# Accreditation of VCE Specialist Mathematics Units 3 and 4

VCE Specialist Mathematics Units 3 and 4 has been designed and published for complying Victorian Certificate of Education 1 January 2016 – 31 December 2018 and implementation of this course commences in January 2018.

VCE Specialist Mathematics Units 3 and 4 are structured to be taken in conjunction with VCE Mathematical Methods Units 3 and 4, or following previous completion of VCE Mathematical Methods Units 3 and 4. The areas of study extend content from VCE Mathematical Methods Units 3 and 4 to include rational and other quotient functions as well as other advanced mathematics topics including complex numbers, vectors, differential equations, mechanics and statistical inference. Study of VCE Specialist Mathematics Units 3 and 4 assumes concurrent study or previous completion of VCE Mathematical Methods Units 3 and 4.

### Area of Study 1

**Functions and graphs**

In this area of study students cover inverse circular functions, reciprocal functions, rational functions and other simple quotient functions, the absolute value function, graphical representation of these functions, and the analysis of key features of their graphs including intercepts, asymptotic behaviour and the nature and location of stationary points, points of inflection, periodicity, and symmetry.

### Area of Study 2

**Algebra**

In this area of study students cover the expression of simple rational functions as a sum of partial fractions; the arithmetic and algebra of complex numbers, including polar form; points and curves in the complex plane; introduction to factorisation of polynomial functions over the complex field; and an informal treatment of the fundamental theorem of algebra.

### Area of Study 3

**Calculus**

In this area of study students cover advanced calculus techniques for analytic and numeric differentiation and integration of a range of functions, and combinations of functions; and their application in a variety of theoretical and practical situations, including curve sketching, evaluation of arc length, area and volume, differential equations and kinematics.

### Area of Study 4

**Vectors**

In this area of study students cover the arithmetic and algebra of vectors, linear dependence and independence of a set of vectors, proof of geometric results using vectors, vector representation of curves in the plane and vector kinematics in one and two dimensions.

### Area of Study 5

**Mechanics**

In this area of study students cover an introduction to Newtonian mechanics, for both constant and variable acceleration.

### Area of Study 6

**Probability and statistics**

In this area of study students cover statistical inference related to the definition and distribution of sample means, simulations and confidence interval.

#### Course Materials

#### Inverse Circular Functions

- Inverse Sine Values
- Inverse Sine Function
- Inverse Cosine Function
- Inverse Cosine Values
- Inverse Cosine Values with a Negative Sign
- Inverse Tangent Values
- Inverse Tangent Function
- General Solutions of Sine Function
- General Solutions of Cosine Function
- General Solutions of Tangent Function
- Substitutions
- Trigonometric Properties
- Symmetries
- Equations
- Inverse Sine Graphs
- Inverse Cosine Graphs
- Inverse Tangent Graphs
- Inverse Function as an Input
- Regular Function as an Input of Inverse Functions

#### Differential Calculus

- Differentiation of Logarithmic Functions
- Definition of Exponential Function
- Differentiation of Sine Function : Chain Rule
- Differentiation of Sine Function : Product Rule
- Differentiation of Sine Function : Quotient Rule
- Differentiation of Cosine Function : Chain Rule
- Differentiation of Cosine Function : Product Rule
- Differentiation of Cosine Function : Quotient Rule
- Differentiation of Tangent Function
- Differentiation of Inverse Sine Functions
- Differentiation of Inverse Cosine Functions
- Differentiation of Inverse Tangent Functions

#### Kinematics

- Velocity and Speed
- at Rest
- Maximum and Minimum Speed
- Distance Travelled
- Velocity and Acceleration
- Working with Velocity
- Maxima and Minima
- Maximum Velocity and Speed
- Changes in Displacement and Velocity
- Displacement-time Graph
- Sketching Graphs from Displacement to Velocity
- Velocity-time Graph
- Acceleration-time Graph

#### Mechanics

- Basic Rates of Change
- Acceleration Due to Gravity
- Forces
- Resultant Forces
- Time and Velocity
- Upwards Resisted Motion
- Downwards Resisted Motion
- Terminal Velocity
- Angular and Tangential Velocity
- Tension
- Conical Pendulum
- Conical Pendulum with Multiple Forces
- Friction Force and Banking Angle
- Direction of Friction Forces
- Resolving Forces Parallel to the Slope