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Table of Contents

Module 1: Limits and Continuity		5
1.1 Introduction	…………………………………………………………………………………………………………………..	5
1.2 The function	…………………………………………………………………………………………………………………..	5
1.2.1 System of axes and ordered pairs	…………………………………………………………………………….	5
1.2.2 Domain and range	…………………………………………………………………………………………….	6
1.2.3 Dependent and independent variables	……………………………………………………………………….	6
1.2.4 Functions	…………………………………………………………………………………………………………..	6
1.2.5 Continuous and discontinuous functions	……………………………………………………………………..	6
1.2.6 Functions and relations	…………………………………………………………………………………………	7
1.2.7 Inverse functions	………………………………………………………………………………………………..	7
1.3 Limits	………………………………………………………………………………………………………………………	8
1.3.1 Increments	……………………………………………………………………………………………………….	8
1.3.2 The concept of the limit	…………………………………………………………………………………………	8
1.3.3 A limit in the form `00\frac{0}{0}00`	……………………………………………………………………………………	11
1.3.4 L Hospital’s rule	………………………………………………………………………………………………..	12
Module 2: Differentiation		19
2.1 Introduction	…………………………………………………………………………………………………………………..	19
2.2 Differentiation from first principles	……………………………………………………………………………………	19
2.2.1 Differentiate from first principles functions in the form `fx=axnfx = ax^nfx=axn`	…………………………………	19
2.2.2 Differentiate from first principles functions in the form `fx=a±bx±cxfx = a\pm bx\pm cxfx=a±bx±cx`	……………………	20
2.2.3 Differentiate from first principles functions in the form `sin⁡x\sin xsinx` and `cos⁡x\cos xcosx`	……………………….	22
2.3 Differentiation techniques	………………………………………………………………………………………………..	23
2.3.1 Trigonometric functions	……………………………………………………………………………………….	23
2.3.2 The chain rule	………………………………………………………………………………………………….	30
2.3.3 Logarithmic differentiation	……………………………………………………………………………………	31
2.3.4 Implicit functions	………………………………………………………………………………………………	33
Module 3: Application of Differentiation		42
3.1 Introduction	…………………………………………………………………………………………………………………..	42
3.2 Determine the roots of cubic polynomials	…………………………………………………………………………	42
3.2.1 The table for `xxx` and `f(x)f(x)f(x)`	…………………………………………………………………………………….	42
3.2.2 Draw the graph	………………………………………………………………………………………………..	43
3.2.3 Find the roots	………………………………………………………………………………………………….	43
3.3 Determine the Turning points of cubic polynomials	…………………………………………………………..	45
3.4 Rates of change	…………………………………………………………………………………………………………..	54
3.4.1 Velocity and acceleration	……………………………………………………………………………………..	54
3.4.2 Related rates of change	……………………………………………………………………………………….	55
Module 4: Integration Techniques		66
4.1 Introduction	…………………………………………………………………………………………………………………..	67
4.2 Indefinite integrals	……………………………………………………………………………………………………….	67
4.2.1 Standard forms of integrals	……………………………………………………………………………………	68
4.2.2 The chain rule	………………………………………………………………………………………………….	69
4.2.3 Other integrals	………………………………………………………………………………………………….	70
4.3 Substitution to transform composite functions	………………………………………………………………….	71
4.4 The integral of the form `∫f′(x)f(x)dx\int f'(x)f(x) dx∫f′(x)f(x)dx`	……………………………………………………………………..	72
4.5 The integral of the form `∫f′(x).fnxdx\int f'(x).f^n x dx∫f′(x).fnxdx`	……………………………………………………………………..	73
4.6 Algebraic fractions	……………………………………………………………………………………………………….	75
4.6.1 Partial fractions	………………………………………………………………………………………………..	75
Module 5: Application of the Definite Integral		89
5.1 Introduction	…………………………………………………………………………………………………………………..	89
5.2 The area between two curves	…………………………………………………………………………………………	94
5.3 Second moment of area	………………………………………………………………………………………………	102
5.3.1 Second moment of area of a rectangular lamina	………………………………………………………..	102
5.3.2 Second moment of area of a circular lamina	…………………………………………………………….	103
Module 6: Differential Equations		114
6.1 Introduction	…………………………………………………………………………………………………………………..	114
6.2 General and particular solutions of differential equations	……………………………………………………	115
Past Examination Papers	………………………………………………………………………………………………..	122

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Module 1
Limits and Continuity

Learning Outcomes
On the completion of this module the student must be able to:

Introduction to the function
Apply the L Hospital’s rule
Describe the conditions of continuity
Determine whether a function is continuous or discontinuous in a specified point

1.1 Introduction
(Open book icon) This module introduces the function and describes the application of the L Hospital’s rule. It also describes the conditions of continuity and explains how to determine whether a function is continuous or discontinuous in a specified point.

1.2 The function
1.2.1 System of axes and ordered pairs
The reason why the three numbered pairs in Figure 1.1 are termed ordered pairs is because the