Instantaneous rate of change
OCR J560 Higher (J560/04–06) expects students to find the gradient of a curve at a specific point and interpret it as a real-world rate. This is the precursor to A-level differentiation.
The tangent method
The gradient at a point P on a curve is defined as the gradient of the tangent to the curve at P.
To find it from a graph:
- Mark the point P on the curve.
- Draw a straight line just touching the curve at P (and not crossing it locally) — this is the tangent.
- Pick two clear points on the tangent (not on the curve!).
- Compute (Δy)/(Δx) between those two points.
The longer the tangent line you draw, the more accurate the gradient reading.
Why it matters
On a distance–time curve, the gradient at a point gives the instantaneous speed at that moment. On a speed–time curve, the gradient gives the acceleration at that instant.
✦Worked example
A distance–time curve passes through (3, 18) where the tangent at this point has been drawn. The tangent passes through (1, 8) and (5, 28).
Gradient = (28 − 8) / (5 − 1) = 20/4 = 5 m/s.
So at t = 3 s the object is travelling at 5 m/s.
Estimation note
Reading a gradient from a hand-drawn tangent has uncertainty. OCR mark schemes typically allow a ±10% tolerance and award A1 for any answer within range, with M1 for a sensible tangent and reading.
Average vs instantaneous rate
- Average rate of change between t = a and t = b: chord gradient = (f(b) − f(a)) / (b − a).
- Instantaneous rate of change at t = a: tangent gradient at t = a.
These differ unless the curve is straight.
OCR mark scheme conventions
- M1 for drawing a sensible tangent at the correct point.
- M1 for picking two readable points on the tangent and computing (Δy)/(Δx).
- A1 for the numerical gradient (within tolerance).
- B1 for stating the rate with units and meaning ("acceleration is 3 m/s² at t = 4 s").
⚠Common mistakes
- Picking points on the curve instead of on the tangent.
- Drawing too short a tangent — small Δx amplifies error.
- Ignoring units when stating the rate.
- Confusing average with instantaneous gradient.
AI-generated · claude-opus-4-7 · v3-ocr-maths-leaves