This is Chapter 3 – Applications of Differential Calculus for the Year 12 Mathematical Methods course.

This chapter takes the differentiation skills developed earlier in the course and applies them to analysing functions, modelling change and solving practical problems. Rather than simply being asked to find a derivative, students must now decide what that derivative reveals and how it can be used within a broader mathematical context.

We begin with equations of tangents, using derivatives to determine the gradient of a curve at a chosen point. This provides a clear geometric interpretation of instantaneous rate of change and reinforces the connection between calculus and coordinate geometry.

Stationary points are then investigated in detail. Students use first derivatives to identify where a function is increasing or decreasing and to locate possible turning points. Second derivatives help classify these points and analyse concavity, while points of inflection require careful consideration of how the curve changes shape.

Kinematics provides one of the most important applications of calculus in the course. Students connect displacement, velocity and acceleration through differentiation and learn to interpret both algebraic and graphical information about motion. Rates-of-change questions extend this thinking into a wider range of scientific and practical contexts.

Optimisation brings several skills together. Students must define suitable variables, construct a mathematical model, determine any restrictions and use calculus to find a maximum or minimum value. These questions are often less direct than routine differentiation and reward careful planning, clear interpretation and disciplined algebra.

Applications of differential calculus appear throughout physics, engineering, economics, biology and business, wherever changing quantities need to be analysed or optimised.

Subchapters:

• 3A: Equations of tangents

• 3B: Stationary points
• 3C: Inflections and curve shape
• 3D: Kinematics
• 3E: Rates of change
• 3F: Optimisation

By the end of this chapter, students will be able to use derivatives to analyse curves, model motion and rates of change, and solve a wide range of optimisation problems.