Calculus
Collection of tutorial exercises for Bachelor of
Engineering Mathematics (EMA105B) students at
Tshwane University Technology compiled
by
Dr M. Aphane with the help of tutors Ms Daphney
Hlotse and Mr Akis Muepu at the Department of
Mathematics and Statistics
2020
Revised : ET Motlotle
,The main purpose of this compilation of tutorial exercise is to collect problems
from different calculus books for students who enrolled for Engineering
Mathematics 1 (EMA105B) at Tshwane University of Technology. The tutorial
guide does not replace the prescribed book. It is still necessary for students to buy
and use the prescribed book.
Calculus is divided into two parts, which are differentiation and integration. Some
applications on differentiation and integration are also included. The guide is
structured in such a way that problems are sorted by topics and some solutions
are provided at the end of each chapter. Some examples and basic introductions
are also provided at the beginning of each chapter.
We trust that you will find the tutorial guide useful and enjoy using it. If you
encounter any errors, incorrect solutions or suggestions on how to improve this
tutorial guide. Feel free to contact us on .
©COPYRIGHT : Tshwane University of Technology
Private Bag X680
PRETORIA
0001
All rights reserved. Apart from any reasonable quotations for the purposes of research
criticism or review as permitted under the Copyright Act, no part of this book may be
reproduced or transmitted in any form or by any means, electronic or mechanical,
including photocopy and recording, without permission in writing from the publisher.
ii
,Contents
Chapter 1 ............................................................................................................................................... 1
Differentiation ................................................................................................................................... 1
1.1 The derivative and the tangent problem ............................................................................. 1
1.2 Derivative of a function ........................................................................................................ 1
1.3 Chain Rule Differentiation ................................................................................................... 3
1.4 Implicit Differentiation ......................................................................................................... 5
1.5 Logarithmic and Some Implicit Differentiation ................................................................. 8
1.6 Higher Order Derivatives ................................................................................................... 11
1.7 Optimization ........................................................................................................................ 12
1.8 Parametric Differentiation ................................................................................................. 17
1.9 Differentiation of Hyperbolic Functions ........................................................................... 20
1.10 Inverse Trigonometric Functions ...................................................................................... 21
1.11 Inverse Hyperbolic Functions ............................................................................................ 27
Chapter 2 ............................................................................................................................................. 30
Applications of Differentiation ...................................................................................................... 30
2.1 L’Hopital’s rule:.................................................................................................................. 30
2.2 Curve Sketching and Tangents and Normal .................................................................... 32
2.3 Newtown – Raphson Method ............................................................................................. 36
2.4 Maclaurin Series ................................................................................................................. 38
2.5 The Binomial Expansions ................................................................................................... 40
Chapter 3 ............................................................................................................................................. 42
Partial differentiation ..................................................................................................................... 42
3.1 Partial derivatives .............................................................................................................. 42
3.2 Clairaut’s Theorem ............................................................................................................ 43
3.3 Critical points ...................................................................................................................... 44
3.4 The second partial test ........................................................................................................ 44
3.5 The total differential ........................................................................................................... 45
3.6 Small change ........................................................................................................................ 46
3.7 Rate of change and chain rule ............................................................................................ 47
3.8 Implicit partial differentiation ........................................................................................... 49
3.9 Application of partial derivatives (Partial Differential equations) ................................. 50
Chapter 4 ............................................................................................................................................. 51
Single Variable Integration ............................................................................................................ 51
4.1 Anti-derivative and indefinite integrals ............................................................................ 51
iii
, Indefinite integrals .......................................................................................................................... 51
4.2 Integration by Substitution ................................................................................................ 53
4.3 Integration of Inverse Trigonometry and Inverse Hyperbolic Functions ..................... 56
4.4 Integration by Partial Fraction.......................................................................................... 57
4.5 Integration by Parts ............................................................................................................ 60
4.6 Trigonometric Integrals ..................................................................................................... 63
4.7 Trigonometric Substitution ................................................................................................ 68
Chapter 5 ............................................................................................................................................. 73
Application of Integration .............................................................................................................. 73
5.1 Mean Value Theorem for Integrals ................................................................................... 73
Theorem (Mean Value Theorem for Integrals) ............................................................................ 73
5.2 Root Mean Square value (RMS value) .............................................................................. 74
5.3 Area between the curves ..................................................................................................... 78
5.4 Motion .................................................................................................................................. 79
5.5 Mix problem: Application of integration .......................................................................... 80
Chapter 6 ............................................................................................................................................. 81
Multiple Integrals............................................................................................................................ 81
6.1 Double Integrals .................................................................................................................. 81
6.2 Triple integral...................................................................................................................... 87
6.3 Line integrals ....................................................................................................................... 88
6.4 Line integrals with Respect to Arc length ......................................................................... 89
6.5 Green’s Theorem ................................................................................................................ 90
6.6 Surface integral ................................................................................................................... 90
References .......................................................................................................................................... 117
iv
Collection of tutorial exercises for Bachelor of
Engineering Mathematics (EMA105B) students at
Tshwane University Technology compiled
by
Dr M. Aphane with the help of tutors Ms Daphney
Hlotse and Mr Akis Muepu at the Department of
Mathematics and Statistics
2020
Revised : ET Motlotle
,The main purpose of this compilation of tutorial exercise is to collect problems
from different calculus books for students who enrolled for Engineering
Mathematics 1 (EMA105B) at Tshwane University of Technology. The tutorial
guide does not replace the prescribed book. It is still necessary for students to buy
and use the prescribed book.
Calculus is divided into two parts, which are differentiation and integration. Some
applications on differentiation and integration are also included. The guide is
structured in such a way that problems are sorted by topics and some solutions
are provided at the end of each chapter. Some examples and basic introductions
are also provided at the beginning of each chapter.
We trust that you will find the tutorial guide useful and enjoy using it. If you
encounter any errors, incorrect solutions or suggestions on how to improve this
tutorial guide. Feel free to contact us on .
©COPYRIGHT : Tshwane University of Technology
Private Bag X680
PRETORIA
0001
All rights reserved. Apart from any reasonable quotations for the purposes of research
criticism or review as permitted under the Copyright Act, no part of this book may be
reproduced or transmitted in any form or by any means, electronic or mechanical,
including photocopy and recording, without permission in writing from the publisher.
ii
,Contents
Chapter 1 ............................................................................................................................................... 1
Differentiation ................................................................................................................................... 1
1.1 The derivative and the tangent problem ............................................................................. 1
1.2 Derivative of a function ........................................................................................................ 1
1.3 Chain Rule Differentiation ................................................................................................... 3
1.4 Implicit Differentiation ......................................................................................................... 5
1.5 Logarithmic and Some Implicit Differentiation ................................................................. 8
1.6 Higher Order Derivatives ................................................................................................... 11
1.7 Optimization ........................................................................................................................ 12
1.8 Parametric Differentiation ................................................................................................. 17
1.9 Differentiation of Hyperbolic Functions ........................................................................... 20
1.10 Inverse Trigonometric Functions ...................................................................................... 21
1.11 Inverse Hyperbolic Functions ............................................................................................ 27
Chapter 2 ............................................................................................................................................. 30
Applications of Differentiation ...................................................................................................... 30
2.1 L’Hopital’s rule:.................................................................................................................. 30
2.2 Curve Sketching and Tangents and Normal .................................................................... 32
2.3 Newtown – Raphson Method ............................................................................................. 36
2.4 Maclaurin Series ................................................................................................................. 38
2.5 The Binomial Expansions ................................................................................................... 40
Chapter 3 ............................................................................................................................................. 42
Partial differentiation ..................................................................................................................... 42
3.1 Partial derivatives .............................................................................................................. 42
3.2 Clairaut’s Theorem ............................................................................................................ 43
3.3 Critical points ...................................................................................................................... 44
3.4 The second partial test ........................................................................................................ 44
3.5 The total differential ........................................................................................................... 45
3.6 Small change ........................................................................................................................ 46
3.7 Rate of change and chain rule ............................................................................................ 47
3.8 Implicit partial differentiation ........................................................................................... 49
3.9 Application of partial derivatives (Partial Differential equations) ................................. 50
Chapter 4 ............................................................................................................................................. 51
Single Variable Integration ............................................................................................................ 51
4.1 Anti-derivative and indefinite integrals ............................................................................ 51
iii
, Indefinite integrals .......................................................................................................................... 51
4.2 Integration by Substitution ................................................................................................ 53
4.3 Integration of Inverse Trigonometry and Inverse Hyperbolic Functions ..................... 56
4.4 Integration by Partial Fraction.......................................................................................... 57
4.5 Integration by Parts ............................................................................................................ 60
4.6 Trigonometric Integrals ..................................................................................................... 63
4.7 Trigonometric Substitution ................................................................................................ 68
Chapter 5 ............................................................................................................................................. 73
Application of Integration .............................................................................................................. 73
5.1 Mean Value Theorem for Integrals ................................................................................... 73
Theorem (Mean Value Theorem for Integrals) ............................................................................ 73
5.2 Root Mean Square value (RMS value) .............................................................................. 74
5.3 Area between the curves ..................................................................................................... 78
5.4 Motion .................................................................................................................................. 79
5.5 Mix problem: Application of integration .......................................................................... 80
Chapter 6 ............................................................................................................................................. 81
Multiple Integrals............................................................................................................................ 81
6.1 Double Integrals .................................................................................................................. 81
6.2 Triple integral...................................................................................................................... 87
6.3 Line integrals ....................................................................................................................... 88
6.4 Line integrals with Respect to Arc length ......................................................................... 89
6.5 Green’s Theorem ................................................................................................................ 90
6.6 Surface integral ................................................................................................................... 90
References .......................................................................................................................................... 117
iv
