Instantaneous rate of change
This is exclusively a Higher topic on Edexcel 1MA1, examined on Papers 2H and 3H. It connects the geometry of a tangent to physical rates such as speed, acceleration and rate of fluid flow.
Why a tangent gives an instantaneous rate
The gradient of a chord between two points on a curve is the average rate of change between them. As the two points move closer together, the chord approaches the tangent at the point, and its gradient approaches the instantaneous rate at that point.
Drawing the tangent (the careful part)
To find the instantaneous rate at a point P on a curve:
- Draw a straight line that touches the curve at P only — it should not cut through.
- Extend the tangent so it covers a wide horizontal range on the graph (this reduces reading error).
- Pick two clear points on the tangent (NOT on the curve) and read off their coordinates.
- Compute m = (y₂ − y₁) / (x₂ − x₁).
- State the answer with units in context.
Contextual interpretations
- Distance-time curve: gradient at a point = instantaneous speed.
- Velocity-time curve: gradient at a point = instantaneous acceleration.
- Cooling curve (temperature vs time): gradient = rate of cooling at that time.
- Population vs time: gradient = rate of growth.
Common Edexcel mark-scheme phrasing
- M1 for drawing a tangent at the correct point.
- M1 for picking two valid coordinates on the tangent.
- A1 for the gradient (often with a tolerance band, e.g. ±0.2).
- B1 for a contextual interpretation with units.
⚠Common mistakes— Common errors
- Reading two points on the curve, not the tangent. The result is not the instantaneous rate.
- Drawing a tangent that crosses the curve twice (it must touch once locally).
- Forgetting units when interpreting the gradient.
- Using the chord between widely separated points and calling it the instantaneous rate.
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