# SACE Stage 1 Mathematics – Differential Calculus, Growth and Decay

3.1 Indices and Index laws
3.2 Exponential functions
3.3 Logarithmic Functions
3.4 Rate of Change
3.5 The Concept of a Derivative
3.6 Computations of Derivatives
3.7 Properties of Derivatives
3.8 Applications of Derivatives

## Growth and Decay

This topic covers studying exponential and logarithmic functions under the unifying idea of modelling growth and decay. Knowing indices enables students to consider exponential functions and appreciate how exponential functions can model actual situations involving growth and decay.

The mathematical models investigated arise from actual growth and decay situations such as human population growth, the growth of bacteria, radioactive decay, and the spread of diseases. By developing and applying these mathematical models, students see how the wider community might use them for analysis, prediction, and planning.

So that actual data can be handled efficiently, technology is used extensively in this topic for graphing and calculation. Much of the technology has the facility to fit curves to data automatically. This allows students to compare their models with a solution from another source.

## Introduction to Differential Calculus

The development of calculus enabled the study of the links between constantly changing variables. Using mathematical modelling from other topics in Stage 1 Mathematics can be extended significantly by exploring rates of change using differentiation.

Rates and average rates of change are introduced, followed by the key concept of the derivative as an ‘instantaneous rate of change’. These concepts are reinforced numerically by calculating difference quotients geometrically as slopes of chords and tangents and algebraically. Calculus is developed to study the derivatives of polynomial functions and other linear combinations of power functions, with simple applications of the derivative to curve sketching, calculating slopes and equations of tangents, determining instantaneous velocities, and solving optimisation problems. The range of functions that can be differentiated and the different uses of derivatives are expanded in Stage 2 Mathematical Methods and Stage 2 Specialist Mathematics.

Source – Subject Outline, South Australian Certificate of Education 2023

SACE Stage 1 Mathematics Courses

SACE Stage 1 Mathematics Syllabus

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