# VCE Mathematical Methods Units 1 and 2 – Calculus

3.1 Rates of Change

3.2 Differentiation

3.3 Application of Differentiation

3.4 Integration

## 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*

VCE Mathematical Methods Units 1 and 2 Courses

VCE Mathematical Methods Units 1 and 2 Syllabus

## Course Content

### Rates of Change

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