Rates of change

The graph of a function is drawn on the grid for 0 ⩽ x ⩽ 4.

Work out the average rate of change of the function between x = 1 and x = 3.

Use the graph to estimate the gradient of the function at x = 2.5. You must show working to support your estimate.

The graph shows the speed of an object during the first 20 seconds of its motion.

Calculate the distance travelled by the object during the 20 seconds.

The graph shows the distance travelled by an object during the first 4 seconds of its motion.

Work out the average speed of this object between 2 and 4 seconds.

Use the graph to estimate the speed of this object at 3 seconds. You must show working to support your estimate.

What happens to the speed of this object during these 4 seconds of motion. Explain how you know.

I know this because........

A straight line, L, is shown below.

Write down the ratio v: h when the gradient of line L is 4.

Find the gradient of line L as a fraction in its simplest form when v: h = 14: 6.

A ball of dough is left to rise before it is baked. The graph shows the height of the ball of dough over the first 20 minutes.

Work out the gradient of the line as a decimal, giving the units of your answer. Show how you work out your answer.

A baker works out the height of the ball of dough at the end of 25 minutes as 14 cm.

Use your gradient to show that the baker could be correct.

What assumption has the baker made?

The graph shows the distance travelled by a particle over the first 20 seconds of its motion.

Show that the average speed of the particle over the first 20 seconds of its motion is 1.8 m/s.

Estimate the speed of the particle at 10 seconds. You must show working to support your estimate.

The graph shows the speed of a car during the first 40 seconds of a journey.

Write down the acceleration of the car between 10 seconds and 25 seconds.

Work out the average speed of the car, in m/s, during the 40 seconds. You must show your working.

An athlete goes for a training run. The graph shows their speed as they run.

Write down the athlete’s acceleration between 400 seconds and 1000 seconds.

Work out the athlete’s average speed, in m/s, during the 1200 seconds. You must show your working.

This graph shows part of a straight line.

Show that the gradient of the line is 2.5.

Write down the equation of the line.

The graph shows the speed of a vehicle during the first 40 seconds of motion.

Calculate the distance travelled by the vehicle during the 40 seconds.

The graph shows the distance travelled by a particle over 6 seconds.

Work out the average speed of the particle between 2 and 4 seconds.

Estimate the speed of the particle at 4 seconds.

The graph shows the cost of electricity with Company A.

Use the information in the graph to estimate the cost of electricity for a customer who uses 450 kwH of electricity.

Company B charges 14.3 pence per kwH of electricity used.

If Company B’s cost of electricity was plotted on the same axes as Company A’s cost of electricity, which line would be steeper? Explain how you know.

Company........ would have the steeper line because........

The graph shows the distance travelled by a particle over 8 seconds.

Estimate the speed of the particle at 5 seconds.

The graph shows the speed of a train during the first 60 seconds of motion.

What is the speed of the train after 9 seconds?

What does the straight line suggest about the speed of the train over the first 15 seconds?

Work out the average speed of the train, in m / s, during the 60 seconds.

The graph shows the speed of a cyclist during 20 seconds of a journey.

Find the acceleration of the cyclist

for the first 4 seconds

between 4 seconds and 14 seconds.

Work out the distance travelled by the cyclist during the 20 seconds.

Viraj cycled from his home to visit his aunt. He drew this graph to show his journey. He stopped at a shop 7 km from his home.

State one assumption that Viraj made when he drew his graph.

For how long did Viraj stop at the shop?

Work out Viraj’s average speed between his home and the shop. Give your answer in metres per minute.

(c)........ metres per minute

How can you tell, without doing any calculations, that Viraj’s average speed between his home and the shop is greater than his average speed between the shop and his aunt?

The graph shows the speed of a tram as it travels from the library to the town hall.

Calculate the deceleration of the tram as it approaches the town hall.

Calculate the distance travelled by the tram between the library and the town hall.

What was the maximum speed of the tram as it travelled between the library and the town hall? Give your answer in kilometres per hour.

The graph shows information about the speed of a vehicle during the final 50 seconds of a journey. At the start of the 50 seconds the speed is k metres per second. The distance travelled during the 50 seconds is 1.35 kilometres.

Work out the average speed of the vehicle during the 50 seconds. Give your answer in metres per second.

Work out the value of k.

Calculate the gradient of the graph in the final 10 seconds of the journey.

Describe what this gradient represents.

This graph shows the world population, in billions, between 1951 and 2015.

Use the graph to estimate the average rate of growth of the world population between 1951 and 2015. Give suitable units for your answer.

The graph shows the speed, v metres per second (m/s), of a car at time t seconds.

Find the speed of the car at t = 7.

It is claimed that the car has accelerated from 0 to 60 miles per hour in the first 10 seconds.

Does the graph support this claim? Show your reasoning.

Use 1 mile = 1.6 kilometres.

Use the graph to estimate the acceleration at t = 7.

The speed of this car is directly proportional to the square of the time.

Find a formula linking v and t.

Georgina says that the graph shows that the speed of the car will continue to increase after 10 seconds.

Make one comment to show that this statement is incorrect.