A chord joins two points on a curve. Its gradient gives the average rate of change between those two points.
A tangent touches the curve at one point. Its gradient gives the instantaneous rate of change at that exact point.
As the two points used to form a chord get closer and closer together, the chord approaches the tangent. This is why drawing a tangent at a point gives a better and better estimate as the tangent is drawn more carefully — the tangent represents the "limit" of average rates over smaller and smaller intervals.
Example: On a curve, the points (1, 5) and (4, 14) give average rate of change = (14 − 5)/(4 − 1) = 9/3 = 3.
The tangent at x = 2.5 (the midpoint) might give a similar instantaneous value, but these are different quantities.
This deep dive covers Chord vs Tangent — Average vs Instantaneous Rate within Gradients of Curves for GCSE Mathematics. Revise Gradients of Curves in Graphs for GCSE Mathematics with 9 exam-style questions and 10 flashcards. This topic appears less often, but it can still be a useful differentiator on mixed-topic papers. It is section 5 of 10 in this topic. Use this deep dive to connect the idea to the wider topic before moving on to questions and flashcards.
Practice questions for Gradients of Curves
How do you find the gradient of a curve at a specific point?
Explain why the gradient of a chord between two points on a curve is only an estimate of the gradient at a point, and how this estimate can be improved.