Algebra and Functions

Indices

• Introduction
• Multiplication rule 1
• Multiplication rule 2
• Division
• Negative indices
• Fractions to negative indices
• Fractional type
• Simplifying (negative powers)
• Expressing terms in the form axⁿ
• Exam questions

Exponential functions

• Exponential functions: what they are and their graphs

Logarithms

• What do we mean by a log?
• Rules of logs
• Simplifying
• Equations
  • Solving equations 1
  • Solving equations 2
    • Change of base type
      • Solving equations 3
  • Solving equations 4
• Inequalities
  • Solving inequalities
  Exam questions

Sequences and series

Recurrence relationships

• Recurrence relationships
• Exam questions

Sigma notation

• Sigma notation
• Exam questions

Arithmetic sequences and series

• Arithmetic sequences and series
• Sum of the first n terms
• Exam questions

Geometric Series

• Introduction
  • Example 1
  • Example 2 (consecutive terms example)
  • Example 3 (investment example)
• Proof of Sum of first n terms, Sn
• Sum to infinity
  • Example
• Exam questions

Binomial Expansion

• Introduction
• nCr method
  • method of expanding example 1
  • method of expanding example 2
  • finding particular terms
• Pascal's triangle method
• Binomial formula
  • Binomial formula method example 1
  • Binomial formula method example 2
• Exam questions

Trigonometry

Trigonometric Ratios

• Trig. ratios for 30, 60 and 90 degrees
• Trig. ratios for 45, 45 and 90 degrees
• Trig. ratios for positive multiples of 30, 45 and 60 degrees
• Trig. ratios for negative multiples of 30, 45 and 60 degrees

Graphs

• Trigonometric graphs

• Transformations
  • Translations
  • Reflections
  • Stretches
  • Combining transformations

Applications

• Area of a triangle
  • Given two sides and an included angle
• Sine Rule
  • Finding the length of a side of a non right angled triangle
  • Finding an angle of a non right angled triangle
• Cosine Rule
  • Finding the length of a side of a non right angled triangle
  • Finding an angle of a non right angled triangle
• Arcs, Sectors and Segments
  • Radians
  • Arc length and sector area
    • Examples
  • Area of a segment
    • Examples
• Exam questions

Equations

• The quadrant rule
  • How to solve sinθ = a positive or negative value in a given range
  • How to solve cosθ = a positive or negative value in a given range
  • How to solve tanθ = a positive or negative value in a given range
  • How to solve multiple angles, example sin2x = a positive value
• Equations that factorise
  • Solving trigonometric equations that factorise 1
  • Solving trigonometric equations that factorise 2
  • Solving trigonometric equations that factorise 3

Identities

• Two basic trigonometric identities: tanθ ≡sinθ/cosθ and sin²θ+cos²θ ≡1
  • Proving trignometric identities 1
  • Proving trignometric identities 2
  • Proving trignometric identities 3
• The identities: cos(θ) ≡ cos(-θ), sin(θ) ≡ -sin(-θ)
• Solving equations using identities
  • Solving trigonometric equations using identities 1
  • Solving trigonometric equations using identities 2
  • Solving trigonometric equations using identities 3
  • Solving trigonometric equations using identities 4
• Exam questions

Differentiation

Introduction

• The gradient function dy/dx
  • Differentiation from 1st principles
• terms of the form axⁿ
  • Extending to root types
  • Extending to fractional types
• Second Derivatives
  • The second derivative
• Exam Questions
  • Exam questions

Tangents and Normals

• Tangents and normals
• Exam Questions
  • Exam questions

Stationary Points

• Stationary points. What are they?
• Finding stationary points (example)
• Nature of a stationary point using the first differential
• Nature of a stationary point using the second differential
• Finding stationary points and their nature example
• Applications
• Exam questions

Increasing and Decreasing functions

• Increasing and decreasing functions

Integration

Introduction

• terms of the form axⁿ
  • fractional and root type terms
• Exam questions

Equations of curves

• Finding the equation of a curve given dy/dx
  • Using f'(x) notation
• Exam questions

Definite Integration

• Definite integration
• Infinite integrals
• Application - Area
  • Area under a curve
  • Finding the area below the x-axis
  • Finding the area above and below the x-axis
  • Finding the area bounded by two curves
• Exam questions

Numerical Integration

• Trapezium rule
  • Type of estimate
• Exam questions