and the x-axis represented degrees. The same for y = cos x
and y = tan x.
On pages A30 and A31 you see the graphs of y = sin x, y = cos x and y = tan x where the x-axis
represents real numbers. In Calculus, if x is a real number, then sin x is found by interpreting x as
x radians. Similarly for the other trigonometric ratios.
15. Example: Find all the values of x in the interval [0, 2π] satisfying 2 cos x + 1 < 0.
Solution: We have to solve cos x < − 12 .
We first solve cos x = − 12 .
π
4π
x = π − π3 = 2π
3 or x = π + 3 = 3 .
Sketch y = cos x.
4π
2π 4π
From the graph, 2π
3 < x < 3 . So the set of solutions is ( 3 , 3 ).
22
Unit 1.5: Functions and their graphs
2 lectures
Textbook: Section 1.1, p. 8.
Learning outcomes: On completion of this unit you should be able to
1. determine whether a rule defines a function.
2. find the intended domain of a function (in case a domain is not specified).
3. write absolute value functions as piecewise defined functions.
4. determine whether a function is even, odd or neither.
Additional notes:
1. Study the definition of a function, its domain, range, independent variable and dependent variable
(pages 8 & 9). We denote the domain and range of a function f by D f and R f , respectively.
2. The set E in the definition of a function f on p. 8 is called the codomain of f .
3. We use the notation f : D → E to denote a function f with domain D and codomain E.
4. We denote the set {(x, y) | x ∈ D, y ∈ E} by D × E.
5. On p. 9 we see that the graph of a function f with domain D is the set of ordered pairs
{(x, f (x)) | x ∈ D} = {(x, y) | y = f (x), x ∈ D}.
Note that for a function f : D → E the graph
{(x, f (x)) | x ∈ D} ⊂ D × E.
completely determines f .
For example, the function f with f (x) = 2x + 3, x ≥ 0, can be thought of as the set
{(x, y) | y = 2x + 3, x ≥ 0}.
Therefore we also refer to f as the function y = 2x + 3, x ≥ 0.
Example: Does x2 + y2 = 1 define a function with domain [−1, 1]?
Solution: The graph {(x, y) | x2 + y2 = 1} is the circle with center the origin and radius 1 and for
example, x = 0 is associated with y = 1 and y = −1. Therefore x2 + y2 = 1 does not define a
function.
(Read about the “vertical line test” on p. 15).
Exercise: Does y2 = x define a function with domain [0, ∞)? Give reasons for your answer.
6. Study Examples 1, 2 and 3 (pp. 9–10) and do the following exercise:
Exercise: Sketch the graph of f (x) = x2 − 2x − 3, 0 < x < 4. What is the domain and what is the
range of f ?
7. If the domain of a function f is not specified (and in most cases it is not specified), then it is
understood that the domain of f is the largest possible set of real numbers x for which f (x) exists.
23
Injective, Surjective and Bijective Functions (p. 16)
8. Definition: Let D and E be sets of real numbers. A function f : D → E is
(i) one-to-one (injective) if for all x1 , x2 ∈ D,
if x1 , x2 then f (x1 ) , f (x2 );
(ii) onto (surjective) if for every y ∈ E there exists x ∈ D so that f (x) = y;
(iii) bijective if it is both one-to-one and onto.
9. Remark: Note the following.
(i) Let D and E be sets of real numbers and f : D → E a function. f is onto if and only if E is
the range of f .
(ii) Every function is a surjection onto its range.
(iii) A function f : D → E is injective if and only if for all x1 , x2 ∈ D, if f (x1 ) = f (x2 ) then
x1 = x2 .
(iv) If a function f is strictly increasing or strictly decreasing then it is one-to-one.
10. Example: The function f : (0, ∞) → (1, ∞) given by f (x) = 1x + 1 is bijective.
First we show that f is injective. Let a, b ∈ (0, ∞). Assume that f (a) = f (b). Then
1
1
+ 1 = + 1.
a
b
Therefore a1 = 1b so that a = b.
1
. Because y > 1, y − 1 > 0.
Now we show that f is surjective. Let y ∈ (1, ∞). Let x = y−1
1
Therefore x = y−1 is well defined and x > 0; that is, x ∈ (0, ∞). We compute f (x):
f (x) =
1
1
y−1
+ 1 = (y − 1) + 1 = y.
Therefore f is surjective.
Because f is both injective and surjective, it is bijective.
24
Piecewise Defined Functions (p. 16)
11. A piecewise defined function has a domain that is divided into pieces and has a different formula
on each of the pieces.
12. Example:
We write f (x) = |2x + 1| without the absolute value symbol:
(
2x + 1
if 2x + 1 ≥ 0
|2x + 1| =
−(2x + 1) if 2x + 1 < 0
(
2x + 1
if x ≥ − 21
Therefore f (x) =
−2x − 1 if x < − 21 .
The domain of f is R.
13. Example: Write f (x) = |x2 − 1| without the absolute value symbol and write down the domain of
f.
( 2
( 2
x −1
if x2 − 1 ≥ 0
x −1
if x ≤ −1 or x ≥ 1
2
|x − 1| =
=
2
2
2
−x + 1 if x − 1 < 0
−x + 1 if − 1 < x < 1
The domain of f is R.
Unit 1.6: New functions from old functions
2 lectures
Textbook: Section 1.3, p. 36.
Learning outcomes: On completion of this unit you should
1. be able to combine two functions f and g to form the new functions f + g, f − g, f g and gf and be
able to determine the domains of these functions.
2. be able to form the composite functions f ◦ g and g ◦ f for given functions f and g and be able to
determine the domains of the new functions.
3. be able to decide what the rules of the functions f and g are if a rule for the composite function
f ◦ g is given.
Additional notes:
1. Given any two functions f and g the following combinations are defined:
• The sum f + g with ( f + g)(x) = f (x) + g(x).
• The difference f − g with ( f − g)(x) = f (x) − g(x).
• The product f g with ( f g)(x) = f (x)g(x).
!
f
f
f (x)
• The quotient with
(x) =
, g(x) , 0.
g
g
g(x)
• The composition f ◦ g with ( f ◦ g)(x) = f (g(x)).
Make sure that you understand what the domains of these functions are.
2. Study Examples 6, 7 and 8 on pages 41 and 42.
25
Theme 2: Limits and Continuity
Unit 2.1: The tangent and velocity problems
1 lecture
Textbook: Section 2.1, pp. 78–81.
Learning outcomes: On completion of this unit you should be able to
1. estimate the slope of a tangent line to a curve by calculating the limit of the slope of secant lines.
Problems: none
Unit 2.2: The limit of a function
1 lecture
Textbook: Section 2.2, p. 83.
Learning outcomes: On completion of this unit you should be able to
1. explain the informal idea of the limit of a function at a point and be able to interpret this idea
graphically.
2. find the limit of a function at a point if the graph of the function is given.
3. explain what is meant by one-sided limits and be able to interpret these ideas graphically.
4. find lim f (x) by using the one-sided limits lim+ f (x) and lim− f (x) (only when necessary).
x→a
x→a
x→a
5. explain the informal definition of the infinite limit lim f (x) = ∞ (and lim f (x) = −∞) and be able
x→a
x→a
to interpret this idea graphically.
Additional notes:
1. In Definition 1 on p. 83 there is, as in Grade 12, an informal and intuitive definition of the meaning
of
lim f (x) = L.
x→a
In Section 2.4 (p. 105) you will see the mathematically precise meaning that will make it possible
to prove certain facts about limits.
26
One-sided Limits
2. The paragraph preceding Definition 2 (p. 86) is important because it shows why for the Heaviside
function
(
0 if t < 0
H(t) =
,
1 if t ≥ 0
lim H(t) does not exist and it introduces the idea of one-sided limits lim+ H(t) = 1 and lim− H(t) =
t→0
t→0
t→0
0.
Infinite Limits
3. Study Definitions 4 and 5 (pp. 89–90) for the meaning of
lim f (x) = ∞ and lim f (x) = −∞.
x→a
x→a
4. Figures 9, 10, 11 and 12 (pp. 89–90) are very helpful.
5. Study Definition 6 (p. 90) for the meaning of a vertical asymptote of a curve y = f (x).
6. Study Examples 7 and 8 (p. 91).
Unit 2.3: Limit laws
1 lecture
Textbook: Section 2.3, p. 94.
Learning outcomes: On completion of this unit you should be able to
1. use the appropriate limit law to find the limit of the sum, difference, product and quotient of two
functions and the constant multiple of a function.
2. use the direct substitution property to find the limits of polynomials and rational functions.
3. determine whether a limit exists by evaluating one-sided limits.
4. write down and use the Squeeze Theorem.
Remark: The limit laws can be proved by making use of the formal definition of a limit. However, we
will not prove these laws in this module.
Unit 2.4: Precise definition of a limit
1 lecture
Textbook: Section 2.4, p. 105.
Learning outcomes: On completion of this unit you should
1. understand the definition of a limit.
2. be able to use the definition of a limit to confirm simple limits.
27
Unit 2.5: Continuity
3 lectures
Textbook: Section 2.5, p. 115.
Learning outcomes: On completion of this unit you should be able to
1. determine whether a given function is continuous at a number a.
2. prove and use the theorems on the continuity of the combination of continuous functions (sum,
difference, constant multiple, product, quotient and composite).
3. explain and use the Intermediate Value Theorem.
Remarks:
1. Ignore any reference to exponents and logarithms in this section.
Additional note:
We give the proof of Theorem 4(3):
If f is continuous at a and c is a constant then c f is continuous at a.
Proof. Suppose f is continuous at a and c is a constant. Then
lim (c f )(x) = lim c · f (x)
x→a
(definition of c f )
x→a
= c · lim f (x)
(by Law 3 on p. 95)
= c · f (a)
( f is continuous at a)
= (c f )(a)
(definition of cf).
x→a
This proves that the function c f is continuous at a.
Unit 2.6: Limits at infinity and horizontal asymptotes
Textbook: Section 2.6, pp. 127–134 (skip pp. 135–137).
Learning outcomes: On completion of this unit you should be able to
1. explain what is meant by the following limits:
lim f (x) = L, lim f (x) = L, lim f (x) = ∞
x→−∞
x→∞
lim f (x) = ∞, lim f (x) = −∞, lim f (x) = −∞.
x→∞
x→−∞
x→∞
x→−∞
2. interpret limits at infinity graphically.
3. find horizontal asymptotes of a function.
28
1 lecture
Theme 3: Differentiation
Unit 3.1: Derivatives and rates of change
1 lecture
Textbook: Section 2.7, p. 140.
Learning outcomes: On completion of this unit you should be able to
1. write down and use the definition of the derivative of a function at a point.
2. find derivatives of functions by using the definition.
3. find the equation of a tangent line to a graph.
Remark: Omit Examples 7 and 8.
Additional notes: We will focus on the slope of a tangent line to curves and not on velocities and rates
of change. Here is a summary of this section:
1. The derivative of a function f at a number a is the limit
f (a + h) − f (a)
h→0
h
f (x) − f (a)
= lim
x→a
x−a
f ′ (a) = lim
(Definition 4, p. 144)
((5), p. 144)
if the limit exists.
2. If f ′ (a) exists, then m = f ′ (a) is the slope of the tangent line to the curve y = f (x) at the point
P(a, f (a)) and the equation of the tangent line is y − f (a) = m(x − a).
3. To find the derivative of a polynomial of degree ≥ 3, one needs the following useful formula that
holds for all real numbers a and b and all positive integers n:
an − bn = (a − b)(an−1 + ban−2 + b2 an−3 + . . . + bn−1 ).
Try to prove this by using the formula for the sum of a geometric progression,
a + ar + ar2 + . . . + arn−1 =
29
a(rn − 1)
.
r−1
Unit 3.2: The derivative as a function
1 lecture
Textbook: Section 2.8, p. 153.
Learning outcomes: On completion of this unit you should
1. be able to write down and use the definition of the derivative function.
2. be able to sketch the graph of the derivative function if the graph of the function is given.
3. be able to write down what it means for a function to be differentiable at a point.
4. be able to write down what it means for a function to be differentiable on an interval.
5. know and be able to prove that a function that is differentiable at a point is continuous at that point
and that the converse of the theorem is not true.
6. know the different notations for the derivative.
7. know the notation for higher derivatives and be able to find higher derivatives.
Additional notes:
1. If f is a function, then the function f ′ that assigns to x the number
f ′ (x) = lim
h→0
f (x + h) − f (x)
, if it exists,
h
is called the derivative of f (Definition 2, p. 153).
2. In Example 1 (p. 153), study the graphs of f and f ′ in Figure 2 (p. 154). Note that if f ′ (x) > 0 for
each x on an interval, then f is increasing on that interval and that if f ′ (x) < 0 on an interval, then
f is decreasing on that interval. (This is a theorem that will be proved in Theme 4).
3. Note that the derivatives of polynomials, root functions, exponential functions (see Unit 3.6),
logarithmic functions (see Unit 3.9) and rational functions are continuous on their domains.
30
Unit 3.3: Differentiation rules
2 lectures
Textbook: Section 3.1 (pp. 174–179) and Section 3.2 (pp. 185–189).
Learning outcomes: On completion of this unit you should be able to
1. write down and use the rule for the derivative of a constant function.
2. write down and use the rule for the derivative of xn (n a natural number).
3. write down and use the rule for the derivative of a constant multiple of a function.
4. write down and use the rule to find the derivative of the sum, difference, product and quotient of
differentiable functions.
Remark: Omit the exponential functions (pp. 177–179) for now.
Additional notes:
1. In Section 3.2 the rules for the derivative of the product and quotient of differentiable functions
are proved. However, you do not need to memorize the proofs below. Just the rules themselves
need to be memorized.
2. The Product Rule: Suppose f and g are differentiable functions. Let P(x) = f (x) · g(x). Then P(x)
is differentiable and
!
!
d
d
d
[P(x)] =
f (x) g(x) + f (x)
g(x)
dx
dx
dx
or equivalently
P′ (x) = f ′ (x)g(x) + f (x)g′ (x).
Proof. First, we look at the quotient below.
P(x + h) − P(x) f (x + h) · g(x + h) − f (x) · g(x)
=
h
h
f (x + h) · g(x + h) − f (x) · g(x + h) + f (x) · g(x + h) − f (x) · g(x)
=
h
g(x + h) − g(x)
f (x + h) − f (x)
· g(x + h) + f (x)
=
h
h
Next, we explain why the following four limits exist.
f (x + h) − f (x)
= f ′ (x)
h
g(x + h) − g(x)
lim
= g′ (x)
h→0
h
lim f (x) = f (x)
lim
h→0
h→0
lim g(x + h) = g(x)
h→0
(given f is differentiable)
(given g is differentiable)
(constant limit law)
(g is differentiable implies g is continuous)
From the above facts and the product limit law, we obtain
lim
h→0
f (x + h) − f (x)
· g(x + h) = f ′ (x) · g(x)
h
g(x + h) − g(x)
lim f (x) ·
= f (x) · g′ (x)
h→0
h
31
(1)
Finally, from the sum limit law, we obtain
lim
h→0
g(x + h) − g(x)
f (x + h) − f (x)
· g(x + h) + f (x) ·
= f ′ (x) · g(x) + f (x) · g′ (x)
h
h
From equation (1) and above we see that lim P(x+h)−P(x)
exists. Therefore P is differentiable and
h
h→0
d
d
d
[P(x)]
=
f
(x)
·
g(x)
+
f
(x)
g(x)
as
desired.
dx
dx
dx
3. Before we prove the quotient rule, we first prove the reciprocal rule.
The Reciprocal Rule: If g is differentiable and g(x) , 0, then
− d g(x)
d 1
= dx 2
dx g(x)
g(x)
or equivalently
!′
1
−g′ (x)
.
(x) =
g
g(x)2
Proof. We have
d 1
= lim
dx g(x) h→0
1
1
g(x+h) − g(x)
h
g(x) − g(x + h)
= lim
h→0 hg(x)g(x + h)
−1
1
g(x + h) − g(x)
=
lim
· lim
g(x) h→0 g(x + h) h→0
h
−1
1
d
=
·
· g(x)
g(x) g(x) dx
(since g is differentiable in x, g is also continuous in x
which means lim g(x + h) = g(x).)
h→0
− d g(x)
= dx 2 .
g(x)
Examples:
!
d 1
−1
•
= 2
dx x
x
!
d
−2x
1
•
= 2
.
2
dx x + 1
(x + 1)2
32
4. The Quotient Rule If f and g are differentiable and g(x) , 0 then
d
d
d f (x) g(x) dx f (x) − f (x) dx g(x)
,
=
dx g(x)
[g(x)]2
that is
!′
f
g(x) f ′ (x) − f (x)g′ (x)
.
(x) =
g
[g(x)]2
Proof. We have
!
d f (x)
d
1
=
f (x) ·
dx g(x) dx
g(x)
!
d
1
d 1
f (x) ·
+ f (x) ·
=
dx
g(x)
dx g(x)

 d
 − dx g(x) 
d
1

=
f (x) ·
+ f (x) · 
dx
g(x)
g(x)2 
(product rule)
(reciprocal rule)
d
d
f (x) f (x) dx
g(x)
−
= dx
g(x)
g(x)2
d
d
f (x) − f (x) dx
g(x)
g(x) dx
=
.
g(x)2
′
′
Remark: Note that if
f (x)
and g(x), as well as f (x) and g (x), are continuous, then it follows from the
d f (x)
quotient rule that dx g(x) is continuous on D f ′ .
g
Unit 3.4: Derivatives of trigonometric functions
1 lecture
Textbook: Section 3.3, pp. 191–197.
Learning outcomes: On completion of this unit you should
sin h
= 1.
h
2. be able to prove the formulas for the derivatives of the six trigonometric functions, except sin and
cos.
1. know that lim
h→0
33
Unit 3.5: The chain rule
2 lectures
Textbook: Section 3.4 pp. 199–205.
Learning outcomes: On completion of this unit you should
1. know the Chain Rule and be able to use the Chain Rule to find the derivative of a composite
function.
Additional notes:
1. Study the Chain Rule on page 200 but ignore the comments on the proof (p. 200) and the proof
on page 205.
2. Study all the examples except Examples 7 and 9.
3. At this point, also omit Formula 5 for the derivative of b x .
4. Here follows a summary of the Chain Rule formulas:
d
[ f (x)]n = n[ f (x)]n−1 · f ′ (x)
dx
d
sin f (x) = (cos f (x)) · f ′ (x)
(b)
dx
d
(c)
cos f (x) = (− sin f (x)) · f ′ (x)
dx
d
(d)
tan f (x) = (sec2 f (x)) · f ′ (x)
dx
d
cosec f (x) = (−cosec f (x) cot f (x)) · f ′ (x)
(e)
dx
d
(f)
sec f (x) = (sec f (x) tan f (x)) · f ′ (x)
dx
d
(g)
cot f (x) = (−cosec2 f (x)) · f ′ (x)
dx
Examples:
(a)
d
sec(x3 ) = sec(x3 ) tan(x3 ) · 3x2
dx
!
!6
!5
d 1
−1
1
2
2
2.
+x =6 +x
+ 2x
dx x
x
x2
1.
3.
d
(x + 1)6 sin4 x = 6(x + 1)5 · sin4 x + (x + 1)6 4 sin3 x cos x
dx
4.
d (x + 1)6 (sin4 x)6(x + 1)5 − (x + 1)6 4 sin3 x cos x
=
dx sin4 x
sin8 x
34
Unit 3.6: Exponential functions and the number e
1 lecture
Source: The notes below and the textbook, sections 1.4 (pp. 45–50) and 3.1 (pp. 179–181).
Learning outcomes: On completion of this unit you should be able to
1. explain what the meaning is of a x when x is a natural number, a rational number and an irrational
number.
2. write down and use the laws of exponents.
3. find the derivatives of expressions containing e f (x) where f (x) is differentiable.
Problems: Exercises 1.4 (p. 52) no. 3, 15, 17, 21, Exercises 3.4 (p. 206) no. 7-55 odd exponential
Additional notes: Except for specific references, these notes are a substitute for the textbook, pages
45–50 and 179–181.
1. The domain of f (x) = a x (pp. 45-50)
The exponential function f (x) = a x (for a real constant a > 0) is in school defined for positive
integers n and rational numbers mn , where n ∈ Z, n > 0 and m ∈ Z, as follows:
(a) an = a.a. . . . a(n factors)
(b) a−n =
1
an
(c) a0 = 1
(d) a /n = x where x ∈ R is such that xn = a
1
(e) am/n = x where x ∈ R is such that xn = am .
By the above definition the domain of f (x) = a x includes all the integers (Z) and even all the
rational numbers (Q).
However, the domain of f (x) = a x can be extended to R. The idea is discussed in the textbook,
Section 1.4 (pages 45–47). A rigorous proof can be found in the course notes for the next Calculus
module WTW124. The idea is as follows: If x is irrational, then it can be approximated as
accurately as we wish by a rational number r. Then ar is an approximation of a x . Consider 2π for
1571
example. Since 3, 142 is an approximation of π, the number 23,142 = 2 500 is an approximation of
15708
2π . The number 23,1416 = 2 5000 is a better approximation. Obviously an even better approximation
is possible.
35
Exponential laws and inequalities
2. If a and b are positive real numbers and x and y any real numbers, then
(a) a x+y = a x ay
ax
ay
x
y
(c) (a ) = a xy
(b) a x−y =
(d) (ab) x = a x b x
(e) If a > 1 then ar > a s whenever r > s
(f) If 0 < a < 1 then ar < a s whenever r > s
The inequalities (e) and (f) above show that f (x) = a x is increasing if a > 1 and decreasing if
0 < a < 1 (see figures 1–6 on pp. 46–48).
The derivative of f (x) = a x (a > 0) (pp. 179–181)
3. With respect to the differentiability of the exponential function we have the following theorem:
Theorem. Let f (x) = a x (a > 0). If f is differentiable at 0, then f is differentiable everywhere and
f ′ (x) = f ′ (0) f (x) for every x ∈ R.
Proof. Assume that f is differentiable at x = 0, that is, assume
f (0 + h) − f (0)
ah − 1
= lim
h→0
h→0
h
h
f ′ (0) = lim
exists. Now
f (x + h) − f (x)
a x+h − a x
a x (ah − 1)
ah − 1
= lim
= lim
= a x lim
h→0
h→0
h→0
h→0
h
h
h
h
f ′ (x) = lim
ah − 1
exists, we have that f ′ (x) = f (x) f ′ (0) which shows that
h→0
h
f ′ (x) exists for every real number x.
Since we suppose that f ′ (0) = lim
ah − 1
exists for any a > 0. Taking this
h→0
h
into account, it follows from the above theorem that all the exponential functions is everywhere
differentiable with derivative
4. We will assume that the important limit f ′ (0) = lim
f ′ (x) = f (x) f ′ (0)
for
f (x) = a x
d x
or more explicitly, dx
a = f ′ (0)a x for f (x) = a x .
Thus it follows for all x that the derivative of any exponential function f at x is proportional to the
value of f at x. (The slope of f at x is proportional to f (x).)
We will shortly determine f ′ (0), from which it will follow that


if 0 < a < 1, then f ′ (0) < 0





if a = 1,
then f ′ (0) = 0





if a > 0,
then f ′ (0) > 0.
36
(2)
The graph of f (x) = a x
5. From items 2e, 2f and (2) above it follows that
(a) if a > 1 then f and f ′ are increasing
(b) if 0 < a < 1 then f and | f ′ | are decreasing
(but f ′ is increasing since f ′′ (x) = f ′ (0)2 a x > 0).
Study figures 1–6 on pp. 46–48 and confirm (5a) and (5b) above.
The number e
6. A rigorously proof for the fact that the graph of one of the exponential functions f (x) = a x has a
ah − 1
slope of 1 at x = 0, that is, f ′ (0) = lim
= 1, can be found in the course notes for the next
h→0
h
Calculus course, WTW124. This is formally stated in the following theorem:
eh − 1
= 1.
h→0
h
Theorem. There exists a number e such that if f (x) = e x , then f ′ (0) = lim
It can be proved that e is irrational and that
e=1+
1
1
1
1
+
+
+
+ ...
1 2·1 3·2·1 4·3·2·1
(This means that better rational approximations for e are achieved by adding more terms in this
“infinite sum”.)
d x
From the above Theorem it follows that for f (x) = e x we have f ′ (x) = f (x) or
e = e x . So, if
dx
f is a differentiable function, it follows by using the Chain Rule that
d f (x)
e
= f ′ (x)e f (x) .
dx
Examples:
(a)
d −x
e = −e−x .
dx
!
2
d sin2 x
sin2 x d
2
(b)
e
=e
sin x = 2esin x sin x cos x.
dx
dx
√
e x
d √x
e = √ .
(c)
dx
2 x
37
Unit 3.7: The derivatives of inverse functions
2 lectures
Source: The notes below and references to the textbook, sections 1.5 (p. 54–57) and 3.6 (only p. 2222–
223).
Learning outcomes: On completion of this unit you should be able to
1. be able to determine the derivatives of inverse functions including those of sin−1 x, tan−1 x and xr
when r is rational.
Additional notes:
The inverse of a function (textbook p. 55)
1. Recall that a function f is called a one-to-one function if it never takes on the same value twice;
that is, for every x1 and x2 in the domain of f ,
if x1 , x2 then f (x1 ) , f (x2 ).
2. Horizontal Line Test. A function is one-to-one if no horizontal line intersects its graph more than
once.
Examples:
(a) y = x2 , x ≥ 0
(b) y = sin x, −π/2 ≤ x ≤ π/2
(c) y = tan x, −π/2 < x < π/2
(d) y = a x , x ∈ R (a > 0).
3. Definition 1. Let f be a one-to-one function with D f = A and R f = B. Then its inverse function
f −1 is the function defined by:
For every y ∈ B, f −1 (y) = x
if and only if
f (x) = y.
This is the same as
f −1 ( f (x)) = x for all x ∈ A and f ( f −1 (y)) = y for all y ∈ B.
Therefore
D f −1 = B
and
38
R f −1 = A.
The Differentiability of Inverse Functions
4. The following theorem (which we can’t prove now) tells us when an inverse function is differentiable.
Theorem. Let f : D → E be a continuous one-to-one function on an interval I, and let f |I : I → E
be the restriction of f on I. That means that f (x) = f |I (x) for all x ∈ D f |I = I. Then
(a) R f |I := J is an interval;
(b) f |I has a continuous inverse f |−1
I , with D f |−1 = J and R f |−1 = I.
I
I
Also, if f is a one-to-one function that is differentiable at a and f ′ (a) , 0 then f −1 is differentiable
at b = f (a).
Remark. The formula for the derivative of f −1 (x), if it exists is easily obtained by what is called
implicit differentiation:
Let g = f −1 . Then f (g(x)) = x for all x ∈ Dg . Therefore
d
d
( f (g(x)) =
x.
dx
dx
By the Chain Rule
f ′ (g(x))g′ (x) = 1
that is,
g′ (x) =
1
f ′ (g(x))
.
The Derivative of arcsin
5. Consider the function
f (x) = sin x, − π/2 ≤ x ≤ π/2 .
Using the above notation, we can write f (x) as follows
f (x) = sin x|[−π/2 ≤x≤π/2 ] .
Then f is continuous and strictly increasing and therefore one-to-one on [−π/2 , π/2 ]. Note that
D f −1 = [−1, 1] and R f −1 = [−π/2 , π/2 ]. The function f −1 is called arcsin or arcsin . Therefore,
y = arcsin x means that sin y = x, y ∈ [−π/2 , π/2 ] and x ∈ [−1, 1].
For example,
arcsin(−1/2) = −π/6 since sin(−π/6) = −1/2
and
√
√
arcsin(1/ 2) = π/4 since sin π/4 = 1/ 2.
39
6. Theorem.
d
1
arcsin x = √
.
dx
1 − x2
Proof. The derivative of the inverse function y = arcsin x exists for every x ∈ (−1, 1) since
y = sin x is differentiable on (−π/2 , π/2 ).
Firstly
sin(arcsin x) = x for every x ∈ (−1, 1).
Therefore, for every x ∈ (−1, 1)
d
d
sin(arcsin x) =
x
dx
dx
d
cos(arcsin x) arcsin x = 1
dx
d
1
(arcsin x) =
.
dx
cos(arcsin x)
But since
[cos(arcsin x)]2 + [sin(arcsin x)]2 = 1 and arcsin x ∈ (−π/2 , π/2 )
it follows that
cos(arcsin x) > 0 and that cos(arcsin x) =
p
1 − [sin(arcsin x)]2 .
Finally,
1
d
arcsin x = √
.
dx
1 − x2
7. In a similar way, you can show
cos−1 = arccos?
d
−1
arccos x = √
. What is the domain and range for
dx
1 − x2
8. By the Chain Rule it now follows that if f is differentiable, then
d
f ′ (x)
arcsin( f (x)) = p
.
dx
1 − ( f (x))2
40
2
2
d
2xe x
.
Example.
arcsin(e x ) = p
2
dx
1 − e2x
The Derivative of arctan
9. Consider the function
f (x) = tan x, − π/2 < x < π/2 .
We can also express f (x) as
f (x) = tan x|(−π/2 ,π/2 ) .
Then f is continuous and strictly increasing and therefore one-to-one on (−π/2 , π/2 ). Note that for
the inverse f −1 , we have D f −1 = (−∞, ∞) and R f −1 = (−π/2 , π/2 ). We refer to f −1 as arctan
Therefore,
y = arctan x means that tan y = x, y ∈ (−π/2 , π/2 ) and x ∈ (−∞, ∞).
√
√
For example, arctan 1 = π/4 since tan π/4 = 1 and arctan(− 3) = −π/3 since tan(−π/3 ) = − 3.
d
1
arctan x =
.
dx
1 + x2
Proof. The derivative of the inverse function y = arctan x exists for every x ∈ (−∞, ∞) since
y = tan x is differentiable on (−π/2 , π/2 ).
10. Theorem.
Firstly tan(arctan x) = x for every x ∈ (−∞, ∞). Therefore, for every x ∈ (−∞, ∞)
d
d
tan(arctan x) =
x
dx
dx
41
that is,
[sec(arctan x)]2
and therefore
d
arctan x = 1
dx
1
d
(arctan x) =
.
dx
[sec(arctan x)]2
But since [sec(arctan x)]2 = 1 + [tan(arctan x)]2 it follows that
d
1
.
arctan x =
dx
1 + x2
11. By the Chain Rule it now follows that if a function f is differentiable, then
d
f ′ (x)
.
arctan( f (x)) =
dx
1 + ( f (x))2
Example.
d
sec x tan x
.
arctan(sec x) =
dx
1 + sec2 x
42
The Derivative of xm/n
d n
12. We have already proved for a positive integer n that dx
x = nxn−1 (see textbook p. 173).
d r
x = rxr−1 , for r ∈ Q, we need the following result:
To proof that dx
Lemma. For any non-zero integer n and real number x > 0,
d 1 1 1 −1
xn = xn .
dx
n
Proof. For n > 0 the function f (x) = xn is differentiable (we know that f ′ (x) = nxn−1 , see textbook
1
p. 173) and strictly increasing and thus 1 − 1. Thus f −1 (x) = x n is differentiable on (0, ∞).
1
Differentiation of (x n )n = x gives
1
n(x n )n−1
d 1
x n = 1.
dx
That is
d 1/n
x = (1/n)x(1/n)−1 .
dx
Assume now n < 0. Let s = −n. By using the chain rule and the quotient rule we have that
d 1 −1
d 1
xn =
(x s )
dx
dx
1
d 1
= −(x s )−2 x s
dx
−2 1 1
= −x s x s −1
s
1 − 1 −1
=− x s
s
1 1 −1
= xn
n
(quotient rule)
s>0
13. Proposition. For any rational number mn
d m/n
x
= (m/n)x(m/n)−1 .
dx
Proof. Clearly the proposition is true for mn = 0, since
d 0
d
x =
1 = 0 = 0x−1 .
dx
dx
m
1
Because x n = (x n )m , the chain rule gives
d m
d 1 m
xn =
(x n )
dx
dx
1
1 1
= m(x n )m−1 ( )x( n )−1
n
m ( m )−( 1 )+( 1 )−1
= ( )x n n n
n
m ( m )−1
= ( )x n .
n
43
Unit 3.8: Implicit differentiation
1 lecture
Source: The notes below and references to the textbook, section 3.5 (p. 209).
Learning outcomes: On completion of this unit you should be able to
1. be able to use implicit differentiation to find the derivatives of a function that is implicitly defined
by an equation in two variables.
Additional notes:
1. In the previous unit we were able to find the derivative of a function (y = f −1 (x)) without an explicit formula for the function in terms of x. The crucial point was that we knew that the derivative
existed.
√
Similarly x2 + y2 = 25 defines
one
function
if
y
>
0
namely
y
=
25 − x2 and another function
√
2
if y < 0, namely y = − 25 − x . But without explicitly solving for y in terms of x, there is a
theorem that assures one that x2 + y2 = 25 defines functions y = f (x) that are differentiable. The
moment one knows that f ′ (x) exists, one differentiates the left hand side and right hand side of
x2 + ( f (x))2 = 25
or
x2 + y2 = 25
2x + 2 f (x) f ′ (x) = 0
or
2x + 2yy′ = 0
x
f (x)
or
x
y′ = − .
y
to get
that is
f ′ (x) = −
Note that the derivative is expressed in terms of x and y.
2. The textbook does not point out the importance of the fact that one must know that the function is
differentiable before one can do implicit differentiation.
44
Unit 3.9: Logarithmic functions and their derivatives
2 lectures
Source: The notes below, and references to the textbook, Section 3.6 (p. 217–221).
Learning outcomes: On completion of this unit you should be able to
1. know the ln and log laws.
2. write down, use and prove the formulas for the derivatives of y = ln x and y = loga x.
3. differentiate y = a x , y = f (x)g(x) by using the properties of y = ln x.
4. use the technique of logarithmic differentiation.
Additional notes:
The Logarithmic Function (see also Section 1.5 p. 57–61)
1. Consider the function
f (x) = a x , x ∈ (−∞, ∞) and a > 0, a , 1.
Then f is continuous and either strictly increasing or strictly decreasing on (−∞, ∞). Note that
D f −1 = (0, ∞) and R f −1 = (−∞, ∞). We call f −1 the logarithmic function with base a and denote
it by loga x. Therefore,
y = loga x means that ay = x, y ∈ (−∞, ∞) and x ∈ (0, ∞).
For example,
log2 64 = 6 since 26 = 64.
Furthermore, since the function y = a x is differentiable on (−∞, ∞), its inverse y = loga x is
differentiable on (0, ∞). We will determine the derivative of loga x after the discussion of the
so-called natural logarithm.
The Natural logarithm
2. The logarithmic function with base e is called the natural logarithm and has a special notation,
namely
loge x = ln x.
Therefore,
y = ln x means that ey = x
and that
ln(e x ) = x, x ∈ (−∞, ∞) and eln x = x, x > 0.
We prove the well-known logarithm laws for the natural logarithm and then show how the same
laws follow from these.
45
Laws for the natural logarithm
3. For all real numbers a, x, y and with x, y > 0,
ln(xy) = ln x + ln y;
ln(x/y) = ln x − ln y;
ln xa = a ln x.
Proof. Let ln x = c and ln y = d. Then x = ec and y = ed and
ln xa = ln(ec )a = ln eca = ac = a ln x.
Prove the other two laws as an exercise.
Change of Base Formula
4. The formula that connects the natural logarithm with the other logarithms:
For any real number x > 0,
loga x =
ln x
.
ln a
Proof. Let loga x = y. Then ay = x and therefore
ln x = ln(ay ) = y ln a,
that is,
y=
ln x
.
ln a
General log laws from ln laws
5. For all real numbers a, b, x and y with a, x and y positive,
loga (xy) = loga x + loga y;
loga (x/y) = loga x − loga y;
Proof.
ln(xy)
ln a
ln x + ln y
=
ln a
ln x ln y
=
+
ln a ln a
= loga x + loga y.
loga (xy) =
Prove the other log laws as an exercise.
46
loga (xb ) = b loga x.
The derivatives of ln x, ln f (x), loga x and loga f (x)
6. Theorem.
d
1
ln x =
dx
x
d
f ′ (x)
(b)
ln f (x) =
dx
f (x)
(a)
d
1
loga x =
dx
x ln a
d
f ′ (x)
(d)
loga f (x) =
dx
f (x) ln a
(c)
Proof of 6a. Since f (x) = e x is differentiable on R, its inverse f −1 (x) = ln x is differentiable on
d
1
d
1
(0, ∞). So, from eln x = x we have eln x ln x = 1. Therefore,
ln x = ln x = .
dx
dx
x
e
Proof of 6b. From the Chain Rule follows that if f is differentiable, then
1 ′
f ′ (x)
d
ln( f (x)) =
f (x) =
.
dx
f (x)
f (x)
d
d ln x
1 d
1 1
loga x =
=
ln x =
· .
dx
dx ln a ln a dx
ln a x
1
d
loga f (x) =
· f ′ (x).
Proof of 6d. From the Chain Rule,
dx
f (x) ln a
Proof of 6c.
Logarithmic differentiation
7. Combining the properties of the functions f (x) = e x and f −1 (x) = ln x with the Chain Rule gives
a powerful tool to prove the existence of derivatives of power functions and to find them. We call
this method logarithmic differentiation.
√
d √3
Our first example is the proof that even for irrational powers of x like x 3 it is true that
x =
dx
√ √3−1.
3x
d r
8. Theorem.
x = rxr−1 for any r ∈ R and x > 0.
dx
Proof. Since xr > 0 it follows that xr falls in the domain of the function ln x for all x > 0. Since
r
f (x) = e x and f −1 (x) = ln x are inverse functions, it follows for all x > 0 that xr = eln(x ) and thus
that
d r
d ln x r
d r ln x
r rxr
x =
e
=
e
= er ln x =
= rxr−1 .
dx
dx
dx
x
x
d
9. Lemma. dx
(ln |x|) = 1x .
d
d
Proof. For x > 0 it follows that dx
ln |x| = dx
ln x = 1x . For x < 0 it follows, by using the chain
d
d
1
rule, that dx
ln |x| = dx
ln(−x) = − −x
= 1x .
47
10. Examples.
1
√
√
1 + 2√
√
d
d ln(x + x)
2 x+1
x
log2 (x + x) =
=
(a)
√ = √
√ .
dx
dx
ln 2
ln 2(x + x) 2 x ln 2(x + x)
(b) Since 2 x > 0 for all x, it follows that
d ln 2 x
d x ln 2
d x
2 =
(e ) =
e
= (ln 2)e x ln 2 = (ln 2)2 x .
dx
dx
dx
(c) Since 3sin x > 0 for all x, it follows that
d sin x
d ln 3 sin x
d (ln 3) sin x
3
=
(e )
=
e
= e(ln 3) sin x (ln 3) cos x = 3sin x (ln 3) cos x
dx
dx
dx
√
(d) Let f (x) = ( x) x . Since f (x) > 0 for all x ∈ D f , that is for all x > 0, it follows for all
x ∈ D f ′ that
√ d
√
√
√
d √ x
d ln √ x x
d x ln √ x
1
1
( x) =
(e
) =
e
= e x ln x x ln x = ( x) x (ln x + x √ · √ ).
dx
dx
dx
dx
x 2 x
(e) Find g′ (x) if g(x) = √
x3
x + 1(x3 + 1)5
.
We can determine the g′ (x) using the quotient rule, but this will be quite cumbersome. Instead we use logarithmic differentiation. Let f (x) = ln x. Note that g(x) takes both positive
and negative values for different values of x. Hence g(x) does not fall in the domain of f for
all x ∈ Dg . The value of the function |g(x)| is bigger than 0 for all non-zero values of x. Thus
it follows for all x ∈ Dg , x , 0 that
ln |g(x)| = ln √
x3
x + 1(x3 + 1)5
|x|3
= ln √
| x + 1||x3 + 1|5
√
= ln |x|3 − ln x + 1 − ln |x3 + 1|5
1
= 3 ln |x| − ln(x + 1) − 5 ln |x3 + 1|.
2
By differentiating on both sides, it follows that
d
g′ (x) 3
1
5(3x2 )
ln |g(x)| =
= −
− 3
.
dx
g(x)
x 2(x + 1) x + 1
Therefore, it follows for all x ∈ Dg′ , x , 0 that
g (x) = √
′
= √
x3
x + 1(x3 + 1)5
3x2
x + 1(x3 + 1)5
3
1
15x2
−
− 3
x 2(x + 1) x + 1
− √
x3
x + 1(x3 + 1)5
!
!
1
15x2
+
.
2(x + 1) x3 + 1
Since the derivative of a polynomial, as well as a root function is continuous, it follows that
the derivative of g(x) is continuous (see remark after the quotient rule). Thus it follows for
all x ∈ Dg′ = Dg (i.e. including x = 0) that
!
x3
1
15x2
3x2
′
g (x) = √
− √
+
.
x + 1(x3 + 1)5
x + 1(x3 + 1)5 2(x + 1) x3 + 1
48
Unit 3.10: Hyperbolic functions
Textbook: Section 3.11, pp. 261–266.
Learning outcomes: On completion of this unit you should
1. be able to write down and use the definitions of the hyperbolic functions y = sinh x, y = cosh x
and y = tanh x.
2. know, be able to prove and use the derivatives of the functions y = sinh x, y = cosh x and y =
tanh x.
3. be able to prove the identities involving sinh and cosh .
49
Theme 4: Applications of Differentiation
Unit 4.1: Maximum and minimum values
2 lectures
Textbook: Section 4.1, pp. 280–286.
Learning outcomes: On completion of this unit you should
1. be able to write down and use the definition of an absolute maximum (global maximum) of a
function.
2. be able to write down and use the definition of an absolute minimum (global minimum) of a
function.
3. be able to write down and use the definition of a local maximum (relative maximum) of a function.
4. be able to write down and use the definition of a local minimum (relative minimum) of a function.
5. know and be able to apply The Extreme Value Theorem (proof in WTW 220).
6. be able to use Fermat’s Theorem.
7. be able to write down and use the definition of a critical point of a function.
8. be able to find the critical points of a function.
9. be able to use the closed interval method to find the absolute extremes of a continuous function
on a closed interval.
10. be able to find the absolute and local extremes of a function if the graph of the function is given.
Remark: It is important to remember that if c is a critical point it does not mean that f (c) is a local
extreme. You have to use the tests in Unit 4.3 to test whether c is a local extreme.
50
Unit 4.2: The Mean Value Theorem
2 lectures
Textbook: Section 4.2, pp. 290–295.
Learning outcomes: On completion of this unit you should
1. be able to write down Rolle’s Theorem and apply it.
2. be able to write down the Mean Value Theorem and apply it.
3. know that a function with a zero derivative on an open interval is a constant function on the
interval.
4. know and be able to prove that f ′ (x) = g′ (x) for all x in (a, b) implies that f (x) = g(x) + c for all
x in (a, b), where c is some constant. (The graph of f is a vertical shift of the graph of g.)
Unit 4.3: How the derivative affects the shape of the graph
2 lectures
Textbook: Section 4.3, pp. 296–305.
Learning outcomes: On completion of this unit you should be able to
1. write down and use the definition of a function that is increasing on an interval.
2. write down and use the definition of a function that is decreasing on an interval.
3. use the Increasing / Decreasing Test.
4. use the First Derivative Test to find the local extremes of a function.
5. write down and use the definition of a function that is concave upward on an interval.
6. write down and use the definition of a function that is concave downward on an interval.
7. apply the Concavity Test.
8. write down and use the definition of an inflection point.
9. use the second derivative to find the inflection point(s) of a function.
10. use the Second Derivative Test to find the local extremes of a function.
Remark: Omit Examples 6, 7 and 8 for now. We will do curve sketching in Unit 4.5.
51
Unit 4.4: Indeterminate forms and L’Hospital’s rule
2 lectures
Textbook: Section 4.4, pp. 309–316.
Learning outcomes: On completion of this unit you should be able to
1. explain what indeterminate limit forms are.
2. write down the different types of indeterminate forms.
3. identify when a limit is of indeterminate form.
4. use the rule of L’Hospital to find certain limits of indeterminate forms.
Remark: Use the following notation to indicate indeterminate forms:
ex
∞
ex
∞
(form
or
type
)
=
lim
(form or type ) = . . .
x→∞ x2
x→∞
∞
2x
∞
Example. lim
Unit 4.5: Curve sketching
2 lectures
Textbook: Section 4.5, pp. 320–327.
Learning outcomes: On completion of this unit you should be able to sketch the graph of a given
function. It is a good test of your knowledge and understanding of Themes 3 and 4.
Procedure: When we ask you to sketch the graph of a function you have to
1. find the domain of the function.
2. find the x− and y− intercepts of the function.
3. determine whether the function is even, odd, periodic, or none of these.
4. find the horizontal asymptotes of the function.
5. find the vertical asymptotes of the function.
6. find the interval(s) on which the function is increasing and the interval(s) on which the function is
decreasing.
7. find the local extremes of the function. (Use number 6.)
8. find the interval(s) on which the function is concave upward and the interval(s) on which the
function is concave downward.
9. find the inflection points of the function, and
10. use the information above to sketch the graph of the function.
52
Unit 4.6: Optimization problems
2 lectures
Textbook: Section 4.7, pp. 336–342.
Learning outcomes: On completion of this unit you should
1. know what is meant by an optimization problem.
2. be able to convert a word problem into a mathematical optimization problem by setting up the
function that has to be minimized or maximized.
3. be able to use the methods in this theme to find absolute extreme values.
4. be able to solve optimization problems such as described in the examples.
Unit 4.7: Antiderivatives
1 lecture
Textbook: Section 4.9, pp. 356–360.
Learning outcomes: On completion of this unit you should be able to
1. write down and use the definition of an antiderivative of a function.
2. find the most general antiderivative F of a function f if an antiderivative of f is known.
3. find the function f if an antiderivative of f (x) and, for some a, f (a) are known.
Remarks:
1. You can omit the section about linear motion (textbook pp. 360–361).
2. You must be able to explain the table on p. 358.
3. Theorem 1 on page 357 is a reformulation of Corollary 7 on page 294.
53
Theme 5: Integration
Unit 5.1: The area problem
1.5 lectures
Textbook: Section 5.1, pp. 372–381.
Learning outcomes: On completion of this unit you should be able to
1. estimate the area of a region that lies under a curve by using the sum of the areas of rectangles.
2. find an upper and lower estimate of the area of a region that lies under a curve by using the sum
of the areas of rectangles.
Remarks: Omit the distance problem (p. 379–381).
Unit 5.2: The definite integral
2.5 lectures
Textbook: Stewart, Section 5.2 pp. 384–394.
Learning outcomes: On completion of this unit you should
1. know that the integral has applications other than finding areas.
2. be able to interpret and use the definition of the definite integral.
3. know what the integrand and limits of integration of a definite integral is.
4. be able to use a Riemann sum to approximate a definite integral.
5. know and be able to use the properties of a definite integral.
Remark: Omit the Midpoint Rule (p. 390–391)
54
Unit 5.3: The Fundamental Theorem of Calculus
2 lectures
Textbook: Section 5.3, pp. 399–405.
Learning outcomes: On completion of this unit you should
1. know and be able to use the Fundamental Theorem of Calculus (Part 2).
Remarks:
1. Do not read the proof of the Fundamental Theorem of Calculus (Part 2) in the textbook. Read the
proof given below.
2. Study Examples 5 to 9 on pp. 403 - 405. Omit the other examples.
3. The Fundamental Theorem of Calculus Part 1 will be treated in WTW124.
Fundamental Theorem of Calculus (Part 2): Let f be continuous on [a, b] and differentiable on (a, b),
and assume f ′ is integrable on [a, b] then
Z b
f ′ (x)dx = f (b) − f (a).
a
Proof. Divide [a, b] into n subintervals of equal length △x = b−a
n and let x0 = a, x1 , x2 , . . . , xn = b be the
endpoints of the subintervals. Note that
f (b) − f (a) =
n
X
[ f (xk ) − f (xk−1 )]
(1)
k=1
Now, f is continuous on each interval [xk−1 , xk ], and differentiable on each interval (xk−1 , xk ), for k =
1, 2, . . . , n. By the Mean Value Theorem
f (xk ) − f (xk−1 ) = f ′ (ck ) △ x
(2)
for some ck ∈ (xk−1 , xk ). Thus by (1) and (2) we obtain
n
X
f (b) − f (a) =
[ f (xk ) − f (xk−1 )] =
k=1
n
X
f ′ (ck ) △ x.
k=1
Taking limits both sides as n → ∞, using the ck ’s as sample points in [xk−1 , xk ] and using the fact that f ′
is integrable on [a, b], we have
Z b
f (x)dx = lim
′
a
n→∞
n
X
f ′ (ck ) △ x = lim [ f (b) − f (a)] = f (b) − f (a),
n→∞
k=1
since f (b) − f (a) is constant and by definition
Z b
a
f ′ (x)dx = lim
n→∞
exists.
55
n
X
k=1
f ′ (ck ) △ x
Unit 5.4: Indefinite integrals
1 lecture
Textbook: Section 5.4, pp. 409–415.
Learning outcomes: On completion of this unit you should
Z
1. know that the notation
f (x) dx is traditionally used for an antiderivative of f.
Z
2. know that
f (x) dx represents an infinite number of functions all differing by a constant.
Z b
3. know that
f (x) dx is a real number and not an antiderivative.
a
4. be able to evaluate indefinite integrals.
Remarks:
1. The tables in the textbook p. 358 and p. 410 are essentially the same.
2. Omit the section on The Net Change Theorem, pages 412–415.
Z
Additional notes: Consider
2x cos(x2 ) dx. By the Chain Rule
d
sin(x2 ) = cos(x2 )2x
dx
and therefore
Z
2x cos(x2 )dx = sin(x2 ) + c.
In general, if F is an antiderivative of f, then
Z
g′ (x) f (g(x)) dx = F(g(x)) + c
since
d
[F(g(x)) + c] = F ′ (g(x))g′ (x) = f (g(x))g′ (x).
dx
Here are a number of examples to illustrate the previous remark:
Z
| f (x)]n+1
1.
[ f (x)]n f ′ (x) dx =
+ c in case n , −1
n+1
Z ′
f (x)
2.
/, dx = ln | f (x)| + c
f (x)
Z
3.
f ′ (x) cos f (x) dx = sin f (x) + c
Z
f ′ (x) sin f (x) dx = − cos f (x) + c
Z
f ′ (x) sec2 f (x) dx = tan f (x) + c
4.
5.
56
Z
f ′ (x) sec f (x) tan f (x) dx = sec f (x) + c
Z
f ′ (x) sinh f (x) dx = cosh f (x) + c
Z
f ′ (x) cosh f (x) dx = sinh f (x) + c
Z
f ′ (x)
dx = arcsin f (x) + c
p
1 − [ f (x)]2
Z
f ′ (x)
dx = arctan f (x) + c
1 + [ f (x)]2
Z
f ′ (x)
dx = sinh−1 f (x) + c
p
2
1 + [ f (x)]
6.
7.
8.
9.
10.
11.
Unit 5.5: Transformation of the integral
3 lectures
Textbook: Section 5.5, pp. 419–425.
Learning outcomes: On completion of this unit you should
1. be able to use the Substitution Rule for indefinite integrals.
2. know and apply the Transformation Theorem (substitution rule for definite integrals).
3. know and apply the theorem about “Integrals of Symmetric Functions”.
Remark: Take care to also transform the bounds of integration when using the Transformation Theorem
(substitution rule) for definite integrals.
57
Appendix I: Differentiation formulas
1. Derivative of a constant function
d
c = 0, c ∈ R
dx
2. Derivative of a power function
d
d n
x = nxn−1 and
[ f (x)]n = n[ f (x)]n−1 f ′ (x) for every real number n
dx
dx
3. Derivative of the natural exponential function
d x
d f (x)
e = e x and
e
= e f (x) f ′ (x)
dx
dx
4. Derivative of any exponential function
d x
d f (x)
a = (ln a)a x and
a
= (ln a)a f (x) f ′ (x), a > 0
dx
dx
5. The derivatives of the trigonometric functions
d
d
sin x = cos x
and
sin f (x) = cos f (x) × f ′ (x)
dx
dx
d
d
cos x = − sin x
and
cos f (x) = − sin f (x) × f ′ (x)
dx
dx
d
d
tan x = sec2 x
and
tan f (x) = sec2 f (x) × f ′ (x)
dx
dx
d
d
cosecx = −cosecx × cot x
and
cosec f (x) = −cosec f (x) × cot f (x) × f ′ (x)
dx
dx
d
d
sec x = sec x × tan x
and
sec f (x) = sec f (x) × tan f (x) × f ′ (x)
dx
dx
d
d
cot x = −cosec2 x
and
cot f (x) = −cosec2 f (x) × f ′ (x)
dx
dx
6. The derivatives of the inverse trigonometric functions
d
d
1
arcsin x =
sin−1 (x) = √
dx
dx
1 − x2
1
d
d
arctan x =
tan−1 (x) =
dx
dx
1 + x2
and
and
7. The derivative of logarithmic functions
d
1
d
f ′ (x)
ln x = , x > 0 and
ln f (x) =
dx
x
dx
f (x)
d
1
loga x =
, x>0
dx
x ln a
8. The derivatives of the hyperbolic functions
d
sinh x = cosh x
dx
d
cosh x = sinh x
dx
d
tanh x = sec h2 x
dx
58
d
f ′ (x)
arcsin f (x) = p
dx
1 − [ f (x)]2
d
f ′ (x)
arctan f (x) =
dx
1 + [ f (x)]2
Appendix II: Differentiation rules
Assume the functions are differentiable.
Derivative of a constant multiple
d
d
[c f (x)] = c[ f (x)]
dx
dx
Sum Rule
"
# "
#
d
d
d
[ f (x) + g(x)] =
f (x) +
g(x)
dx
dx
dx
Difference Rule
"
# "
#
d
d
d
[ f (x) − g(x)] =
f (x) −
g(x)
dx
dx
dx
Product Rule
"
#
"
#
d
d
d
[ f (x) × g(x)] =
f (x) × g(x) + f (x) ×
g(x)
dx
dx
dx
Quotient Rule
"
#
d
g(x)]
[ d f (x)] × g(x) − f (x) × [ dx
d f (x)
= dx
2
dx g(x)
[g(x)]
The Chain Rule
d
d
( f ◦ g)(x) =
f (g(x)) × g′ (x)
dx
dx
In Leibniz notation:
If y = f (u) and u = g(x) then
dy dy du
=
×
dx du dx
59
Appendix III: List of standard indefinite integrals
1. Integral of a constant function
Z
c dx = cx + C, c ∈ R
2. Integral of a power function
Z
Z
1
1 n+1
x
+ C and
[ f (x)]n f ′ (x)dx =
[ f (x)]n+1 + C for each real number
xn dx =
n+1
n+1
n , −1
3. Integral of the natural exponential function
Z
Z
x
x
e dx = e + C and
e f (x) f ′ (x)dx = e f (x) + C
4. Integral of any exponential function
Z
Z
1 x
1 f (x)
a x dx =
a + C and
a f (x) f ′ (x)dx =
a
+ C, a > 0
ln a
ln a
5. The integrals of the trigonometric functions
Z
Z
sin xdx = − cos x + C
and
sin( f (x)) f ′ (x))dx = − cos( f (x)) + C
Z
Z
cos xdx = sin x + C
and
cos( f (x)) f ′ (x)dx = sin( f (x)) + C
Z
Z
sec2 xdx = tan x + C
and
sec2 ( f (x)) f ′ (x)dx = tan( f (x)) + C
Z
Z
cosecx cot xdx = −cosecx + C and
cosec( f (x)) cot( f (x)) f ′ (x)dx = −cosec( f (x)) + C
Z
Z
sec x tan xdx = sec x + C
and
sec( f (x)) tan( f (x)) f ′ (x)dx = sec( f (x)) + C
Z
Z
cosec2 xdx = − cot x + C
and
cosec2 ( f (x)) f ′ (x)dx = − cot( f (x)) + C
6. The integrals of the inverse trigonometric functions
Z
1
dx = arcsin(x) + C = sin−1 (x) + C
√
2
1−x
Z
f ′ (x)
dx = arcsin( f (x)) + C = sin−1 ( f (x)) + C
p
2
1 − [ f (x)]
Z
1
dx = arctan(x) + C = tan−1 (x) + C
1 + x2
Z
f ′ (x)
dx = arctan( f (x)) + C = tan−1 ( f (x)) + C
1 + [ f (x)]2
7. The integrals of logarithmic functions
Z
Z ′
1
f (x)
dx = ln |x| + C
and
dx = ln | f (x)| + C
x
f (x)
60
8. The integrals of hyperbolic functions
Z
Z
sinh xdx = cosh x + C
and
sin h( f (x)) f ′ (x)dx = cosh( f (x)) + C
Z
Z
cosh xdx = sinh x + C
sec h xdx = tanh x + C
2
Z
cos h( f (x)) f ′ (x)dx = sinh( f (x)) + C
Z
sec h2 ( f (x)) f ′ (x)dx = tanh( f (x)) + C
and
and
61
FLY@UP encourages students to make use of the available UP resources in order to finish their
degrees in the shortest possible time.

Think carefully before dropping modules (after the closing date for amendments or
cancellation of modules).

Make responsible choices with your time and work consistently.

Aim for a good semester mark. Don’t rely on the examination to pass.
Faculty student advisors
Library
www.library.up.ac.za
Student
Counselling Unit
SA Depression and
Anxiety Group
(SADAG)
CROSS ROAD Help
Centre
Student Health
Services
The Careers Office
Provides counselling and therapeutic
support to students.
Offers two 24-hour telephonic counselling
and referrals to traumatised staff and
students.
Counselling of vulnerable women that find
themselves in a pregnancy crisis.
Promotes and assists students with health
and wellness.
Provides support for UP students and
graduates as they prepare for their careers.
012 420 2333
Department of
Security Services
24-hour Operational Management Centre
24-hour Operational Manager cell
Crisis Line
Department of
Student Affairs
Enquiries concerning studies,
accommodation, food, funds, social
activities and personal problems.
Identifies and provides training of student
peer counsellors.
012 420-2310 / 012 420-2760.
083 654 0476.
0800 006 428
012 420 2371/4001
Roosmaryn Building, Hatfield
campus
012 420 4391
Centre for
Sexualities, AIDS
and Gender
Disability Unit
Fees and funding
IT Helpdesk
Green Route
Provides specialised services to students
with disabilities.
www.up.ac.za/studentportals
www.up.ac.za/student-fees
0800 747 747
Zelda – 082 6600 554
012 420 5233/3423
careerservices@up.ac.za
012 420 2315
012 420 2064
012 420 3111
012 420 3051
From 18:00 till 06:00 Security Officers are available to escort you (on foot) to
and from your residence or campus anywhere east of the Hatfield campus
through to the LC de Villiers terrain.
Departure point is next to the ABSA ATM next to the Merensky Library.
Phone the Operational Management Centre if you need a Security Officer to
fetch you from the residence to the campus.
FLY@UP | The Finish Line is Yours | www.up.ac.za/fly@up | email: fly@up.ac.za
Academic support

Tutors allocated to the module

the NAS Faculty Student Advisor (FSA)
FSA
Mpho Mmadi
Clusters allocated
- Physical Sciences
- Mathematical sciences
- Consumer science
Contact detail
Hatfield Campus, Mathematics Building Office 1-29
012 420 6740
Whatsapp: 0660658179
mpho.mmadi@up.ac.za
Rebecca Fulbeck
- Physical Sciences
- Mathematical sciences
- Consumer science
Hatfield Campus Akanyang Building, Ground Floor
Next to Tuks shop/Vida-e
012 420 2238
Whatsapp: 0660688379
rebecca.fulbeck@up.ac.za
Dolly Ayob
- Physical Sciences
- Mathematical sciences
- Consumer science
Hatfield Campus Akanyang Building, Ground Floor
Next to Tuks shop/Vida-e
012 420 4990
Whatsapp: 0660255425
dolly.ayob@up.ac.za
Chandre O'Reilly
(nee Dreyer)
- Physical Sciences
- Mathematical sciences
- Consumer science
Hatfield Campus, Mathematics Building Office 1-30
012 420 3096
WhatsApp: 0660643923
chandre.dreyer@up.ac.za
Students with special needs:
The University of Pretoria is keen to accommodate students with special needs. These needs are
catered for without influencing academic standards. It is, however, the student’s responsibility to
inform the lecturer in good time of any special needs in order to ensure effective communication in
this regard. The student should contact the Disability Unit and speak to the lecturer.
E-learning support:
Report a problem you experience to the Student Help Desk.
o
Approach the assistants at the help desks (adjacent to the Student Computer Laboratories in
IT Building, NW2, CBT, etc).
o
Visit the Open Labs situated in the Informatorium Building to report the problem at the
offices of the Student Help Desk.
o
Call 012 420 3837 (IT Support for Students)
o
Student Related Problems (Clickup, Registration, Passwords) – 012 420 3111
o
Email studenthelp@up.ac.za
DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS
Vision
The Department of Mathematics and Applied Mathematics strives to be internationally
recognised for academic excellence through the depth of its research and teaching, and to
be locally relevant through its role in the development of the community it serves.
Mission
The Department is an academic unit of the University of Pretoria, entrusted with the
development of mathematical skills, knowledge and insights. Its mission is to
 be actively, visibly and notably involved in research at the forefront of the
mathematical fields in which it has strength and expertise
 offer postgraduate training, up to doctoral and postdoctoral level, in its chosen fields
of research expertise
 at undergraduate and honours level, engage in mathematical training in support of its
own and other academic programmes of the University
Through its activities, the Department intends to
 contribute to the deep understanding of complex mathematical structures and their
applications
 deliver graduates with considerable mathematical skills and the desire to be involved
in problem solving
 intellectually and materially enhance the community and South African society by its
relevant research outputs and involvement in projects that depend on its
mathematical expertise