Introduction to calculus.
Build calculus from average change and limiting tangents, then use polynomial derivatives to analyse graphs, optimise models and describe motion.
Work through the topic
Learn the method, follow the examples, then try the worksheet.
- 01
Average rates and secant gradients
Measure change over an interval and interpret the gradient of the joining secant.
- 02
Instantaneous rates and limiting secants
Approach an instantaneous gradient using shorter secants and a limiting difference quotient.
- 03
Derivative notation and the derivative function
Distinguish a derivative function from the rate at a specified input.
- 04
Differentiating powers and polynomials
Differentiate powers and polynomials term by term, including coefficients and constants.
- 05
Tangent and normal lines
Construct tangent and normal equations, including horizontal and vertical cases.
- 06
Derivative sign and increasing/decreasing intervals
Use the sign of a derivative to describe increasing and decreasing intervals.
- 07
Stationary points
Locate stationary points and distinguish maxima, minima and stationary non-turning points.
- 08
First-derivative graph analysis
Combine intercepts, derivative signs and stationary points in a justified polynomial sketch.
- 09
Simple optimisation
Model a constrained practical quantity in one variable and justify its optimum.
- 10
Displacement, velocity and acceleration models
Use displacement derivatives to describe velocity, acceleration, speed and direction changes.
Topic review
Revisit the key ideas, then choose a practice paper.