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Ratio, variation and rates
Lesson 34 of 75
Challenge

Average and estimated rates

Objective. Compare chord, interval and tangent rates.

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Learn: the key idea

Average and estimated rates compare the chord gradient between two points on a curve, the gradient over a smaller interval, and the tangent gradient at a single point, showing how estimates improve as the interval narrows. The chord gradient between (a, f(a)) and (b, f(b)) equals (f(b) − f(a))/(b − a), giving an average rate over that whole interval, whereas narrowing the interval around a point gives estimates approaching the true instantaneous rate. For a curve with f(1) = 3 and f(4) = 15, the chord gradient (15−3)/(4−1) = 4 estimates the rate near that region.

Worked example

A curve passes through (2, 5) and (6, 29). Find the average rate of change between these points, then explain why it may differ from the instantaneous rate at x = 2.

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Bar chart of the values 2, 5, 6, 29, 2
  1. 1Average rate = (change in y) ÷ (change in x)
  2. 2Average rate = (29 − 5) ÷ (6 − 2) = 24 ÷ 4
  3. 3Simplify: average rate = 6
  4. 4This may differ from the instantaneous rate at x = 2 because the curve's gradient can change across the interval

Answer: 6 (the chord gradient only estimates the rate; curvature makes it differ from the tangent gradient at x = 2)

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Common mistake. Students sometimes treat a chord gradient over a wide interval as an accurate instantaneous rate at one endpoint, which can be very different from the true tangent gradient there. Always state that a chord only gives an average, and use a narrower interval or tangent for a more accurate instantaneous estimate.

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