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GCSE Maths revision · Higher: number and algebra

Rates of change from a curve

Higher only

The idea

The average rate of change between two points is the gradient of the chord joining them. The rate of change at one point is the gradient of the tangent there.

  1. Average rate: draw the chord and find its gradient.
  2. Rate at a point: draw the tangent to the curve at that point.
  3. Find the tangent's gradient using two points far apart on it.

Watch out At a single instant you need a tangent, not a chord.

See it worked

Worked exampleCalculator allowed2 marks

The graph shows the volume of water, VV litres, in a tank tt seconds after a tap is opened. A tangent has been drawn to the curve at t=6t = 6. It passes through (2,7)(2, 7) and (10,31)(10, 31).

Use the tangent to estimate the rate at which the volume is increasing at t=6t = 6.

  1.  31−710−2\frac{31-7}{10-2}
  2. The instantaneous rate of change is estimated by the gradient of the tangent.
  3. Use its two stated points: (31 −- 7)/(10 −- 2) = 24/8 = 3 litres per second.

Answer 33 litres/second

Your turn

Question 1Calculator allowed4 marks

The temperature, T ∘CT\ ^{\circ}\text{C}, of a cup of tea tt minutes after it is made is given by T=80×0.9t+20T = 80 \times 0.9^t + 20.

(a) Work out the average rate at which the tea cools during the first 5 minutes, in ∘C^{\circ}\text{C} per minute. Give your answer to 2 decimal places.

(b) Is the tea cooling faster at t = 0 or at t = 5? Give a reason.

(b)
Question 2Non-calculator3 marks

Volume V litres is modelled by V = 4t24t^{2} + 3, where t is time in seconds. The tangent at t = 3.75 has gradient 30 litres per second. By what percentage is this instantaneous rate greater than the average rate from t = 1 to t = 5? You must show your working.

Question 3Non-calculator5 marks

A model has V=3t2+7V=3t^2+7, with volume V in litres and time t in seconds. At t = 4 a tangent has gradient 24 litres per second.

(a) Find the average rate of increase from t = 1 to t = 5.

(b) By what percentage is the tangent rate greater than this average?

In past papers

29 official questions in 29 papers since 2017. The latest:

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