## Unit 1

In this area of study, students cover constant and average rates of change and an introduction to the instantaneous rate of change of a function in familiar contexts, including graphical and numerical approaches to estimating and approximating these rates of change.

This area of study includes:

- average and instantaneous rates of change in a variety of practical contexts and informal treatment of instantaneous rate of change as a limiting case of the average rate of change
- interpretation of graphs of empirical data with respect to the rate of change such as temperature or pollution levels over time, motion graphs and the height of water in containers of different shapes that are being filled at a constant rate, with informal consideration of continuity and smoothness
- use of the gradient of a tangent at a point on the graph of a function to describe and measure the instantaneous rate of change of the function, including consideration of where the rate of change is positive, negative, or zero, and the relationship of the gradient function to features of the graph of the original function.

## Unit 2

In this area of study, students cover first principles approach to differentiation, differentiation and anti-differentiation of polynomial functions and power functions by rule, and related applications including the analysis of graphs.

This area of study includes:

- graphical and numerical approaches to approximating the value of the gradient function for simple polynomial functions and power functions at points in the domain of the function
- the derivative as the gradient of the graph of a function at a point and its representation by a gradient function
- notations for the derivative of a function: \( f^{\prime}, \displaystyle \frac{dy}{dx}, \frac{d}{dx} f(x), D_x(f) \)
- first principles approach to differentiation of \( f(x) = x^n ,n \in Z \), and simple polynomial functions
- derivatives of simple power functions and polynomial functions by rule
- applications of differentiation, including finding instantaneous rates of change, stationary values of functions, local maxima or minima, points of inflection, analysing graphs of functions, solving maximum and minimum problems and solving simple problems involving straight-line motion
- notations for an anti-derivative, primitive or indefinite integral of a function: \( F(x), \displaystyle \int f(x)dx \)
- use of a boundary condition to determine a specific anti-derivative of a given function
- anti-differentiation as the inverse process of differentiation and identification of families of curves with the same gradient function, including the application of anti-differentiation to solving simple problems involving straight-line motion.

*source – VCE Mathematics Study Design*

Lessons

#### Constant Rates

- Video - Definition of Rates of Change (4:55)
- Topic - Choosing Constant Rates
- Topic - Graphs of Constant Rates
- Topic - Constant Rates of Change: Positive, Negative or Zero
- Topic - Constant Rates of Change: Behaviours
- Topic - Calculating Constant Rates of Change
- Topic - Constant Rates of Change: Rules
- Topic - Constant Rates of Change: Empty Swimming Pool

#### Limits at Constants

- Video - Understanding Limits at Constant Values (6:58)
- Topic - Limits at Constant Values
- Topic - Limits at Constant Values involving Common Factors
- Topic - Limits at Constant Values involving Differences of Squares
- Topic - Limits at Constant Values involving Factorise
- Topic - Limits at Constant Values involving Quadratic Factorise

#### Limits at Infinity

- Video - Understanding Limits at Infinity (6:20)
- Video - Calculating Limits at Infinity (7:51)
- Topic - Limits at Infinity
- Topic - Limits at Infinity involving Linear Fractions
- Topic - Limits at Infinity involving Quadratic Fractions
- Topic - Limits at Infinity involving Fractions Higher Degree Numerators

#### First Principles

- Video - Definition of First Principles (4:20)
- Video - Differentiation of x^2 from First Principles (2:12)
- Video - Differentiation of 3x^2 from First Principles (2:48)
- Video - Differentiation of Quadratic Polynomials from First Principles (4:27)
- Video - Differentiation of (x+1)^2 from First Principles (3:55)
- Video - Differentiation of x^3 from First Principles (2:05)
- Video - Differentiation of Square Root of x from First Principles (3:04)
- Topic - First Principles: Linear Expressions
- Topic - First Principles: Quadratic Expressions
- Topic - First Principles: Cubic Expressions
- Topic - First Principles: Surd Expressions
- Topic - First Principles: Rational Expressions

#### First Principles at a Given x-Value

- Topic - First Principles at Given x Values: Linear Expressions
- Topic - First Principles at a Given x Value: Quadratic Expressions
- Topic - First Principles at a Given x Value: Cubic Expressions
- Topic - First Principles at Given x Values: Surd Expressions
- Topic - First Principles at Given x Values: Rational Expressions

#### Basic Differentiation Rules

- Video - Differentiation of Constant Terms (2:14)
- Video - Differentiation of Linear Functions (3:16)
- Video - Methods of Differentiation (5:01)
- Video - Differentiation of Negative Exponents (5:12)
- Video - Differentiation of Non-Monic Negative Exponents (8:13)
- Video - Simplifying Expressions before Differentiation (6:37)
- Video - Differentiation of Polynomial Functions (3:48)
- Video - Differentiation of Functions in Terms of Specific Letters (2:43)
- Video - Differentiation of Functions in Terms of Specific Letters involving Additions (4:37)
- Video - Differentiation of Functions in Terms of Specific Letters involving Multiplications (6:20)
- Video - Equations involving Differentiation (1:51)

#### Higher Derivatives

- Video - Fundamental of Higher Derivatives (5:34)
- Video - Higher Derivatives involving Chain Rule (4:59)
- Video - Higher Derivatives involving Surds (5:27)
- Video - Higher Derivatives involving Quotient Rule (2:40)
- Video - Higher Derivatives involving Product Rule (6:16)
- Topic - Higher Derivatives of Quadratic Polynomials
- Topic - Higher Derivatives of Cubic Polynomials
- Topic - Higher Derivatives using Chain Rule

#### Turning Points

- Video - Coordinates of Turning Points (7:30)
- Video - Identifying Nature of Turning Points from Derivative Curves (5:25)
- Video - Identifying Nature of Turning Points using Tables (5:08)
- Video - Identifying Stationary Points (8:17)
- Video - Understanding Nature of Stationary Points (5:31)
- Video - No Stationary Points (1:50)
- Video - Condition of Being Stationary Points (2:51)
- Video - Maximum and Minimum Turning Points (6:49)
- Video - Stationary Points by their Shapes (7:34)
- Video - Maximum and Minimum Turning Points of Quartic Functions (7:54)

#### Curves of Rational Functions in Quadratics by Quadratic Form

- Video - Understanding Curve Sketching involving Symmetry (1:24)
- Video - Even Functions Odd Functions (1:49)
- Video - Differentiation using Quotient Rule (2:29)
- Video - Turning Points of Rational Functions (1:17)
- Video - Nature of Turning Points of Rational Functions (3:03)
- Video - x-Intercepts of Rational Functions (0:58)
- Video - Vertical Asymptotes in Rational Functions (1:53)
- Video - Horizontal Asymptotes in Rational Functions (4:52)
- Video - Curve Sketching of Rational Functions involving Symmetry (1:53)

#### Gradient Functions

- Video - Principle of Graphing Gradient Functions (5:08)
- Video - Gradient Function Graphs of Straight Lines (5:23)
- Video - Gradient Function Graphs of Parabola (8:13)
- Video - Gradient Function Graphs of Cubic Functions (10:46)
- Video - Gradient Function Graphs of Hyperbola (13:22)
- Video - Gradient Function Graphs of Surds (12:19)
- Video - Gradient Function Graphs of Periodic Functions (8:15)
- Video - Gradient Function Graphs of Quartic Functions (8:25)
- Video - Gradient Function Graphs of Semicircle (2:24)
- Video - Gradient Function Graphs of Exponential Functions (2:11)
- Video - Gradient Function Graphs of Logarithmic Functions (1:47)

#### Rates of Change

- Video - Basic Rates of Change in Polynomial Functions (4:08)
- Video - Rates of Change of Population (9:29)
- Video - Rates of Change of Volumes in terms of Quantity (6:32)
- Video - Rates of Change of Volumes in terms of Ratio (4:47)
- Video - Related Rate of Change involving Surface Area and Volume (6:29)
- Video - Related of Rates of Change involving Radius and Volume (3:22)
- Video - Related of Rates of Change involving Pythagoras Theorem (4:14)
- Video - Eliminating Height Pronumerals in Rates of Change (5:14)
- Video - Eliminating Radius Pronumerals in Rates of Change (6:15)
- Video - Rate of Change Involving Angles (7:01)

#### Optimisation

- Video - Maxima and Minima of Quadratic Functions (4:16)
- Video - Maxima and Minima of Cubic Functions 1 (8:47)
- Video - Maxima and Minima of Cubic Functions 2 (4:02)
- Video - Maxima and Minima in 2D Applications (4:23)
- Video - Maxima and Minima in 3D Applications 1 (6:09)
- Video - Maxima and Minima in 3D Applications 2 (3:53)
- Video - Absolute Maximum and Minimum (8:59)
- Video - Minimum Speed of the Motion (1:43)
- Video - Maximum Velocity Given Displacement (1:35)

#### Differentiation and Displacement, Velocity and Acceleration

- Video - Velocity and Speed (3:04)
- Video - Kinematics given Velocity (9:49)
- Video - Kinematics given Displacement (10:51)
- Topic - Working with Displacement
- Topic - Moving Forward or Backward
- Topic - Obtaining Velocity from Displacement
- Topic - Obtaining Acceleration from Velocity
- Topic - Obtaining Acceleration from Displacement
- Topic - Obtaining Displacement from Velocity
- Topic - Obtaining Displacement from Acceleration
- Topic - Understanding Velocity and Speed
- Topic - Total Distance Travelled of Linear Displacement
- Topic - Total Distance Travelled of Quadratic Displacement
- Topic - Total Distance Travelled of Cubic Displacement

#### Integration of Power Functions

- Video - Principle of Integration of Power Function (2:49)
- Video - Integration of Power Functions (1:56)
- Video - Integration of Power Functions involving Fractions (2:23)
- Video - Integration of Power Functions Surds (1:06)
- Video - Integration of Power Functions (5:32)
- Video - Integration of Reciprocal Functions (1:35)
- Video - Integration of Irrational Functions (4:32)
- Video - Integration of Reciprocal Irrational Functions (4:41)

#### Particular Values

- Video - Finding Integral Constants (7:27)
- Video - Finding Integral Constants involving Point of Inflections (4:45)
- Video - Finding Integral Constants involving Second Derivatives (2:33)
- Topic - Finding Integral Constants from Linear Expressions
- Topic - Integral Constants by Non-Monic Expressions from Linearity
- Topic - Finding Integral Constants from Quadratics
- Topic - Integral Constants using Non-Monic Expressions from Quadratics
- Topic - Finding Two Integral Constants from Linear Expressions