Instantaneous rate of change

Objective. Interpret tangent gradients in physical contexts.

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Learn: the key idea

Instantaneous rate of change is the gradient of a curve at a single exact point, found using the tangent line drawn at that point rather than between two points. In physical contexts this represents, for example, the exact speed at one moment on a distance–time graph, read as the gradient of the tangent touching the curve there. If a tangent at t = 3 passes through (1, 10) and (5, 30), its gradient 20/4 = 5 gives the instantaneous rate at t = 3.

Worked example

A tangent to a distance–time curve at t = 4 seconds passes through the points (2, 6) and (6, 22) on the tangent line. Find the instantaneous rate of change at t = 4.

1 $Instantaneous rate equals the gradient of the tangent line$
2 $Gradient = (change in distance) \div (change in time)$
3 $Gradient = (22 - 6) \div (6 - 2) = 16 \div 4$
4 $Simplify: gradient = 4$

Answer: 4 metres per second

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Common mistake. A common confusion is calculating the gradient between two points on the curve itself and calling it instantaneous, when that actually gives an average rate. Always draw or identify the tangent line touching the curve at just the one point in question.

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