Rates of change

The graph shows the velocity, v m/s, of a particle t seconds after it starts to move.

Work out an estimate of the gradient of the graph at t = 3 You must show how you get your answer.

What does this gradient represent?

Work out an estimate for the distance the particle travelled in the first 6 seconds. Use 3 strips of equal width.

The graph shows the volume of water, V litres, in a tank at time t seconds.

What does the gradient of this graph represent?

The graph shows the volume of water, V litres, in a tank at time t seconds.

What does the gradient of this graph represent?

Here is a table of values of x and y.

Nadia says that y is directly proportional to x because the value of y increases by 4 as the value of x increases by 2

Is Nadia correct? You must give a reason for your answer.

w is directly proportional to the square root of t. w = 140 when t = 64

Calculate the value of w when t = 7.84

On the axes below, sketch a graph to show the relationship between w and t.

To cook rice the number of cups of rice (x): the number of cups of water (y) = 4: 5

Use this information to draw a graph to show the relationship between the number of cups of rice and the number of cups of water needed to cook rice.

Find the gradient of the line drawn in part (a).

Explain what this gradient represents.

An object falls from rest. Here is the distance‑time graph for the distance (d metres) fallen by the object t seconds after it starts to fall.

Work out an estimate for the gradient of the graph at t = 3 You must show how you get your answer.

The graphs of y against x represent four different types of proportionality. Match each type of proportionality in the table to the correct graph.

Here is a distance-time graph.

Find an estimate of the gradient of the graph at time 2.5 seconds. You must show how you get your answer.

What does the gradient of the graph represent?

The graph below gives the volume, in litres, of water in a container t seconds after the water starts to fill the container.

Calculate an estimate for the gradient of the graph when t = 17.5 You must show how you get your answer.

Describe fully what the gradient in part (a) represents.

Here is a speed-time graph.

Work out an estimate of the gradient of the graph at t = 2

What does the area under the graph represent?

An electricity company charges the same fixed amount for each unit of electricity used. David uses this graph to work out the total cost of the electricity he has used.

Work out the gradient of the straight line.

What does the gradient of this line represent?

Using the axes below, sketch a graph to represent the statement y is directly proportional to x

Using the axes below, sketch a graph to represent the statement y is inversely proportional to x

The graph gives the volume of water, in litres, in a container at time t seconds after the water started to flow out of the container.

Using the graph, work out an estimate for the rate at which the water is flowing out of the container when t = 12 You must show your working.

_........ litres per second

The graph gives information about the volume, v litres, of petrol in the tank of Jim’s car after it has travelled a distance of d kilometres.

Find the gradient of the graph.

Interpret what the gradient of the graph represents.

Here is a speed-time graph for a car.

Work out an estimate for the distance the car travelled in the first 30 seconds.

Is your answer to part (a) an underestimate or an overestimate of the actual distance the car travelled in the first 30 seconds? Give a reason for your answer.

Julian used the graph to answer this question. Work out an estimate for the acceleration of the car at time 60 seconds. Here is Julian’s working. acceleration = speed ÷ time = 13 ÷ 60

Julian’s method does not give a good estimate of the acceleration at time 60 seconds.

Explain why.

The graph shows information about part of a cyclist’s journey.

Work out an estimate of the speed, in m/s, of the cyclist at time 6 seconds.

A car moves from rest. The graph gives information about the speed, v metres per second, of the car t seconds after it starts to move.

Calculate an estimate of the gradient of the graph at t = 15

Describe what your answer to part (i) represents.

Work out an estimate for the distance the car travels in the first 20 seconds of its journey. Use 4 strips of equal width.

Here are four graphs.

The graphs represent four different types of function f. Match each description of the function in the table to the letter of its graph.

The distance-time graph shows information about part of a car journey.

Use the graph to estimate the speed of the car at time 5 seconds.

The graph shows the volume of liquid (L litres) in a container at time t seconds.

Find the gradient of the graph.

Explain what this gradient represents.

The graph intersects the volume axis at L = 4

Explain what this intercept represents.