Instantaneous rate of change: gradient at a point
For a straight line, the gradient is constant. For a curve, the gradient varies — at each point it tells you the instantaneous rate of change at that moment.
What "instantaneous" means
On a distance–time graph that's a curve, the average speed over an interval is (Δd)/(Δt) — but the speed at a single instant requires the gradient at the point, not over an interval.
Drawing a tangent
The gradient at a point on a curve = gradient of the tangent line at that point. To estimate it manually:
- Draw a tangent line that touches the curve at the point and just kisses it (doesn't cross).
- Pick two points on the tangent (not the curve) far apart for accuracy.
- Compute (Δy)/(Δx).
Worked example: at point P on a distance–time curve, you draw a tangent passing through (3, 5) and (7, 21). Speed at P = (21 − 5)/(7 − 3) = 16/4 = 4 m/s (or whatever the units are).
Sketch tips
- Use a ruler to draw the tangent.
- Pick points where coordinates are easy to read.
- Check the tangent doesn't cross the curve nearby.
Average vs instantaneous rate
| Method | |
|---|---|
| Average rate | Chord between two points: (y₂ − y₁)/(x₂ − x₁) |
| Instantaneous rate | Tangent at a single point — gradient of the tangent |
Worked example: a velocity–time curve at t = 5 s has tangent through (4, 8) and (8, 20). Acceleration at t = 5 s = (20 − 8)/(8 − 4) = 3 m/s².
⚠Common mistakes
- Treating the curve like a chord — using (y₂ − y₁)/(x₂ − x₁) on the curve gives the average rate, not instantaneous.
- Drawing a chord, not a tangent — tangent touches, chord cuts.
- Reading tangent points from the curve instead of from the tangent line.
- Choosing tangent points too close — small denominator inflates errors.
- Forgetting units — m/s for speed; m/s² for acceleration; £ per unit for unit cost.
➜Try this— Quick check
A tangent at point Q is drawn passing through (1, 10) and (5, 2). Gradient at Q = (2 − 10)/(5 − 1) = −2.
AI-generated · claude-opus-4-7 · v3-deep-ratio