VCE Mathematical Methods Units 1 and 2 Calculus
Course

VCE Mathematical Methods Units 1 and 2 – Calculus

3.1 Rates of Change
3.2 Differentiation
3.3 Application of Differentiation
3.4 Integration

34 Lessons

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

RATES OF CHANGE

Constant Rates

Average Rates of Change

Gradient using Rates of Change

Limits at Constants

Limits at Infinity

DIFFERENTIATION

First Principles

First Principles at a Given x-Value

Basic Differentiation Rules

Higher Derivatives

Increasing and Decreasing Functions

Turning Points

Points of Inflection and Concavity

Curves of Rational Functions in Quadratics by Linear Form

Curves of Rational Functions in Quadratics by Quadratic Form

Gradient Functions

Sketching Curves from Derivatives

APPLICATION OF DIFFERENTIATION

Rates of Change

Related Rates

Optimisation

Maxima and Minima with Trigonometry

Sketching Graphs Containing Stationary Points

Kinematics using Differentiation

Velocity and Acceleration

Differentiation and Displacement, Velocity and Acceleration

INTEGRATION

Basic Integration Rules

Basic Integration Rules involving Algebra

Integration of Power Functions

Particular Values

Area Under a Curve

Graphs of Antiderivatives of Functions