lOMoAR cPSD| 45211451
MAT1512 EXAM PACK 2023
Calculus A (Mathematics)
CALCULUS A
,
, lOMoAR cPSD| 45211451
iii
MAT1512/1/2010-2012
Contents
Page
Introduction vii
How to use the study guide viii
Keys to success in studying Mathematics ix Preparing for the examination ix
CHAPTER 0: Preliminaries 1
1. Background 1
2. Learning outcomes 1
3. Principles of problem solving 2
4. Summary 3
5. The way forward 3
CHAPTER 1: Limits and Continuity 4
1. Background 4
2. Learning outcomes 4
3. Prescribed reading 5
4. The limit 5
4.1 Introduction to the limit concept 5
4.2 Definition of a limit: left- and right-hand limits 6
5. Worked examples 6
I. Limits as x → c (c ∈R) 7
II. Limits as x →±∞ 10
III. Limits involving absolute values 14
IV. Left-hand and right-hand limit 16
V. Limits involving trigonometric functions 19
VI. The Squeeze Theorem 23
, lOMoAR cPSD| 45211451
iv
VII. The ε−δ definition of a limit 26
VIII. Continuity 29
Key points 38
CHAPTER 2: Differentiation of different types of functions 39
1. Background 39
2. Learning outcomes 39
3. Prescribed reading 40
4. The deriviative 40
4.1 Introducing the derivative 40
4.2 Definition of the derivative 42
5. Worked examples 42
I. Differentiation from first principles (definition of the derivative) 44
II. Basic differentiation formulas 46
(a) The power rule 46
(b) The product rule 47
(c) The quotient rule 48
(d) The chain rule 49
(e) Combinations of rules 50
III. Derivatives of trigonometric functions and inverse trigonometric functions 52
(a) Derivatives of trigonometric functions 52
(b) Derivatives of inverse trigonometric functions 56
IV. Derivatives of exponential and logarithmic functions 57
(a) The exponential function 57
(b) The logarithmic function 58
(c) Examples of the exponential function 59
(d) Examples of the logarithmic function 60
V. Logarithmic differentiation 62
(a) The simplification of functions 62
(b) Functions of the form f(x) = g(x)h(x) 64
VI. Implicit differentiation 66
VII. Tangents and normal lines 71
MAT1512 EXAM PACK 2023
Calculus A (Mathematics)
CALCULUS A
,
, lOMoAR cPSD| 45211451
iii
MAT1512/1/2010-2012
Contents
Page
Introduction vii
How to use the study guide viii
Keys to success in studying Mathematics ix Preparing for the examination ix
CHAPTER 0: Preliminaries 1
1. Background 1
2. Learning outcomes 1
3. Principles of problem solving 2
4. Summary 3
5. The way forward 3
CHAPTER 1: Limits and Continuity 4
1. Background 4
2. Learning outcomes 4
3. Prescribed reading 5
4. The limit 5
4.1 Introduction to the limit concept 5
4.2 Definition of a limit: left- and right-hand limits 6
5. Worked examples 6
I. Limits as x → c (c ∈R) 7
II. Limits as x →±∞ 10
III. Limits involving absolute values 14
IV. Left-hand and right-hand limit 16
V. Limits involving trigonometric functions 19
VI. The Squeeze Theorem 23
, lOMoAR cPSD| 45211451
iv
VII. The ε−δ definition of a limit 26
VIII. Continuity 29
Key points 38
CHAPTER 2: Differentiation of different types of functions 39
1. Background 39
2. Learning outcomes 39
3. Prescribed reading 40
4. The deriviative 40
4.1 Introducing the derivative 40
4.2 Definition of the derivative 42
5. Worked examples 42
I. Differentiation from first principles (definition of the derivative) 44
II. Basic differentiation formulas 46
(a) The power rule 46
(b) The product rule 47
(c) The quotient rule 48
(d) The chain rule 49
(e) Combinations of rules 50
III. Derivatives of trigonometric functions and inverse trigonometric functions 52
(a) Derivatives of trigonometric functions 52
(b) Derivatives of inverse trigonometric functions 56
IV. Derivatives of exponential and logarithmic functions 57
(a) The exponential function 57
(b) The logarithmic function 58
(c) Examples of the exponential function 59
(d) Examples of the logarithmic function 60
V. Logarithmic differentiation 62
(a) The simplification of functions 62
(b) Functions of the form f(x) = g(x)h(x) 64
VI. Implicit differentiation 66
VII. Tangents and normal lines 71
