P3 Tutorials

Algebra and Functions

Algebraic Long Division

  1. Algebraic long division

Factor and Remainder Theorems

  1. The factor theorem
  2. How to solve a cubic equation
  3. Exam Questions - Factor theorem
  4. The remainder theorem
  5. Exam Questions - Remainder theorem

Modulus Functions, Equations and Inequalities

  1. The modulus function
  2. Graphing y=|f(x)|
  3. Graphing y=f(|x|)
  4. Exam Questions - Modulus functions graphing
  5. Modulus equations
  6. Exam Questions - Modulus equations
  7. Modulus inequalities
  8. Exam Questions - Modulus inequalities

Rational Expressions

Partial Fractions

  1. Partial fractions
  2. Denominator contains 2 or 3 linear factors
  3. Denominator contains repeated factors
  4. Exam Questions - Partial fractions
  5. Denominator contains 1 linear and 1 quadratic factor

Sequences and Series

Binomial Expansion

  1. Exam Questions - Binomial expansion for rational and negative powers
  2. Exam Questions - Partial fractions with the binomial expansion

Logarithmic and Exponential Functions

Exponential Functions and Logarithms

  1. Exponential functions: what they are and their graphs
  2. What do we mean by a log?
  3. Rules of logs
  4. Logarithms - Change of Base
  5. Simplifying and expanding
  6. Converting between Logarithmic and Power (Exponential) equations
  7. Modelling exponential growth and decay
  8. Exponential and log equations
  9. Simultaneous equations
  10. Solving inequalities
  11. Exam Questions - Logarithms

The Exponential Function ex and Natural Log Functions

  1. The exponential function ex
  2. Sketching exponential graphs based on transformations
  3. The natural logarithmic function, ln x
  4. Exam Questions - Natural log functions

Modelling Curves of the form y=kxn and y=kax

  1. Modelling curves - Converting to linear form

Trigonometry

Sec θ, Cosec θ and Cot θ

  1. Trig functions sec θ, cosec θ and cot θ
  2. Graphs of sec θ, cosec θ and cot θ

Identities & Equations – Pythagorean Type

  1. sin²x + cos²x ≡1 , 1 + tan²x ≡ sec²x , 1 + cot²x ≡ cosec²x
  2. Solving equations using Pythagorean identities

Identities – Addition type

  1. sin(A±B), cos(A±B) and tan(A±B)
  2. Using the Addition formulae to get exact values
  3. Proving identities using the addition formulae
  4. Identities - Addition type - Equations

Identities & Equations – Double angle type

  1. Identities for sin2A, cos2A and tan2A
  2. Examples using double angle identities
  3. Solving equations using double angle identities
  4. Exam Questions - Double Angles

Identities – Half angle type

  1. Examples using half angle identities

Identities & Equations – Harmonic Formulae

  1. Harmonic Identities Rsin(x ± α), Rcos(x ± α)
  2. Equations using harmonic identities
  3. Harmonic identities - Max and Min
  4. Exam Questions - Harmonic identities and equations

Differentiation

Standard Differentials

  1. Exponential function ex
  2. The natural log function, ln(x)
  3. The trig functions sin(x), cos(x) and tan(x)

The Chain Rule

  1. Chain rule: Polynomial to a rational power
  2. Chain rule: Exponential types
  3. Chain rule: Natural log types
  4. Chain rule: Trigonometric types

The Product and Quotient Rules

  1. The product rule
  2. The quotient rule

More Standard Differentials

  1. The trig functions, sec(x), cosec(x) and cot(x)

Differentiating arc tan(x)

  1. Differentiating arctan(x/a) or inverse tan(x/a)
  2. Differentiating arctan(x/a) or inverse tan(x/a) example

The Reciprocal Function of dx/dy

  1. The reciprocal function of dy/dx

Parametric Functions

  1. Differentiation: Parametric functions
  2. Tangents and normals
  3. Stationary points
  4. Exam Questions - Parametric functions

Implicit Functions

  1. Implicit functions
  2. Implicit functions with product rule
  3. Tangents and normals
  4. Stationary points
  5. Exam Questions - Implicit functions

Integration

Integration – Common Functions

  1. Integration:(ax+b)n types
  2. Exam Questions - Integration:(ax+b)n types
  3. Integrating exponential functions ex, eax and e(ax+b)
  4. Exam Questions - Integrating exponential functions ex, eax and e(ax+b)
  5. Integrating reciprocal functions 1/x and 1/(ax+b)
  6. Exam Questions - Integrating reciprocal functions 1/x and 1/(ax+b)
  7. Integrals of the form : f '(x)/f(x)
  8. Integrals of the form : f'(x)ef(x)

Integrals of Trigonometric Functions

  1. Integrals of sin x, cos x, sec² x
  2. Integrals of the form sin(ax+b), cos(ax+b), sec² (ax+b) types
  3. Integrals Using Trigonometric Identities
  4. sin2x and cos2x types
  5. Exam Questions - Trigonometric types

Integrals involving Partial fractions

  1. Integrals involving partial fractions
  2. Exam Questions - Integrals involving partial fractions

Integration by Substitution

  1. Integration by substitution
  2. Square root types
  3. Integration of trigonometric functions by substitution
  4. Integration of exponential types by substitution
  5. Integration by substitution using limits
  6. Integration of trigonometric functions by substitution with limits
  7. Exam Questions - Integration by substitution

Integrating Products of the form f[g(x)]g'(x) by inspection

  1. Integrating products of the form f[g(x)]g'(x) by inspection

Integration by Parts

  1. Integration by parts
  2. Integration by parts (ln types)
  3. Integration by parts using limits
  4. Proof of the formula - Integration by parts
  5. Exam Questions - Integration by parts

Mixed Examples & Exam Question – Integration

  1. Integration - General Methods
  2. Mixed Examples - Integration
  3. Exam Questions - Integration

Numerical Integration

  1. Trapezium rule
  2. Exam Questions - Trapezium rule

Numerical Solution of Equations

Solution of Equations by Numerical methods

  1. Graphical methods
  2. Change of sign
  3. Iteration
  4. Exam Questions - Iteration

Vectors

Vector Equations of Lines

  1. Vector equation of a line
  2. Angle between two lines
  3. Parallel lines
  4. Intersecting and skew lines
  5. Exam Questions - Parallel intersecting and skew lines
  6. Closest point to a line and shortest distance from the origin
  7. Shortest distance of a point to a line

Lines

  1. Cartesian form of a line
  2. Vector product form of a line
  3. Shortest distance between two skew lines

Planes

  1. Parametric vector form of a plane
  2. Scalar product forms of a plane
  3. Cartesian form of a plane
  4. Finding the point of intersection between a line and a plane
  5. The equation of the line of intersection between two non parallel planes
  6. Shortest distance from a point to a plane
  7. The angle between two planes

First Order Linear Differential Equations

Separating the variables

  1. Differential Equations - Finding a general and a particular solution
  2. Working with constants in log types
  3. Differential Equations - Exponential and trig type

Forming differential equations

  1. Direct proportion type
  2. Inverse proportion type
  3. Newton's law of cooling
  4. Exam Questions – Forming differential equations

Complex Numbers

Complex Numbers

  1. Real and imaginary numbers
  2. Addition, subtraction and multiplying complex numbers and simplifying powers of i
  3. Complex conjugates
  4. Division of a complex number by a complex number
  5. Argand diagrams
  6. Modulus and argument of a complex number
  7. Solving problems with complex numbers
  8. Square roots of a complex number
  9. Solving quadratic equations with complex roots
  10. Solving cubic equations
  11. Solving quartic equations
  12. Exam Questions - Complex numbers
  13. Geometrical effects of conjugating a complex number
  14. Exam Questions - Finding roots
  15. Exponential Form (Euler's relation)
  16. Multiplication and division rules for mod and argument of two complex numbers
  17. Exam Questions - Further complex numbers
  18. Complex numbers specimen questions (mixed exam boards)

Further Complex Numbers

  1. Exponential Form (Euler's relation)
  2. Multiplication and division rules for mod and argument of two complex numbers
  3. Exam Questions - Further complex numbers
  4. Complex numbers specimen questions (mixed exam boards)

Loci in the Complex Plane

  1. The locus of a point moving in a circle
  2. The locus of a point moving along a perpendicular bisector
  3. The locus of a point moving along a half-line
  4. The locus of a point moving on the arc of a circle
  5. Using complex numbers to represent regions on an Argand diagram
  6. Exam Questions - Loci in the complex plane