IGCSE Maths / Calculus
Variable rate of change revision notes
Edexcel International GCSE Maths (4MA1), chapter 28: Calculus, specification point 3.4a. Free lesson.
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A curve’s gradient changes from point to point, so its rate of change must be measured at a named moment using a tangent. The sign of the gradient tells you whether the quantity is increasing or decreasing.
Why a curve has no single gradient
A straight line has the same steepness all the way along, so one triangle gives its gradient wherever you draw it. A curve can be steep in one place and nearly flat in another. Its rate of change therefore varies as you move along it. A question asking for the gradient usually refers to a straight line. For a curve, the question must name a point, such as at t = 1. A curve has a gradient at a point, not one gradient for the whole curve.
How to estimate the rate of change at t = 1
The ball’s height is modelled by h = 20t − 5t², where h is height in metres and t is time in seconds. To estimate how fast the height is changing at one moment, use the tangent: a straight line that touches the curve at that point and follows its direction there. The curve rises, reaches its highest point, then falls. The tangent at t = 1 slopes upwards, so the ball is still rising.
What a negative rate means at t = 3
At t = 3, the tangent slopes down as time moves forwards. Its gradient is negative because the ball’s height is decreasing. The minus sign is meaningful: it shows the direction of change, not an error. A positive rate means the height is increasing, so the ball is rising. A negative rate means the height is decreasing, so the ball is falling. The size of the rate tells you how fast the height is changing.
When the rate of change is zero
At the ball’s highest point, it has stopped rising but has not yet started falling. The tangent there is flat, so its gradient and the rate of change of height are zero. For this ball, that moment is t = 2. A flat tangent means the rate of change is 0 m/s at that instant.