1 Sets
1.1 Set Operations
1.2 The Algebra of Sets
1.3 Practice Problems
2 Sets of Numbers
2.1 Rational Numbers
2.2 Irrational Numbers
2.3 Real Numbers
2.4 Practice Problems
3 Complex Numbers
3.1 The Imaginary Unit
3.2 Complex Numbers, Real and Imaginary Parts
3.3 Multiplication
3.4 Modulus
3.5 The Complex Conjugate
3.6 Division
3.7 Practice Problems
4 Surds
4.1 Rationalising the Denominator
4.2 Practice Problems
5 Binary Operations
5.1 Identity and Inverse Elements
5.2 Practice Problems
6 Relations
6.1 Relations
6.2 Functions
6.3 Domain and Range
6.4 One-to-One Functions
6.5 Even and Odd functions
6.6 Inverse Functions
6.7 Composite Functions
6.8 Practice Problems
7 Quadratic Functions
7.1 Roots of a Quadratic Equation
7.2 Quadratic Equation
7.3 The Discriminant
7.4 Quadratic Functions
7.5 Practice Problems
8 Equations
8.1 Extraneous Roots
8.2 Equations Reducible to Quadratic Form
8.3 Equations Involving Absolute Value
8.4 Practice Problems
9 Functions of the Form \(f(x)=\sqrt {x}\)
10 Inequalities
10.1 Linear Inequalities
10.2 Absolute Value Inequalities
10.3 Quadratic Inequalities
10.4 Rational Inequalities
10.5 Practice Problems
11 Graphs of Absolute Value Functions
11.1 Practice Problems
12 Trigonometry
12.1 Trigonometric Functions
12.2 Radian Measure
12.3 Identities
12.4 Graphs
12.5 Period, Amplitude, Phase Shift
12.5.1 Period
12.5.2 Amplitude
12.6 Practice Problems
13 Limits
13.1 Limits Of The Form \(\dfrac {0}{0}\)
13.2 Standard Limits
13.3 Limits Of The Form \(\dfrac {\infty }{\infty }\)
13.4 Some Standard Limits:
13.5 Continuity
13.6 Practice Problems
14 Derivatives
14.1 Chain Rule
14.2 Product Rule
14.3 Quotient Rule
14.4 Implicit Differentiation
15 Polynomials
15.1 Remainder Theorem
15.2 Synthetic Division
15.3 Practice Problems
16 Exponential and Logarithm
16.1 The Natural Logarithm Function
16.2 Laws of Logarithm
16.3 Change of Bases
16.4 Equations of The Form \(a^x=b\)
16.5 Derivatives of Logarithmic Function
16.6 Practice Problems
17 Hyperbolic Functions
17.1 Graphs
17.2 Identities for Hyperbolic Functions
17.3 Osborn’s Rule
17.4 Practice Problems
18 Derivatives of Inverse Trigonometric Functions
18.1 Domain Restrictions that make Trigonometric Functions One-to-One
18.2 Differentiation of Inverse Trigonometric Functions
18.3 Practice Problems
19 Curve Sketching
20 Mathematical Induction
20.1 Practice Problems
21 Binomial Expansion
21.1 The Binomial Series
21.2 Practice Problems
22 The Straight Line and the Circle
22.1 Properties of Circles
22.2 The Equation and Proportional of Tangents to a Circle
22.3 The Distance of a Point from a Line
22.4 Division of a Line in a Given Ratio
22.4.1 Internal Division of a Line Segment
22.4.2 External Division of the Line Segment
22.5 Concentric Circles
22.6 Contact of Circles
22.7 Practice Problems
23 Integration
23.1 Fundamental Theorem of Calculus
23.2 Methods of Integration
23.2.1 Integration of powers
23.2.2 Integration by Parts
23.2.3 Integration by Substitution
23.3 Trigonometric Substitution
23.4 Partial Fractions
23.5 Area Under the Curve
24 Matrices
24.1 Operations On Matrices
24.2 Transpose Matrix
24.3 Determinants
24.4 Inverse Of Matrix
24.5 Solving Systems Of Equations
24.6 Practice Problems
25 Polar Form of Complex Numbers
25.1 Multiplication/Division of Complex Numbers
25.2 De Moivres Theorem
25.3 The \(n^{\text {th}}\) Root of any Complex Number
26 Vectors
26.1 Introduction
26.2 Modulus of a Vector
26.3 Unit vector
26.4 Scalar Multiplication of a Vector
26.5 Parallel Vectors
26.6 Position Vectors
26.7 The Scalar Product of Two Vectors
26.8 The Vector Product