Average and instantaneous rates of change

3 exam-style questions. Answers and the mark for each step are on the last page.

Question 1Calculator · 2 marks
(a) A curved graph shows the volume of water V litres in a tank against time t seconds. At t = 6, a drawn tangent passes through (2, 7) and (10, 31). Use this tangent to estimate the rate at which the volume is increasing at t = 6. (2)
Question 2Non-calculator · 3 marks
(a) Volume V litres is modelled by V = $4t^{2}$ + 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. (3)
Question 3Non-calculator · 5 marks
A ball is thrown upwards. Its height, $h$ metres, after $t$ seconds is $h = 20t - 5t^2$ .
(a) Work out the average rate of change of height between $t = 1$ and $t = 3$ . (2)
(b) Interpret your answer to part (a). (1)
(c) Use a chord from $t = 2$ to $t = 2.1$ to estimate the rate of change of height at $t = 2$ . (2)

Answers and marks

(a) $3$ litres/second

(a) $25$ %

(a) $0$ m/s

(b) The ball is at the same height at t = 1 and t = 3: it rises and then falls back.

C1 Saying the ball is at the same height at both times (up then down), so the average rate is zero although it moves.

(c) $-0.5$ m/s

Average and instantaneous rates of change

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