Instantaneous rate of change

For a curved real-context graph, the gradient changes from point to point. The gradient at a single point is the INSTANTANEOUS rate of change at that moment.

The tangent method

To estimate the gradient at a point P on a curve:

1. Identify P on the curve carefully.
2. Draw a TANGENT — a straight line that just touches the curve at P with the same slope.
3. Choose two clear points on the tangent (NOT on the curve), at least one division apart.
4. Compute m = (y₂ − y₁) / (x₂ − x₁).

Where the inaccuracy lives

Tangent-drawing introduces error. WJEC mark schemes typically allow ± 0.5 from the published value, depending on the question's precision. Drawing a careful tangent through P is worth M1; the gradient calculation earns A1.

Comparing instantaneous and average

• Average rate over [a, b]: gradient of the CHORD from (a, f(a)) to (b, f(b)).
• Instantaneous rate at a: gradient of the TANGENT at (a, f(a)).

These are usually different unless the function is linear.

Real-context interpretations

• Distance–time curve: tangent gradient = instantaneous SPEED.
• Speed–time curve: tangent gradient = instantaneous ACCELERATION.
• Cooling curve (temperature vs time): tangent gradient = instantaneous rate of cooling.
• Population–time curve: tangent gradient = instantaneous growth rate.

Sign conventions

A negative tangent gradient means the quantity is DECREASING at that moment. State this in any interpretation.

Common WJEC question

"By drawing a tangent, estimate the speed at t = 5 seconds. State your answer to an appropriate degree of accuracy."

Method M1 — visible tangent through (5, f(5)). Calculation M1 — two readable points and a quotient. Answer A1 — sensible value with units. Accuracy B1 — 2–3 s.f. with units.

WJEC exam tip

DRAW the tangent on the printed graph in pencil so examiners can see it. Verbal claims of "the gradient is 4" without a visible tangent line score zero — the tangent is the M1.