# P3 Tutorials

## Algebra and Functions

## Algebraic Long Division

## Factor and Remainder Theorems

## Rational Expressions

## Sequences and Series

## Logarithmic and Exponential Functions

## Exponential Functions and Logarithms

- Exponential functions: what they are and their graphs
- What do we mean by a log?
- Rules of logs
- Logarithms - Change of Base
- Simplifying and expanding
- Converting between Logarithmic and Power (Exponential) equations
- Modelling exponential growth and decay
- Exponential and log equations
- Simultaneous equations
- Solving inequalities
- Exam Questions - Logarithms

## The Exponential Function e^{x} and Natural Log Functions

## Modelling Curves of the form y=kx^{n} and y=ka^{x}

## Trigonometry

## Sec θ, Cosec θ and Cot θ

## Identities & Equations – Pythagorean Type

## Identities – Addition type

## Identities & Equations – Double angle type

## Identities – Half angle type

## Identities & Equations – Harmonic Formulae

## Differentiation

## Standard Differentials

## The Chain Rule

## The Product and Quotient Rules

## More Standard Differentials

## Differentiating arc tan(x)

## The Reciprocal Function of dx/dy

## Parametric Functions

## Integration

## Integration – Common Functions

- Integration:(ax+b)
^{n}types - Exam Questions - Integration:(ax+b)
^{n}types - Integrating exponential functions e
^{x}, e^{ax}and e^{(ax+b)} - Exam Questions - Integrating exponential functions e
^{x}, e^{ax}and e^{(ax+b)} - Integrating reciprocal functions 1/x and 1/(ax+b)
- Exam Questions - Integrating reciprocal functions 1/x and 1/(ax+b)
- Integrals of the form : f '(x)/f(x)
- Integrals of the form : f'(x)e
^{f(x)}

## Integrals of Trigonometric Functions

## Integrals involving Partial fractions

## Integration by Substitution

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

## Integration by Parts

## Mixed Examples & Exam Question – Integration

## Numerical Integration

## Numerical Solution of Equations

## Solution of Equations by Numerical methods

## Vectors

## Vector Equations of Lines

## First Order Linear Differential Equations

## Separating the variables

## Forming differential equations

## Complex Numbers

## Complex Numbers

- Real and imaginary numbers
- Addition, subtraction and multiplying complex numbers and simplifying powers of i
- Complex conjugates
- Division of a complex number by a complex number
- Argand diagrams
- Modulus and argument of a complex number
- Solving problems with complex numbers
- Square roots of a complex number
- Solving quadratic equations with complex roots
- Solving cubic equations
- Solving quartic equations
- Exam Questions - Complex numbers
- Geometrical effects of conjugating a complex number
- Exam Questions - Finding roots
- Exponential Form (Euler's relation)
- Multiplication and division rules for mod and argument of two complex numbers
- Exam Questions - Further complex numbers
- Complex numbers specimen questions (mixed exam boards)