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Chord vs Tangent — Average vs Instantaneous Rate

Part of Gradients of Curves · GCSE GCSE Mathematics revision

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.

Topic position

Section 5 of 10

Practice

9 questions

Recall

10 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.

Keep building this topic

Read this section alongside the surrounding pages in Gradients of Curves. That gives you the full topic sequence instead of a single isolated revision point.

Interpreting the Gradient in Context Improving the Accuracy of Tangent Estimates (Higher)How Fast is Fast Right Now?Gradients of Curves — Essential Concepts

Practice Questions for Gradients of Curves

How do you find the gradient of a curve at a specific point?

A. Draw a chord joining two points on the curve and find its gradient
B. Draw a tangent to the curve at that point and find the gradient of the tangent
C. Find the average of the y-values on either side of the point
D. Divide the y-coordinate by the x-coordinate of the point

1 markfoundation

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.

2 markshigher

Quick Recall Flashcards

What is a tangent to a curve?

A straight line that touches the curve at exactly one point and has the same gradient as the curve at that point. It does NOT cross through the curve at that point — it only touches it. The gradient of the tangent = the gradient of the curve at that point. Used to estimate the rate of change at an instant.

Steps to estimate the gradient of a curve at a point

1. Mark the point on the curve 2. Place a ruler so it just TOUCHES the curve at that point (tangent) 3. Make the tangent line extend well across the graph 4. Choose two clear points on the tangent line 5. Calculate: gradient = (y2 - y1)/(x2 - x1) Tip: use points far apart on the tangent for greater accuracy.

9 questions on Gradients of Curves — practise free

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Interpreting the Gradient in Context

Improving the Accuracy of Tangent Estimates (Higher)