© 2016 University of South Africa All rights reserved Printed and published by the University of South Africa Mucklneuk, Pretoria MAT2691/1/2017
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2018 70402418 Layout done by the Department
MAT2691
UNISA
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CONTENTS
PAGE
Introduction iv
MODULE 1 DIFFERENTIATION STUDY GUIDE 1
Learning Unit 1 Standard Differential Coefficients 1 Learning Unit 2 Implicit Functions 15 Learning Unit 3 Logarithmic Differentiation 23 Learning Unit 4 Successive Differentiation 32 Learning Unit 5 Inverse Trigonometric Functions 40 Learning Unit 6 Hyperbolic Functions 50 Learning Unit 7 Inverse Hyperbolic Functions 61 Learning Unit 8 Parametric Equations 76 Learning Unit 9 Maximum and Minimum Values, Points of Inflection 84 Learning Unit 10 Practical Applications 112 Learning Unit 11 Partial Differentiation 138 Learning Unit 12 Small Increments and Rates of Change 158 Learning Unit 13
Maclaurin’s Series
170
MODULE 2 INTEGRATION STUDY GUIDE 2
Learning Unit 1 Standard Integrals 1 Learning Unit 2 Trigonometric Integrals 13 Learning Unit 3 Trigonometric Integrals II 26 Learning Unit 4 Hyperbolic Functions 36 Learning Unit 5 Inverse Functions 43 Learning Unit 6
Integration using “t

formulae”
57 Learning Unit 7 Trigonometric Substitutions 70 Learning Unit 8 Integration by Parts 81 Learning Unit 9 Areas 96 Learning Unit 10 Volumes 112 Learning Unit 11 Applications 128 Learning Unit 12 Numerical Integration 140
MODULE 3 LINEAR ALGEBRA STUDY GUIDE 2
Learning Unit 1 Matrices: Definition and Classification 155 Learning Unit 2 Matrix Operations 166 Learning Unit 3 Inverses of Matrices 179 Learning Unit4 Systems of Linear Equations 193
MAT2691
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INTRODUCTION: MATHEMATICS II (Engineering)
Welcome to the module, Mathematics II for Engineering. This material has been compiled to serve the mathematical needs of learners engaged in engineering courses. Learners from other fields who want to broaden their mathematical knowledge will also benefit from this course. Very little prior mathematical knowledge is assumed, and hence, the study guide is useful to a very wide audience.
Purpose of this module
This module will be useful to students in developing basic skills that can be applied to the natural and engineering sciences. Students credited with this module will have an understanding of the basic ideas of algebra and calculus in handling problems related to the following
: Cramer’s rule to solve systems of linear equations,
the complex number system, binomial theorem, and basic differentiation and integration. The focus is on building strong algebraic skills that will support the development of analytical skills, which are crucial for problem solving in more advanced mathematics and related subjects. This module will support students in their studies in the field of engineering and the physical sciences, as part of a diploma. The style of the study guide makes it suitable for selfstudy. Discipline and hard work result in success. The next section provides a framework to ensure effective study.
Where do you start?
Learn the following and memorise it:
Your student number
The module code is MAT2691
Do the following on the internet:
Log on to https://my.unisa.ac.za
Register on myUNISA and claim your mylife email.
Log on weekly to check for any new announcements. It is essential that you register as the lecturer posts announcements, which you are alerted to via email.
Organise your workspace. You need:
A place to write
Paper and a pen to try examples, do activities and exercises
A scientific calculator, for example, a Sharp EL531 LH or a Casio
fx
82L
Ordinary tools for drawing
The tutorial letter for Mathematics II
Consult your tutorial letter to obtain information on
Assignments
How to obtain a year mark
Examinations
Due dates