GCSE · Graphs of functions

Rates of change from graphs (tangents, chords and the area under a graph)Exam-style questions

Read a rate off a curved graph: the gradient of a chord is the average rate of change over an interval, the gradient of a tangent is the rate at one instant, and the units on the axes say what that rate is (a speed, an acceleration, litres per minute). Then the other half: the area under a speed–time graph is the distance travelled, found exactly from straight pieces or estimated with strips, with a reason why the estimate is too big or too small.

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20 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The velocity–time graph shows the first $40$ seconds of a train's journey.(a) Work out the acceleration of the train in the first $10$ seconds. [2]

(b) Work out the total distance travelled in the $40$ seconds. [3]
Level 3 · 5 marksTry it
2
Maya is running a race. As she passes a marker post, she starts to sprint for the finish.

The graph shows the distance, $d$ metres, that she runs in the $6$ seconds after she passes the post.(a) Work out Maya's average speed for the first $4$ seconds after she passes the post.

(2 marks)
(b) Work out an estimate of Maya's speed $4$ seconds after she passes the post.

You must show how you get your answer.

(3 marks)
(c) Explain why your answers to parts (a) and (b) are different.

(1 mark)
Level 3 · 6 marksTry it
3
A sledge slides down a hill. The hill gets steeper as the sledge goes down it.

The graph shows the speed, $v$ m/s, of the sledge $t$ seconds after it passes a tree.(a) Use $3$ strips of equal width to estimate the distance the sledge travels between $t = 0$ and $t = 6$

(3 marks)
(b) Is your answer to part (a) an overestimate or an underestimate?

Give a reason for your answer.

(1 mark)
Level 3 · 4 marksTry it
4
Water flows into an empty tank at a constant rate of $3000$ cm³ per minute.

The tank is made from two cuboids, one on top of the other.

The graph shows the depth of the water in the tank until the tank is full.(a) Work out the rate at which the depth of the water increases during the first $10$ minutes.

(2 marks)
(b) Work out the area of the base of the bottom cuboid.

(2 marks)
(c) Without doing any more calculation, explain how the graph shows that the top cuboid has a smaller base than the bottom cuboid.

(1 mark)
Level 3 · 5 marksTry it
5
Rain falls for $6$ hours during a storm.

The graph shows how the rate of rainfall changed during the storm.(a) Show that $32$ mm of rain falls during the storm.

(3 marks)
(b) Work out the mean rate of rainfall for the whole storm.

Give your answer in millimetres per hour, correct to $1$ decimal place.

(2 marks)
Level 3 · 5 marksTry it

2 more questions on rates of change from graphs (tangents, chords and the area under a graph)

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