Estimate the gradient of a curve with a tangent

By UK GCSE Mathematics · independent study · myAudience

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Estimate the gradient of a curve with a tangent

A curve does not generally have one gradient for its whole length. To estimate its gradient at a point, draw a tangent that follows the local direction there. Calculate rise over run using two well-separated points on that tangent, not two arbitrary points on the curve.

A secant joins two points on a curve and gives an average rate over their interval. A tangent gives a local graphical estimate. Confusing these can give the wrong interpretation when the rate changes.

Units still come from vertical change divided by horizontal change. On a distance-time graph, a tangent gradient estimates instantaneous speed. Accuracy depends on drawing and scale; this graphical method does not require differentiating a formula.

Worked example: A drawn tangent on a distance-time graph passes through (2 s, 5 m) and (6 s, 17 m). Estimate speed.

Calculate the distance change: 12 m.

Calculate the time change: 4 s.

Divide change in distance by change in time.

Answer: Approximately 3 m/s.

Check your understanding: Must both points used for the calculation lie on the original curve?

Answer: No; they lie on the drawn tangent.

Why: The tangent line supplies the local gradient estimate.