Inequalities

Fractional Inequalities

• Example 1
  • Solution by graphical methods
  • Solution by analytical methods
• Example 2
  • Solution by a graphical method 1 | graphical method 2
  • Solution by analytical method

Modulus Inequalities (fractional types)

• Example 1
  • Solution by graphical methods
• Example 2
  • Solution by a graphical method

Series

Method of Differences

• Method of differences
• Example using factorials

Further complex numbers

• Modulus-argument form
• Exponential Form (Euler's relation)
• Multiplication rule for the mod and argument of two complex numbers
• Division rule for the mod and argument of two complex numbers
  • Mixed examples

de Moivre's Theorem

• de Moivre's theorem
  • Simplifying example
  • Expansion example
• Expressing sin nθ and cos nθ in terms of sinθ and cosθ
  • Introduction
• Example
• Expessing sinⁿθ and cosⁿθ in terms of sinkθ and coskθ
  • Introduction - new identities you will need
  • Expressing cosⁿθ in terms of coskθ
  • Expressing sinⁿθ in terms of sinkθ when n is odd
  • Expressing sinⁿθ in terms of coskθ when n is even

• nth roots of a complex number
  • Example - cube roots of 1
  • Example - 4th roots of a complex number

Loci in the complex plane

• The locus of a point moving in a circle : | z- z₁| = r
• The locus of a point moving along a perpendicular bisector : | z-z₁ | = | z-z₂ |
• The locus of a point moving along a half-line : arg(z-z₁) = θ
• The locus of a point moving on the arc of a circle : arg[ (z-z₁) / (z-z₂) ] = θ

• Using complex numbers to represent regions on an Argand diagram

(still to be completed - on hold at present)

• Transformations of the complex plane

First order differential equations

Separating the variables (Revision)

• Finding a general solution and a particular solution
• Working with constants in log types
• Exponential and trig type (a little more challenging)

Family of curves

• separating the variables and sketching a family of curves

Exact equations (integrating factors)

• Exact equations where one side is the exact derivative of a product
• Solving equations of the form dy/dx + Py = Q using an integrating factor
  • Examples

Substitution types

• Using substitution to reduce a differential equation to a known form

Second order differential equations

Equations of the form a d²y/dx² + b dy/dx + cy = 0

• Introduction
• Solving equations where b² - 4ac > 0
• Solving equations where b² - 4ac = 0
• Solving equations where b² - 4ac < 0

Equations of the form a d²y/dx² + b dy/dx + cy = f(x)

• General solutions
  • Introduction where f(x) = k (constant types)
  • where f(x) = kx (linear types)
  • where f(x) = kx² (quadratic types)
  • where f(x) = ke^px (exponential types)
  • where f(x) = m cosωx + n sinωx (trig types)
  • Special types of particular integrals
    • f(x) = k (constant types)
    • f(x) = ke^px (exponential types)
• Particular solutions
  • Using boundary conditions to solve differential equations

Using a Substitution

• Using substitution to reduce a differential equation to a known form

Maclaurin and Taylor series

Maclaurins series

• Maclaurin's series expansion
  • series expansion for e^x
  • series expansion for sin(x) and cos(x)
  • series expansion for ln(1+x)
  • further series

(still to be completed on hold whilst past papers are being done)

• Taylor's expansion
• Solution to differential equations using Taylors series

Polar coordinates

• (still to be completed)
• Polar and Cartesian coordinates
• Polar and Cartesian equations of curves
• Sketching polar equations
• Area bounded by a polar curve
• Finding equations of tangents parallel and perpendicular to the initial line