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MAT1512 EXAM PACK 2023
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MAT1512EXAM
PACK
2023
QUESTIONS WITH
ANSWERS/ USEFUL
NOTES AND
EXAMPLES
, lOMoARcPSD|24668432
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
VII. The ε − δ definition of a limit 26
VIII. Continuity 29
Key points 38
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, lOMoARcPSD|24668432
iv
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
VIII. The Mean Value Theorem 78
Key points 81
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, lOMoARcPSD|24668432
v MAT1512/1/2010-2012
CHAPTER 3: Integration 82
1. Background 82
2. Learning outcomes 82
3. Prescribed reading 83
4. Worked examples 83
I. Antiderivatives 84
II. The definite integral and the Fundamental Theorem of Calculus — Part I 86
III. The definite integral and the area between the curve and the x-axis 87
IV. The definite integral and the area under the curve 90
V. The Mean Value Theorem for definite integrals 94
VI. The Fundamental Theorem of Calculus — Part II 96
VII. Integration in general 99
VIII. Indefinite integrals 100
IX. Integration by substitution 101
(a) Indefinite integrals 101
(b) Definite integrals 106
(c) Steps for integration by substitution 108
X. Integration of exponential and logarithmic functions 109
XI. Review of formulas and techniques of integration <Author:
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Key points 115
CHAPTER 4: First—order Differential Equations, Growth and Decay 116
and Partial derivatives
1. Background 116
2. Learning outcomes 116
3. Prescribed reading 117
4. Worked examples 117
I. First-order differential equations 117
(a) The solution of differential equations 117
(b) Separable differential equations 117
(c) Initial-value problems 122
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