Estimating Gradients & Areas under Graphs (Edexcel GCSE Maths: Higher)

Here is a speed-time graph for a train.

Work out an estimate for the distance the train travelled in the first 20 seconds.
Use 4 strips of equal width.

Is your answer to (a) an underestimate or an overestimate of the actual distance the train travelled?
Give a reason for your answer.

The graph shows information about the velocity, m/s, of a parachutist seconds after leaving a plane.

Work out an estimate for the acceleration of the parachutist at = 6

Work out an estimate for the distance fallen by the parachutist in the first 12 seconds after leaving the plane.
Use 3 strips of equal width.

The graph shows the speed of a car, in metres per second, during the first 20 seconds of a journey.

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

Is your answer to part (a) an underestimate or an overestimate of the actual distance the car travelled in the first 20 seconds? Give a reason for your answer.

Here is a speed-time graph for a car.

Work out an estimate for the distance the car travelled in the first 10 seconds. Use 5 strips of equal width.

Is your answer to (a) an underestimate or an overestimate of the actual distance? Give a reason for your answer.

The curve  is shown on the grid.

By drawing a suitable tangent, find an estimate of the gradient of the curve when x = 1.

A point lies on the curve.
The x  co-ordinate of is negative.
The gradient of the tangent at is 0.

Write down the co-ordinates of .

Ellie runs a race.
The graph shows Ellie's speed in the first 10 seconds after the start of the race.

By drawing a suitable tangent, work out the acceleration when = 9.
Give the units of your answer.

Describe what happens to Ellie after 7 seconds.

Olympic medallist Usain runs in a race.
The graph shows his speed, in metres per second (m/s), during the first 10 seconds of the race.

Use the graph to find how long it took Usain to reach his top speed.

Work out an estimate for Usain's acceleration at 2 seconds.
Give the units of your answer.

Calculate the difference in Usain's acceleration between 2 and 6 seconds.

The diagram shows the graph of for 

Use the graph to solve the equation .

By drawing a suitable tangent, find an estimate of the gradient of the graph when x = 1.

Clare emptied a tank and recorded the depth of water each minute.

Plot the graph of depth against time.

Work out the average rate of decrease of the depth of the water in Clare's tank between = 0 and = 10.

Find an estimate for the gradient at = 7.

Give an interpretation of part (c).

Part of the curve with equation  is shown on the grid.

Find an estimate for the gradient of the curve at the point where 
Show your working clearly.

Liquid is leaking out of a container.

The graph shows the depth of the liquid for 60 seconds.

Use the graph to work out an estimate of the rate of decrease of depth at 10 seconds.

You must show your working.

...........................................cm/s

The graph shows the speed, in metres per second, of a particle over the first four seconds of motion.

Use the graph to estimate the distance travelled by the particle in the four seconds.

............................................... metres

The velocity-time graph shows the first 40 seconds of a car in a race.

Work out the average acceleration for the first 40 seconds. Give the units of your answer.

Estimate the time during the 40 seconds when the instantaneous acceleration = the average acceleration. You must show your working on the graph.

The graph gives information about the variation in the temperature of an amount of water that is left to cool from 80° C.  

Work out an estimate for the rate of decrease of temperature at = 300.

Work out the average rate of decrease of the temperature of the water between  = 0 and = 800.

The instantaneous rate of decrease of the temperature of the water at time seconds is equal to the average rate of decrease of the temperature of the water between = 0 and = 800. Find an estimate for the value of . You must show how you got your answer.

The diagram shows part of the graph of  

By drawing a suitable straight line, use your graph to find estimates for the solutions of 

is the point on the graph of where 

Calculate an estimate for the gradient of the graph at the point .

Leo runs for 12 seconds.

The graph shows his speed.

Show that the distance he runs is less than 67.5 metres.

Work out his average acceleration for the first 9 seconds.

State the units of your answer.

A car moves from rest.

The graph gives information about the speed, metres per second, of the car seconds after it starts to move.

i) Calculate an estimate of the gradient of the graph at = 15

[3]

ii) Describe what your answer to part (i) represents.

[1]

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

The diagram shows the graph of  for 

The point on the curve has  coordinate 2

Use the graph to find an estimate for the gradient of the curve at .

Hence find an equation of the tangent to the curve at . Give your answer in the form  .