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.
Chord vs Tangent — Average vs Instantaneous Rate
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.
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.